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BEGIN:VEVENT
SUMMARY:Renming Song (University of Illinois Urbana-Champaign)
DTSTART;VALUE=DATE-TIME:20200623T130000Z
DTEND;VALUE=DATE-TIME:20200623T140000Z
DTSTAMP;VALUE=DATE-TIME:20240715T174606Z
UID:NonLocalOperators/1
DESCRIPTION:Title: Factorizations and estimates of Dirichlet heat kernels for non-l
ocal operators with critical killings\nby Renming Song (University of
Illinois Urbana-Champaign) as part of Non-local operators\, probability an
d singularities\n\n\nAbstract\nIn this talk I will discuss heat kernel est
imates for critical perturbations \nof non-local operators. To be more pre
cise\, let $X$ be the reflected \n$\\alpha$-stable process in the closure
of a smooth open set $D$\, and \n$X^D$ the process killed upon exiting $D$
. We consider potentials of the \nform $\\kappa(x)=C\\delta_D(x)^{-\\alpha
}$ with positive $C$ and the \ncorresponding Feynman-Kac semigroups. Such
potentials do not belong \nto the Kato class. We obtain sharp two-sided es
timates for the heat \nkernel of the perturbed semigroups. The interior es
timates of the \nheat kernels have the usual $\\alpha$-stable form\, while
the boundary \ndecay is of the form $\\delta_D(x)^p$ with non-negative \n
$p\\in [\\alpha-1\, \\alpha)$ depending on the precise value of the \ncons
tant $C$. Our result recovers the heat kernel estimates of both \nthe cens
ored and the killed stable process in $D$. Analogous \nestimates are obtai
ned for the heat kernel of the Feynman-Kac \nsemigroup of the $\\alpha$-st
able process in \n${\\mathbf R}^d\\setminus \\{0\\}$ through the potential
$C|x|^{-\\alpha}$. \n\nAll estimates are derived from a more general resu
lt described as follows: \nLet $X$ be a Hunt process on a locally compact
separable metric space in \na strong duality with $\\widehat{X}$. Assume t
hat transition densities of \n$X$ and $\\widehat{X}$ are comparable to th
e function $\\widetilde{q}(t\,x\,y)$ \ndefined in terms of the volume of b
alls and a certain scaling function. \nFor an open set $D$ consider the ki
lled process $X^D$\, and a critical \nsmooth measure on $D$ with the corre
sponding positive additive functional \n$(A_t)$. We show that the heat ke
rnel of the the Feynman-Kac semigroup \nof $X^D$ through the multiplicativ
e functional $\\exp(-A_t)$ admits the \nfactorization of the form \n${\\ma
thbf P}_x(\\zeta >t)\\widehat{\\mathbf P}_y(\\widehat{\\zeta}>t)\\widetild
e{q}(t\,x\,y)$.\n\nThis talk is based on a joint paper with Soobin Cho\, P
anki Kim and Zoran Vondracek.\n
LOCATION:https://researchseminars.org/talk/NonLocalOperators/1/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Alexei Kulik (Wrocław University of Science and Technology)
DTSTART;VALUE=DATE-TIME:20200630T130000Z
DTEND;VALUE=DATE-TIME:20200630T140000Z
DTSTAMP;VALUE=DATE-TIME:20240715T174606Z
UID:NonLocalOperators/2
DESCRIPTION:Title: Moment bounds for dissipative semimartingales with heavy jumps\nby Alexei Kulik (Wrocław University of Science and Technology) as par
t of Non-local operators\, probability and singularities\n\n\nAbstract\nTh
e talk is based on a joint research with Ilya Pavlyukevich. We show that i
f the jumps of an Ito-semimartingale $X$ admit a finite $p$-moment\, $p>0$
\,\nthe radial part of its drift is dominated at $\\infty$ by $-|X|^\\kapp
a$ for some $\\kappa\\geq -1$\, and the balance condition $p+\\kappa>1$ ho
lds true\, then\nunder some further minor technical assumptions\n$\\sup_{t
\\geq 0} \\mathbb{E} |X_t|^{p_X}<\\infty$ for each $p_X\\in(0\,p+\\kappa-1
)$. The upper bound $p+\\kappa-1$ is generically optimal.\nThe proof is ba
sed on the extension of the method of Lyapunov functions to the semimartin
gale framework.\n\nOur study of the uniform-in-time moment estimates is st
rongly motivated by needs of the Stochastic Averaging/Homogenization theor
y for Levy driven multi-scale systems\, which will be discussed in the tal
k\, as well.\n
LOCATION:https://researchseminars.org/talk/NonLocalOperators/2/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Tomasz Jakubowski (Wrocław University of Science and Technology)
DTSTART;VALUE=DATE-TIME:20200707T130000Z
DTEND;VALUE=DATE-TIME:20200707T140000Z
DTSTAMP;VALUE=DATE-TIME:20240715T174606Z
UID:NonLocalOperators/3
DESCRIPTION:Title: Critical Schrödinger perturbations of fractional Laplacian\
nby Tomasz Jakubowski (Wrocław University of Science and Technology) as p
art of Non-local operators\, probability and singularities\n\n\nAbstract\n
Let $p(t\,x\,y)$ be the fundamental solution of the equation $\\partial_t
u(t\,x) = \\Delta^{\\alpha/2} u(t\,x)$.\nI will consider the integral equa
tion\n$$\n\\tilde{p}(t\,x\,y) = p(t\,x\,y) + \\int_0^t \\int_{\\mathbb{R}^
d} p(t-s\,x\,z) q(z) \\tilde{p}(s\,z\,y) dz ds\,\n$$\nwhere $q(z) = \\frac
{\\kappa}{|z|^{\\alpha}}$ and $\\kappa$ is some constant. The function $\\
tilde{p}$ solving this equation will be called the Schrödinger perturbati
ons of the function $p$ by $q$. I will present the results concerning th
e estimates of the function $\\tilde{p}$ in both cases $\\kappa>0$ and $\\
kappa<0$.\n
LOCATION:https://researchseminars.org/talk/NonLocalOperators/3/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Franziska Kühn (Technical University of Dresden)
DTSTART;VALUE=DATE-TIME:20200714T130000Z
DTEND;VALUE=DATE-TIME:20200714T140000Z
DTSTAMP;VALUE=DATE-TIME:20240715T174606Z
UID:NonLocalOperators/4
DESCRIPTION:Title: A maximal inequality for martingale problems and applications\nby Franziska Kühn (Technical University of Dresden) as part of Non-loc
al operators\, probability and singularities\n\n\nAbstract\nMartingale pro
blems aim to characterize stochastic processes by their martingale propert
ies. A famous example is Lévy's characterization theorem which characteri
zes Brownian motion by its first two conditional moments. More generally\,
a wide class of Markov processes and stochastic differential equations ca
n be described using martingale problems.\n\nIn this talk\, we study marti
ngale problems associated with Lévy-type operators. We present a maximal
inequality\, which goes back to R. Schilling\, and discuss some variants o
f it. We show that the maximal inequality has many useful applications in
the study of distributional and path properties of the corresponding stoch
astic process\, e.g. criteria for non-explosion in finite time\, existence
of moments\, ...\n
LOCATION:https://researchseminars.org/talk/NonLocalOperators/4/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Panki Kim (Seoul National University)
DTSTART;VALUE=DATE-TIME:20200721T130000Z
DTEND;VALUE=DATE-TIME:20200721T140000Z
DTSTAMP;VALUE=DATE-TIME:20240715T174606Z
UID:NonLocalOperators/5
DESCRIPTION:Title: Estimates on transition densities of subordinators with jumping
density decaying in mixed polynomial orders\nby Panki Kim (Seoul Natio
nal University) as part of Non-local operators\, probability and singulari
ties\n\n\nAbstract\nIn this talk\, we discuss the sharp two-sided estimate
s on the transition densities for subordinators whose Lévy measures are a
bsolutely continuous and decaying in mixed polynomial orders. Under a weak
er assumption on Lévy measures\, we also discuss a precise asymptotic beh
aviors of the transition densities at infinity. Our results cover geometri
c stable subordinators\, Gamma subordinators and much more. This is a join
t work with Soobin Cho.\n
LOCATION:https://researchseminars.org/talk/NonLocalOperators/5/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Krzysztof Bogdan (Wrocław University of Science and Technology)
DTSTART;VALUE=DATE-TIME:20200728T130000Z
DTEND;VALUE=DATE-TIME:20200728T140000Z
DTSTAMP;VALUE=DATE-TIME:20240715T174606Z
UID:NonLocalOperators/6
DESCRIPTION:Title: Nonlinear nonlocal Douglas identity\nby Krzysztof Bogdan (Wr
ocław University of Science and Technology) as part of Non-local operator
s\, probability and singularities\n\n\nAbstract\nI will present results fr
om the joint work with Tomasz Grzywny\, Katarzyna Pietruska-Pałuba\, Artu
r Rutkowski with the same title (available at https://arxiv.org/abs/2006.0
1932 ). We give Hardy-Stein and Douglas identities for specific nonlinear
nonlocal Sobolev-Bregman integral forms with unimodal Lévy measures. We p
rove that the corresponding Poisson integral defines an extension operator
for the Sobolev-Bregman spaces. The results generalizes to the setting of
$L^p$ spaces the earlier results of the authors\, obtained for the (quadr
atic) Dirichlet forms and $L^2$ spaces.\n
LOCATION:https://researchseminars.org/talk/NonLocalOperators/6/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Victoria Knopova
DTSTART;VALUE=DATE-TIME:20200811T130000Z
DTEND;VALUE=DATE-TIME:20200811T140000Z
DTSTAMP;VALUE=DATE-TIME:20240715T174606Z
UID:NonLocalOperators/7
DESCRIPTION:Title: Construction and heat kernel estimates of general stable-like Ma
rkov processes\nby Victoria Knopova as part of Non-local operators\, p
robability and singularities\n\n\nAbstract\nStarting with a non-symmetric
$\\alpha$-stable- like pseudo-differential operator $L$ defined on the
test functions\, we show that the corresponding martingale problem is wel
l-posed\, and its solution is a strong Markov process which admits a tran
sition probability density. We investigate the structure of this density
in the vicinity of the starting point. In particular\, we show that due
to the non-symmetry the respective density is not necessarily bounded\, an
d one needs additional assumptions on the Lévy-type kernel of the opera
tor in order to get a point-wise upper bound on the transition probabilit
y density.\n
LOCATION:https://researchseminars.org/talk/NonLocalOperators/7/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Xicheng Zhang (Wuhan)
DTSTART;VALUE=DATE-TIME:20200915T130000Z
DTEND;VALUE=DATE-TIME:20200915T140000Z
DTSTAMP;VALUE=DATE-TIME:20240715T174606Z
UID:NonLocalOperators/8
DESCRIPTION:Title: Singular HJB equations with applications to KPZ on the real line
\nby Xicheng Zhang (Wuhan) as part of Non-local operators\, probabilit
y and singularities\n\n\nAbstract\nI will talk about the Hamilton-Jacobi-B
ellman equations with distribution-valued coefficients\, which is not wel
l-defined in the classical sense and shall be understood by using paracont
rolled distribution method introduced by Gubinelli-Imkeller-Perkowski. By
a new characterization of weighted Hölder space and Zvonkin's transformat
ion we prove some new a priori estimates\, and therefore\, establish the g
lobal well-posedness for singular HJB equations. As an application\, the g
lobal well-posedness for KPZ equations on the real line in polynomial weig
hted Hölder spaces is obtained without using Cole-Hopf's transformation.
This is a joint work with Rongchan Zhu and Xiangchan Zhu.\n
LOCATION:https://researchseminars.org/talk/NonLocalOperators/8/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Zoran Vondraček (Zagreb)
DTSTART;VALUE=DATE-TIME:20200922T130000Z
DTEND;VALUE=DATE-TIME:20200922T140000Z
DTSTAMP;VALUE=DATE-TIME:20240715T174606Z
UID:NonLocalOperators/9
DESCRIPTION:Title: On the potential theory of Markov processes with jump kernels de
caying at the boundary\nby Zoran Vondraček (Zagreb) as part of Non-lo
cal operators\, probability and singularities\n\n\nAbstract\nIn this talk\
, I will consider some potential theory of the process $Y$ on an open set
$D\\subset \\mathbb{R}^d$ associated with a pure jump Dirichlet form whose
jump kernel has the form $J(x\,y)=B(x\,y)|x-y|^{-d-\\alpha}$\, $0<\\alpha
<2$. Here $B(x\,y)$ -- the boundary term -- depends on $\\delta_D(x)\, \\d
elta_D(y)$ and $|x-y|$\, and is allowed to approach 0 at the boundary. Thi
s is in contrast with previous works where $B(x\,y)$ is assumed to be boun
ded between two positive constants\, which can be viewed as a uniform elli
pticity condition for non-local operators. The conditions and the form of
the boundary term $B(x\,y)$ are motivated by jump kernels of some subordin
ate killed Lévy processes.\n\nWe prove that non-negative harmonic functio
ns of the process satisfy the Harnack inequality and Carleson's estimate.
Furthermore\, in case when $D$ is the half-space we investigate when the b
oundary Harnack principle holds. This is joint work with Panki Kim (Seoul
National University) and Renming Song (University of Illinois).\n
LOCATION:https://researchseminars.org/talk/NonLocalOperators/9/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Angkana Rüland (Heidelberg)
DTSTART;VALUE=DATE-TIME:20200929T130000Z
DTEND;VALUE=DATE-TIME:20200929T140000Z
DTSTAMP;VALUE=DATE-TIME:20240715T174606Z
UID:NonLocalOperators/10
DESCRIPTION:Title: Uniqueness\, stability and single measurement recovery for the
fractional Calderón problem\nby Angkana Rüland (Heidelberg) as part
of Non-local operators\, probability and singularities\n\n\nAbstract\nIn t
his talk I discuss a nonlocal inverse problem\, the\nfractional Calderón
problem. This is an inverse problem for a\nfractional Schrödinger equatio
n in which one seeks to recover\ninformation on an unknown potential by ex
terior measurements. In the\ntalk\, I prove uniqueness and stability of th
e "infinite data problem"\nand then address the recovery question. This al
so yields surprising\ninsights on the uniqueness properties of the invers
e problem\, in that it\nturns out that a single measurement suffices to un
iquely recover the\npotential.\n\nThese properties are based on the very s
trong unique continuation and\napproximation properties of fractional Schr
ödinger operators\, which are\nof independent interest and which I also d
iscuss in the talk.\n\nThis is based on joint work with T. Ghosh\, M. Salo
and G. Uhlmann.\n
LOCATION:https://researchseminars.org/talk/NonLocalOperators/10/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Hongjie Dong (Brown)
DTSTART;VALUE=DATE-TIME:20201006T130000Z
DTEND;VALUE=DATE-TIME:20201006T140000Z
DTSTAMP;VALUE=DATE-TIME:20240715T174606Z
UID:NonLocalOperators/11
DESCRIPTION:Title: Evolutionary equations with nonlocal time derivatives\nby H
ongjie Dong (Brown) as part of Non-local operators\, probability and singu
larities\n\n\nAbstract\nI will present some recent results about fractiona
l parabolic and wave equations with nonlocal Caputo time derivatives. Unde
r various vanishing mean oscillation (VMO) conditions on the leading coeff
icients\, we obtained weighted and mixed-norm Sobolev estimates in the who
le space\, half space\, or domains.\n\nThis is based on joint work with Do
yoon Kim (Korea University) and Yanze Liu (Brown University).\n
LOCATION:https://researchseminars.org/talk/NonLocalOperators/11/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Daesung Kim (Illinois Urbana-Champaign)
DTSTART;VALUE=DATE-TIME:20201013T130000Z
DTEND;VALUE=DATE-TIME:20201013T140000Z
DTSTAMP;VALUE=DATE-TIME:20240715T174606Z
UID:NonLocalOperators/12
DESCRIPTION:Title: Quantitative isoperimetric inequalities arising from stochastic
processes\nby Daesung Kim (Illinois Urbana-Champaign) as part of Non-
local operators\, probability and singularities\n\n\nAbstract\nIt is well
known that isoperimetric type inequalities hold for a large class of quant
ities arising from Brownian motion. Banuelos and Mendez-Hernandez showed t
hat such inequalities can be extended to a wide class of Levy processes. A
stability question is if the inequality will be about to achieving the eq
uality when a given domain is close to being a ball. This question can be
answered by quantitative improvement of such inequalities in terms of the
asymmetry. In this talk\, we discuss the quantitative isoperimetric inequa
lities for the expected lifetime of Brownian motion and $\\alpha$-stable p
rocesses\, and some related open problems.\n
LOCATION:https://researchseminars.org/talk/NonLocalOperators/12/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Igor Verbitsky (Missouri)
DTSTART;VALUE=DATE-TIME:20201020T140000Z
DTEND;VALUE=DATE-TIME:20201020T150000Z
DTSTAMP;VALUE=DATE-TIME:20240715T174606Z
UID:NonLocalOperators/13
DESCRIPTION:Title: Pointwise estimates of positive solutions to linear and semilin
ear equations with nonlocal operators\nby Igor Verbitsky (Missouri) as
part of Non-local operators\, probability and singularities\n\n\nAbstract
\nRecent results will be presented involving sharp global pointwise estima
tes of positive solutions to some linear and semilinear partial different
ial equations and inequalities with nonlocal operators satisfying various
forms of the maximum principle or domination principle. In particular\, e
quations of the type\n\\[\n(-\\Delta)^{\\frac{\\alpha}{2}} u = g(u) \\sigm
a +\\mu \\quad \\text{in} \\\, \\\,\n\\Omega\, \\quad u=0 \\\, \\\, \\\, \
\text{in} \\\, \\\, \\Omega^c\,\n\\]\nwith measure coefficients $\\sigma$\
, $\\mu$\, where $g(u)=u^q$ and $0< \\alpha < n$ in certain domains $\\Ome
ga \\subseteq {\\mathbb{R}}^n$\, or Riemannian manifolds\, with positive G
reen's function will be discussed.\n\nJoint work with Alexander Grigor'yan
.\n
LOCATION:https://researchseminars.org/talk/NonLocalOperators/13/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jie-Ming Wang (Beijing)
DTSTART;VALUE=DATE-TIME:20201103T140000Z
DTEND;VALUE=DATE-TIME:20201103T150000Z
DTSTAMP;VALUE=DATE-TIME:20240715T174606Z
UID:NonLocalOperators/14
DESCRIPTION:Title: Boundary Harnack Principle for Diffusion with Jumps\nby Jie
-Ming Wang (Beijing) as part of Non-local operators\, probability and sing
ularities\n\n\nAbstract\nFor $d\\geq 3$\, consider the operator ${\\mathca
l L}^{\\bf b}={\\mathcal L}^0+b_1\\cdot \\nabla+{\\mathcal S}^{b_2}$\,\nwh
ere ${\\mathcal L}^0$ is a second order elliptic operator in non-diverge
nce form\,\nthe function $b_1$ belongs to some Kato class and\n$$\n{\\mat
hcal S}^{b_2} f(x):=\n\\int_{{\\mathbb R}^d} \\left( f(x+z)-f(x)- \\nabla
f(x) \\cdot\nz\\\, {\\mathbb 1}_{{|z|\\leq 1}} \\right) b_2(x\, z)J_0(z)
\\\,dz\, \\quad f\\in C_b^2({\\mathbb R}^d)\,\n$$\nwhere $J_0(z)$ satisfie
s that there exist positive constants $c_1\, c_2$ and $0<\\beta_1\\leq \\b
eta_2 <2$ such that\n $$c_1 (|z_2|/|z_1|)^{d+\\beta_1} \\leq J_0(z_1)/J_0
(z_2)\n\\leq c_2 (|z_2|/|z_1|)^{d+\\beta_2}\n\\quad {f\\!or\\\, any}\\quad
z_1\, z_2\\in {\\mathbb R}^d \\quad{with}\\quad 0<|z_1|\\leq |z_2|\,$$\n
$b_2(x\, z)$ is a real-valued bounded function\non ${\\mathbb R}^d\\time
s {\\mathbb R}^d$ satisfying for each $x\\in {\\mathbb R}^d$\,\n$ b_2(x\,
\\cdot )\\geq 0$ a.e. on ${\\mathbb R}^d$\, and\n$$\n1_{\\beta_2=1} \\
int_{r<|z|\\leq R}z b_2(x\, z) J_0(z)\\\,dz=0 \\quad {f\\!or\\\, every}\\q
uad x\\in {\\mathbb R}^d\n\\quad {and}\\quad 0< r < R < \\infty.\n$$\nUnde
r the uniformly ellipticity condition and Hölder condition on the diffusi
on coefficient $a_{ij}\,$\nthere exists a conservative Feller process $X^{
\\bf b}$ with its infinitesimal generator ${\\mathcal L}^{\\bf b}$.\nWe gi
ve the two-sided Green function estimates of $X^{\\bf b}$ on a bounded $C^
{1\,1}$ domain $D$ and further establish the Martin integral representatio
n of harmonic function with respect to $X^{\\bf b}$ on the domain $D$.\nU
sing the Green function estimates and the Martin integral formula in $D$\
, the Harnack principle and the boundary Harnack principle with explicit b
oundary decay rate for the operator ${\\mathcal L}^{\\bf b}$ under some mi
ld conditions\nare established.\nThis talk is based on a joint work with
Professor Z.-Q. Chen.\n
LOCATION:https://researchseminars.org/talk/NonLocalOperators/14/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jian Wang (Fujian)
DTSTART;VALUE=DATE-TIME:20201110T140000Z
DTEND;VALUE=DATE-TIME:20201110T150000Z
DTSTAMP;VALUE=DATE-TIME:20240715T174606Z
UID:NonLocalOperators/15
DESCRIPTION:Title: Heat kernel upper bounds for symmetric Markov semigroups\nb
y Jian Wang (Fujian) as part of Non-local operators\, probability and sing
ularities\n\n\nAbstract\nIt is well known that Nash-type inequalities for
symmetric Dirichlet forms are equivalent to on-diagonal heat kernel upper
bounds for associated symmetric Markov semigroups. In this talk\, we show
that both imply (and hence are equivalent to) off-diagonal heat kernel upp
er bounds under some mild assumptions. Our approach is based on a new gen
eralized Davies's method. Our results extend that by Carlen-Kusuoka-Strooc
k for Nash-type inequalities with power order considerably and also extend
that by Grigor'yan for second order differential operators on a complete
non-compact manifold.\n\nThe talk is based on a joint work with Z.-Q. Chen
(Seattle)\, P. Kim (Seoul) and T. Kumagai (Kyoto).\n
LOCATION:https://researchseminars.org/talk/NonLocalOperators/15/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Tadeusz Kulczycki (Wroclaw)
DTSTART;VALUE=DATE-TIME:20201117T140000Z
DTEND;VALUE=DATE-TIME:20201117T150000Z
DTSTAMP;VALUE=DATE-TIME:20240715T174606Z
UID:NonLocalOperators/16
DESCRIPTION:Title: On weak solution of SDE driven by inhomogeneous singular Lévy
noise.\nby Tadeusz Kulczycki (Wroclaw) as part of Non-local operators\
, probability and singularities\n\n\nAbstract\nWe study the stochastic dif
ferential equation\n$dX_t = A_t(X_{t-}) \\\, dZ_t$\, $ X_0 = x$\,\nwhere $
Z_t = (Z_t^{(1)}\,\\ldots\,Z_t^{(d)})^T$ and for each $i \\in \\{1\,\\ldot
s\,d\\}$ $Z_t^{(i)}$ is a one-dimensional\, symmetric $\\alpha_i$-stable p
rocess\, where $\\alpha_i \\in (0\,2)$. Under appropriate conditions on $\
\alpha_1\,\\ldots\,\\alpha_d$ and on matrices $A_t$ we prove existence an
d uniqueness of the weak solution of the above SDE\, which will be shown t
o be a time-inhomogeneous Markov process. We also provide a representation
of the transition probability density of this process as a sum of explici
tly given ‘principal part’\, and a ‘residual part’ subject to a se
t of estimates showing that this part is negligible in a short time. The t
alk is based on a joint work with A. Kulik and M. Ryznar.\n
LOCATION:https://researchseminars.org/talk/NonLocalOperators/16/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Longmin Wang (Nankai)
DTSTART;VALUE=DATE-TIME:20201201T140000Z
DTEND;VALUE=DATE-TIME:20201201T150000Z
DTSTAMP;VALUE=DATE-TIME:20240715T174606Z
UID:NonLocalOperators/17
DESCRIPTION:Title: Branching Random Walks on Hyperbolic Spaces\nby Longmin Wan
g (Nankai) as part of Non-local operators\, probability and singularities\
n\n\nAbstract\nThe branching Brownian motion on the hyperbolic plane with
binary\nfission at rate $\\lambda > 0$ exhibits a phase transition in\n$\\
lambda$: For $\\lambda \\leq 1/8$ the number of particles in any\ncompact
region is eventually $0$\, w.p.1\, but for $\\lambda > 1/8$\nthe number of
particles in any open region grows to $\\infty$\nw.p.1. Lalley and Sellke
(1987) proved that in the subcritical and\ncritical case ($\\lambda \\leq
1/8$) the set $\\Lambda$ of all limit\npoints in the boundary circle at $
\\infty$ consisting of particle\ntrails is a Cantor set\, while in the sup
ercritical case ($\\lambda\n>1/8$) the set $\\Lambda$ has full Lebesgue me
asure. For $\\lambda\n\\leq 1/8$ the Hausdorff dimension of $\\Lambda$ is
at most $1/2$\nand has critical exponent $1/2$ near the critical value $\
\lambda =\n1/8$. In this talk we will prove the same type of phase transit
ion\noccurs for branching random walks on hyperbolic spaces.\n
LOCATION:https://researchseminars.org/talk/NonLocalOperators/17/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Mateusz Kwaśnicki (Wroclaw)
DTSTART;VALUE=DATE-TIME:20201124T140000Z
DTEND;VALUE=DATE-TIME:20201124T150000Z
DTSTAMP;VALUE=DATE-TIME:20240715T174606Z
UID:NonLocalOperators/19
DESCRIPTION:Title: Harmonic extensions\, operators with completely monotone kernel
s\, and traces of 2-D diffusions\nby Mateusz Kwaśnicki (Wroclaw) as p
art of Non-local operators\, probability and singularities\n\nAbstract: TB
A\n
LOCATION:https://researchseminars.org/talk/NonLocalOperators/19/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jerome Goldstein (Memphis)
DTSTART;VALUE=DATE-TIME:20201208T150000Z
DTEND;VALUE=DATE-TIME:20201208T160000Z
DTSTAMP;VALUE=DATE-TIME:20240715T174606Z
UID:NonLocalOperators/20
DESCRIPTION:Title: The Boderline between Some Good Problems and the Corresponding
Bad Problems\nby Jerome Goldstein (Memphis) as part of Non-local opera
tors\, probability and singularities\n\n\nAbstract\nWe will discuss three
problems in PDE for which existence or nonexistence of\ncertain kinds of e
quations is a delicate issue. Many coauthors are involved\, and the\nprobl
ems are related to each other.\n\nThe first problem involves work from the
1970s about uniqueness for certain\nill posed problems involving the Eule
r-Poisson-Darboux equation. The number of\ninitial conditions required for
uniqueness involves the size of negative parameter in\nthe singular term
and the definition of solution.\n\nThe second problem involves the Schröd
inger operator with the inverse square\npotential multiplied by a constant
c. The spectrum of this operator on $L^2(\\mathbb{R}^n)$ is\neither $\\ma
thbb{R}$ or $\\mathbb{R}^+$\, depending on the choice of c: In the 1980s\,
it was proved that the corresponding heat equation has instantaneous blow
up and no positive solutions\nwhen the spectrum is $\\mathbb{R}$. The cor
responding result is true when Euclidean space is\nreplaced by the Heisenb
erg group\; this was proved in 2020.\n\nThe final problem is nonlinear and
involves the parabolic problem for the fast\ndiffusion equation or the p-
Laplacian heat equation\, perturbed in various ways\, on\nEuclidean space
or on a Riemannaian manifold. In some cases one can show the\nabsence of n
onnegative solutions (except for the zero function).\n
LOCATION:https://researchseminars.org/talk/NonLocalOperators/20/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Moritz Kassmann (Bielefeld)
DTSTART;VALUE=DATE-TIME:20210112T140000Z
DTEND;VALUE=DATE-TIME:20210112T150000Z
DTSTAMP;VALUE=DATE-TIME:20240715T174606Z
UID:NonLocalOperators/21
DESCRIPTION:Title: Heat kernel estimates for mixed systems of diffusions and jump
processes\nby Moritz Kassmann (Bielefeld) as part of Non-local operato
rs\, probability and singularities\n\n\nAbstract\nWe prove sharp heat kern
el estimates for symmetric Markov processes that are independent copies of
one-dimensional jump or diffusion processes. The result can be seen as a
robustness result for heat kernels like the one of Aronson (1968) for dif
fusions or the one of Chen/Kumagai (2003) for isotropic jump processes. Th
e talk is based on a joint work together with Jaehoon Kang (KAIST).\n
LOCATION:https://researchseminars.org/talk/NonLocalOperators/21/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Guohuan Zhao (Bielefeld)
DTSTART;VALUE=DATE-TIME:20201215T140000Z
DTEND;VALUE=DATE-TIME:20201215T150000Z
DTSTAMP;VALUE=DATE-TIME:20240715T174606Z
UID:NonLocalOperators/22
DESCRIPTION:Title: Regularity properties of jump diffusions with irregular coeffic
ients\nby Guohuan Zhao (Bielefeld) as part of Non-local operators\, pr
obability and singularities\n\n\nAbstract\nIn this talk\, I plan to presen
t some results about the regularity properties of strong solutions to SDEs
driven by Lévy processes with irregular drift coefficients. In short\, I
will show the Malliavin differentiability of the unique strong solutions
as well as the differentiability of the stochastic flows with respect to t
he spatial variable. Meanwhile\, I will also talk about the Schauder's est
imate for the resolvent equations corresponding to the SDEs.\n
LOCATION:https://researchseminars.org/talk/NonLocalOperators/22/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Carina Geldhauser (Lund)
DTSTART;VALUE=DATE-TIME:20210119T140000Z
DTEND;VALUE=DATE-TIME:20210119T150000Z
DTSTAMP;VALUE=DATE-TIME:20240715T174606Z
UID:NonLocalOperators/23
DESCRIPTION:Title: The fractional Green function in atmospheric turbulence models<
/a>\nby Carina Geldhauser (Lund) as part of Non-local operators\, probabil
ity and singularities\n\n\nAbstract\nIn this talk we discuss a family of d
iscrete models for atmospheric turbulence\, often called point vortex mode
ls.\n\nWe state some of it basic properties and show how we can derive an
effective PDE\, the so-called mean field limit\, from the discrete Hamilto
nian system\, by using a variational principle. Furthermore\, we discuss
the usage and interpretation of these models in statistical physics.\n\nTh
e content of this talk is based joint work with Marco Romito (Uni Pisa).\n
LOCATION:https://researchseminars.org/talk/NonLocalOperators/23/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Damir Kinzebulatov (Quebec)
DTSTART;VALUE=DATE-TIME:20210126T140000Z
DTEND;VALUE=DATE-TIME:20210126T150000Z
DTSTAMP;VALUE=DATE-TIME:20240715T174606Z
UID:NonLocalOperators/25
DESCRIPTION:Title: Fractional Kolmogorov operator and desingularizing weights\
nby Damir Kinzebulatov (Quebec) as part of Non-local operators\, probabili
ty and singularities\n\n\nAbstract\nThe subject of this talk are sharp two
-sided bounds on the heat kernel of the fractional Laplacian perturbed by
a Hardy-type drift\, which we establish by transferring the operator to an
appropriate weighted space with singular weight. The talk is based on joi
nt papers with Yu.A.Semenov and K.Szczypkowski.\n
LOCATION:https://researchseminars.org/talk/NonLocalOperators/25/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Vanja Wagner (Zagreb)
DTSTART;VALUE=DATE-TIME:20210202T140000Z
DTEND;VALUE=DATE-TIME:20210202T150000Z
DTSTAMP;VALUE=DATE-TIME:20240715T174606Z
UID:NonLocalOperators/26
DESCRIPTION:Title: Semilinear equations for non-local operators: beyond the fracti
onal Laplacian\nby Vanja Wagner (Zagreb) as part of Non-local operator
s\, probability and singularities\n\n\nAbstract\nWe study semilinear probl
ems in general bounded open sets for non-local operators with exterior and
boundary conditions\, where the operators are more general than the fract
ional Laplacian. We also give results in case of bounded $C^{1\,1}$ open s
ets. The talk is based on joint work with Ivan Biočić and Zoran Vondrač
ek.\n
LOCATION:https://researchseminars.org/talk/NonLocalOperators/26/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Luis Silvestre (The University of Chicago)
DTSTART;VALUE=DATE-TIME:20210316T140000Z
DTEND;VALUE=DATE-TIME:20210316T150000Z
DTSTAMP;VALUE=DATE-TIME:20240715T174606Z
UID:NonLocalOperators/27
DESCRIPTION:Title: Regularity estimates for the Boltzmann equation without cutoff<
/a>\nby Luis Silvestre (The University of Chicago) as part of Non-local op
erators\, probability and singularities\n\n\nAbstract\nWe study the regula
rization effect of the inhomogeneous Boltzmann equation without cutoff. We
obtain a priori estimates for all derivatives of the solution depending o
nly on bounds of its hydrodynamic quantities: mass density\, energy densit
y and entropy density. As a consequence\, a classical solution to the equa
tion may fail to exist after a certain time T only if at least one of thes
e hydrodynamic quantities blows up. Our analysis applies to the case of mo
derately soft and hard potentials. We use methods that originated in the s
tudy of nonlocal elliptic and parabolic equations: a weak Harnack inequali
ty in the style of De Giorgi\, and a Schauder-type estimate.\n
LOCATION:https://researchseminars.org/talk/NonLocalOperators/27/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Gisèle Goldstein (The University of Memphis)
DTSTART;VALUE=DATE-TIME:20210323T140000Z
DTEND;VALUE=DATE-TIME:20210323T150000Z
DTSTAMP;VALUE=DATE-TIME:20240715T174606Z
UID:NonLocalOperators/28
DESCRIPTION:Title: On Thomas-Fermi Theory and Extensions\nby Gisèle Goldstein
(The University of Memphis) as part of Non-local operators\, probability
and singularities\n\n\nAbstract\nOf concern to quantum chemists and solid
state physicists is the approximate numerical computation of the ground st
ate wave function\, and the ground state energy and density for molecular
and other quantum mechanical systems. Since the number of molecules in bul
k matter is of the order of 1026\, direct computation is too cumbersome or
impossible in many situations. In 1927\, L. Thomas and E. Fermi proposed
replacing the ground state wave function by the ground state density\, whi
ch is a function of only three variables. Independently\, each found an ap
proximate expansion for the energy associated with a density. (The wave fu
nction uniquely determines the density\, but not conversely.)\n\nA computa
tionally better approximate expansion was found in the 1960’s by W. Kohn
and his collaborators\; for this work Kohn got the Nobel Prize in Chemist
ry in 1998. A successful attempt to put Thomas-Fermi theory into a rigorou
s mathematical framework was begun in the 1970’s by E. Lieb and B. Simon
and was continued and expanded by Ph. Benilan\, H. Brezis and others. Ver
y little rigorous mathematics supporting Kohn density functional theory is
known. In this talk I will present a survey of rigorous Thomas-Fermi theo
ry\, including recent developments and open problems (in the\ncalculus of
variations and semilinear elliptic systems).\n
LOCATION:https://researchseminars.org/talk/NonLocalOperators/28/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Kazuhiro Kuwae (Fukuoka University)
DTSTART;VALUE=DATE-TIME:20210330T130000Z
DTEND;VALUE=DATE-TIME:20210330T140000Z
DTSTAMP;VALUE=DATE-TIME:20240715T174606Z
UID:NonLocalOperators/29
DESCRIPTION:Title: Lp-Kato class measures for symmetric Markov processes under hea
t kernel estimates\nby Kazuhiro Kuwae (Fukuoka University) as part of
Non-local operators\, probability and singularities\n\n\nAbstract\nI will
talk on the coincidence of two classes of $L^p$-Kato class measures\nin th
e framework of symmetric Markov processes admitting upper and lower estima
tes of heat kernel under mild conditions. One class of $L^p$-Kato class me
asures is defined by the $p$-th power of positive order resolvent kernel\,
another is defined in terms of the $p$-th power of Green kernel depending
on some exponents related to the heat kernel estimates. We also prove tha
t $q$-th integrable functions on balls with radius $1$ having uniformity o
f its norm with respect to centers are of $L^p$-Kato class if $q$ is great
er than a constant related to $p$ and the constants appeared in the upper
and lower estimates of the heat kernel. These are complete extensions of s
ome results\nby Aizenman-Simon and the recent results by the second named
author in the framework of Brownian motions on Euclidean space. We further
give necessary and sufficient conditions\nfor a Radon measure with Ahlfor
s regularity to belong to $L^p$-Kato class. Our results can be applicable
to many examples\, for instance\, symmetric (relativistic) stable processe
s\, jump processes on $d$-sets\, Brownian motions on Riemannian manifolds\
, diffusions on fractals and so on.\nJoint work with Takahiro Mori.\n\nThe
details can be seen in https://arxiv.org/abs/2008.10934\n
LOCATION:https://researchseminars.org/talk/NonLocalOperators/29/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Qi Zhang (University of California\, Riverside)
DTSTART;VALUE=DATE-TIME:20210504T130000Z
DTEND;VALUE=DATE-TIME:20210504T140000Z
DTSTAMP;VALUE=DATE-TIME:20240715T174606Z
UID:NonLocalOperators/31
DESCRIPTION:Title: Time analyticity and reversibility of some parabolic equations<
/a>\nby Qi Zhang (University of California\, Riverside) as part of Non-loc
al operators\, probability and singularities\n\n\nAbstract\nWe describe a
concise way to prove time analyticity for solutions of parabolic equations
including the heat and Navier Stokes equations. In some cases\, results u
nder sharp conditions are obtained. An application is a necessary and suff
icient condition for the solvability of the backward heat equation which i
s ill-posed\, helping to remove an old obstacle in control theory.\n\nPart
of the work is joint with Hongjie Dong\, which is related to earlier join
t work with F. H. Lin.\n
LOCATION:https://researchseminars.org/talk/NonLocalOperators/31/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Nicolai Krylov (University of Minnesota)
DTSTART;VALUE=DATE-TIME:20210416T140000Z
DTEND;VALUE=DATE-TIME:20210416T150000Z
DTSTAMP;VALUE=DATE-TIME:20240715T174606Z
UID:NonLocalOperators/32
DESCRIPTION:Title: (joint with Montreal-Quebec Analsyis Seminar)\nby Nicolai K
rylov (University of Minnesota) as part of Non-local operators\, probabili
ty and singularities\n\n\nAbstract\nFind out more details:\n\nhttps://rese
archseminars.org/seminar/MathematicalAnalysis\n\nhttps://www.math.mcgill.c
a/jakobson/analysish/seminar.html\n
LOCATION:https://researchseminars.org/talk/NonLocalOperators/32/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jamil Chaker (Bielefeld University)
DTSTART;VALUE=DATE-TIME:20210420T130000Z
DTEND;VALUE=DATE-TIME:20210420T140000Z
DTSTAMP;VALUE=DATE-TIME:20240715T174606Z
UID:NonLocalOperators/33
DESCRIPTION:Title: On nonlocal operators with anisotropic kernels\nby Jamil Ch
aker (Bielefeld University) as part of Non-local operators\, probability a
nd singularities\n\n\nAbstract\nIn this talk we study a class of (linear a
nd nonlinear) integro-differential operators with anisotropic and singular
kernels. We present local robust regularity estimates for weak solutions
in the general framework of bounded measurable coefficients. \nThe results
in this talk are based on joint works with Moritz Kassmann\, Minhyun Kim
and Marvin Weidner.\n
LOCATION:https://researchseminars.org/talk/NonLocalOperators/33/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Tomasz Grzywny (Wroclaw University of Science and Technology)
DTSTART;VALUE=DATE-TIME:20210615T130000Z
DTEND;VALUE=DATE-TIME:20210615T140000Z
DTSTAMP;VALUE=DATE-TIME:20240715T174606Z
UID:NonLocalOperators/35
DESCRIPTION:Title: Subordinated Markov processes: estimates for heat kernels and G
reen functions\nby Tomasz Grzywny (Wroclaw University of Science and T
echnology) as part of Non-local operators\, probability and singularities\
n\n\nAbstract\nLet (M\, d) be a metric space and μ a Radon measure on M.
Assume that {S(t)}_{t\\in T} is a Markov process on M such that its transi
tion function is absolutely continuous with μ\, where T is a set of non-n
egative integers or a set of non-negative real numbers. By A we denote the
semigroup generator associated with the transition function of {S(t)}. F
or a Bernstein function f we define a new semigroup with generator -f(-A)
that is a semigroup for the Markov process {S(K_t)}\, where {K_t} is a sub
ordinator on T associated with the function f. During the talk\, there wi
ll be discussed estimates of the haet kernel/transition function and Gree
n function of {S(K_t)}. The proofs are elementary and do not use estimates
for transition probability of the subordinator. The talk is based on joi
nt work with Bartosz Trojan.\n
LOCATION:https://researchseminars.org/talk/NonLocalOperators/35/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Timur Yastrzhembskiy (Brown University)
DTSTART;VALUE=DATE-TIME:20210601T130000Z
DTEND;VALUE=DATE-TIME:20210601T140000Z
DTSTAMP;VALUE=DATE-TIME:20240715T174606Z
UID:NonLocalOperators/37
DESCRIPTION:Title: Global $L_p$-estimates for kinetic Kolmogorov-Fokker-Planck equ
ation\nby Timur Yastrzhembskiy (Brown University) as part of Non-local
operators\, probability and singularities\n\n\nAbstract\nWe study the de
generate Kolmogorov equation (also known as kinetic Fokker-Planck equatio
n) in both nondivergence and divergence forms:\n$$\n \\partial_t u - v \\c
dot D_x u - a^{i j} (z) D_{v_i v_j} u + \\lambda u = f\,\n$$\n$$\n \\parti
al_t u - v \\cdot D_x u - D_{v_i} (a^{i j} (z) D_{v_j} u) + \\lambda u = D
_{v_i} f_i + f_0.\n$$\nThe leading coefficients are merely measurable in $
t$ and satisfy the VMO condition in $x\, v$ with respect to some quasi-m
etric. We also assume boundedness and uniform nondegeneracy of $a$. We pro
ve global a priori estimates in some weighted mixed-norm Lebesgue spaces a
nd solvability results. Our proof does not rely on kernel estimates.\n\nTh
e talk is based on a joint work with Hongjie Dong\n
LOCATION:https://researchseminars.org/talk/NonLocalOperators/37/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Stjepan Šebek (University of Zagreb)
DTSTART;VALUE=DATE-TIME:20210608T130000Z
DTEND;VALUE=DATE-TIME:20210608T140000Z
DTSTAMP;VALUE=DATE-TIME:20240715T174606Z
UID:NonLocalOperators/38
DESCRIPTION:Title: Limit theorems for a stable sausage\nby Stjepan Šebek (Uni
versity of Zagreb) as part of Non-local operators\, probability and singul
arities\n\n\nAbstract\nIn this talk\, we study fluctuations of the volume
of a stable sausage defined via a d-dimensional rotationally invariant alp
ha-stable process. As the main results\, we establish a central limit theo
rem and functional central limit theorem (in the case when d/alpha > 3/2)
with a standard one-dimensional Brownian motion in the limit\, and Khintch
ine’s and Chung’s laws of the iterated logarithm (in the case when d/a
lpha > 9/5).\n
LOCATION:https://researchseminars.org/talk/NonLocalOperators/38/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Mario Maurelli (Università degli Studi di Milano)
DTSTART;VALUE=DATE-TIME:20210427T130000Z
DTEND;VALUE=DATE-TIME:20210427T140000Z
DTSTAMP;VALUE=DATE-TIME:20240715T174606Z
UID:NonLocalOperators/39
DESCRIPTION:Title: Regularization by noise for transport PDEs: two results\nby
Mario Maurelli (Università degli Studi di Milano) as part of Non-local o
perators\, probability and singularities\n\n\nAbstract\nWe say that a regu
larization by noise phenomenon occurs for a possibly ill-posed differentia
l equation if this equation becomes well-posed (in a pathwise sense) under
addition of a suitable noise term. The long-term aim of regularization by
noise is to show this phenomenon for PDEs coming from physics\, especiall
y fluid dynamics.\n\nIn this talk we consider regularization by noise for
transport-type PDEs. A transport-type PDE is a prototype for many PDEs fro
m physics and takes the form\n$$\n\\partial_t u(t\,x) +b(t\,x\,u)\\cdot\\n
abla u(t\,x) =0\,\\quad t>0\,x\\in\\mathbb{R}^d\,u(t\,x)\\in\\mathbb{R}\,\
\qquad (1)\n$$\nwith $b$ given vector field. We focus on two cases\,\n$$\n
b(t\,x\,u) = \\tilde b(t\,x) \\text{ and } b(t\,x\,u)=\\tilde b(t\,x)u\,\\
qquad (2)\n$$\ncorresponding respectively to the linear transport equation
and a scalar conservation law. For irregular vector fields $\\tilde{b}$\,
the corresponding deterministic transport equations (1) are in general il
l-posed. We add to the transport equation the so-called transport noise\,
namely we consider the stochastic PDE\n$$\ndu(t\,x\,\\omega) +b\\cdot\\nab
la u\\\,dt +\\nabla u\\circ dW =0\,\\\,t>0\,x\\in\\mathbb{R}^d\,u\\in\\mat
hbb{R}\, \\qquad (3)\n$$\nwhere $W$ is a $d$-dimensional Brownian motion a
nd $\\circ$ stands for Stratonovich integration. We show that\, for $b$ of
the form (2) for some classes of irregular vector fields $\\tilde{b}$\, t
he corresponding stochastic transport PDEs (3) are well-posed.\n\nThe proo
fs are based on a combination of the renormalization argument by DiPerna-L
ions and some parabolic bounds.\n\nThis talk is based on the works Attanas
io-Flandoli 2011 and Gess-M. 2018.\n
LOCATION:https://researchseminars.org/talk/NonLocalOperators/39/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Katarzyna Pietruska-Pałuba (University of Warsaw)
DTSTART;VALUE=DATE-TIME:20210511T130000Z
DTEND;VALUE=DATE-TIME:20210511T140000Z
DTSTAMP;VALUE=DATE-TIME:20240715T174606Z
UID:NonLocalOperators/40
DESCRIPTION:Title: Properties of the integrated density of states for random Schr
ödinger operators driven by nonlocal processes\nby Katarzyna Pietrusk
a-Pałuba (University of Warsaw) as part of Non-local operators\, probabil
ity and singularities\n\n\nAbstract\nWe will discuss asymptotic properties
of the integrated density of states for random systems whose hamiltonians
are driven by nonlocal processes\, and the random field is either of Pois
sonian or alloy type. In both cases the IDS exhibits unusually fast decay
near the bottom of the spectrum (called the Lifschitz singularity). If tim
e permits\, we will also discuss the relation of the asymptotic behaviour
of the IDS to the long-time behaviour of solutions of the nonlocal parabol
ic Anderson problem.\n
LOCATION:https://researchseminars.org/talk/NonLocalOperators/40/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Tomasz Klimsiak (Institute of Mathematics Polish Academy of Scienc
es)
DTSTART;VALUE=DATE-TIME:20210525T130000Z
DTEND;VALUE=DATE-TIME:20210525T140000Z
DTSTAMP;VALUE=DATE-TIME:20240715T174606Z
UID:NonLocalOperators/41
DESCRIPTION:Title: Schrödinger equations with smooth measure potential and genera
l measure data\nby Tomasz Klimsiak (Institute of Mathematics Polish Ac
ademy of Sciences) as part of Non-local operators\, probability and singul
arities\n\n\nAbstract\nWe study equations driven by Schrödinger operators
consisting of a self-adjoint Dirichlet operator and a singular potential\
, which belongs to a class of positive Borel measures absolutely continuou
s with respect to a capacity generated by the operator. In particular\, we
cover positive potentials exploding on a set of capacity zero. The right-
hand side of equations is allowed to be a general bounded Borel measure. T
he class of self-adjoint Dirichlet operators is quite large. Examples incl
ude integro-differential operators with the local part of divergence form.
\n\nWe give a necessary and sufficient condition for the existence of a s
olution\, and prove some regularity and stability results.\n
LOCATION:https://researchseminars.org/talk/NonLocalOperators/41/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Minhyun Kim (Bielefeld University)
DTSTART;VALUE=DATE-TIME:20210622T130000Z
DTEND;VALUE=DATE-TIME:20210622T140000Z
DTSTAMP;VALUE=DATE-TIME:20240715T174606Z
UID:NonLocalOperators/42
DESCRIPTION:Title: The concentration-compactness principle for the nonlocal anisot
ropic $p$-Laplacian of mixed order\nby Minhyun Kim (Bielefeld Universi
ty) as part of Non-local operators\, probability and singularities\n\n\nAb
stract\nIn this talk\, we introduce a new class of operators with an ortho
tropic structure having different exponents of integrability and different
orders of differentiability. We prove a robust Sobolev-type inequality an
d establish the existence of minimizers of the Sobolev quotient. The metho
d for the existence of minimizers is based on the concentration-compactnes
s principle which we extend to this class of operators. One consequence of
the main result is the existence of a nontrivial nonnegative solution to
the corresponding critical problem.\n
LOCATION:https://researchseminars.org/talk/NonLocalOperators/42/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Nikola Sandrić (University of Zagreb)
DTSTART;VALUE=DATE-TIME:20210629T130000Z
DTEND;VALUE=DATE-TIME:20210629T140000Z
DTSTAMP;VALUE=DATE-TIME:20240715T174606Z
UID:NonLocalOperators/43
DESCRIPTION:Title: Periodic homogenization of linear degenerate PDEs\nby Nikol
a Sandrić (University of Zagreb) as part of Non-local operators\, probabi
lity and singularities\n\n\nAbstract\nIn this talk\, we discuss functional
CLT for a class of degenerate diffusion processes with periodic coefficie
nts\, thus generalizing the already classical results in the context of un
iformly elliptic diffusions. As an application\, we obtain periodic homoge
nization of a class of linear degenerate elliptic and parabolic PDEs. The
talk is based on joint work with Ivana Valentić.\n
LOCATION:https://researchseminars.org/talk/NonLocalOperators/43/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Wolfgang Arendt (Ulm University)
DTSTART;VALUE=DATE-TIME:20210706T130000Z
DTEND;VALUE=DATE-TIME:20210706T140000Z
DTSTAMP;VALUE=DATE-TIME:20240715T174606Z
UID:NonLocalOperators/44
DESCRIPTION:Title: Variational Methods for the Dirichlet-to-Neumann Operator and F
ractional Powers\nby Wolfgang Arendt (Ulm University) as part of Non-l
ocal operators\, probability and singularities\n\n\nAbstract\nUsually the
DtN operator is considered as a pseudo-differential operator on C-infinity
domains.\nWe will present a variational approach which works for Lipschit
z domains and even domains with finite surface. The same approach allows o
ne to give a functional framework of the Caffarelli-Silvestre extension le
ading to the fractional laplacian. We will realize the fractional power of
any sectorial operator on a Hilbert space (in the sense of Kato) as a DtN
operator with precise description of the domains.\n\nReferences:\n\nW. Ar
endt\, A.F.M. ter Elst: The Dirichlet-to-Neumann operator on C. Ann. Sc. N
orm. Super. Pisa Cl. Sci. 20 (2020) 1169-1196\n\nW. Arendt\, A.F.M. ter El
st\, M. Warma: Fractional powers of sectorial operators via the Dirichlet-
to-Neumann operator.\nComm. PDE 43 (2018) 1-24\n\nJ. Galé\, P. Miana\, P.
R. Stinga: Extension problem and fractional operators: semigroups and wave
equation. J. Evol. Eqn. 13 (2013) 343-368.\n
LOCATION:https://researchseminars.org/talk/NonLocalOperators/44/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Yehuda Pinchover (Technion – Israel Institute of Technology)
DTSTART;VALUE=DATE-TIME:20210713T130000Z
DTEND;VALUE=DATE-TIME:20210713T140000Z
DTSTAMP;VALUE=DATE-TIME:20240715T174606Z
UID:NonLocalOperators/45
DESCRIPTION:Title: Optimal Hardy inequalities for Schrodinger operators on graphs<
/a>\nby Yehuda Pinchover (Technion – Israel Institute of Technology) as
part of Non-local operators\, probability and singularities\n\n\nAbstract\
nFor a given subcritical discrete Schrödinger operator $H$ on a weighted
infinite graph $X$\, we construct a Hardy-weight $w$ which is optimal in t
he following sense. The operator $H − \\lambda w$ is subcritical in $X$
for all $\\lambda < 1$\, null-critical in $X$ for $\\lambda = 1$\, and sup
ercritical near any neighborhood of infinity in $X$ for any $\\lambda>1$.
Our results rely on a criticality theory for Schrödinger operators on gen
eral weighted graphs.\n\nThis is a joint work with Matthias Keller and Fel
ix Pogorzelski.\n
LOCATION:https://researchseminars.org/talk/NonLocalOperators/45/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Yana Butko (Universität des Saarlandes)
DTSTART;VALUE=DATE-TIME:20210720T130000Z
DTEND;VALUE=DATE-TIME:20210720T140000Z
DTSTAMP;VALUE=DATE-TIME:20240715T174606Z
UID:NonLocalOperators/46
DESCRIPTION:Title: Stochastic solutions of generalized time-fractional evolution e
quations\nby Yana Butko (Universität des Saarlandes) as part of Non-l
ocal operators\, probability and singularities\n\n\nAbstract\nThis is a jo
int work with Christian Bender\, Saarland University. We consider a gener
al class of integro-differential evolution equations which includes the go
verning equation of the generalized grey Brownian motion and the time- and
space-fractional heat equation:\n\n$$u(t\,x) = u_0(x) + \\int_0^t k(t\,s)
Lu(s\,x)ds\, \\qquad t>0\,\\quad x\\in\\mathbb{R}^d\, \\qquad (1)$$\n\nwhe
re $L$ is a pseudo-differential operator associated to a Lévy process and
\n$k(t\,s)$\, $0 < s < t < \\infty$\, is a general memory kernel. Such eq
uations arise in models of anomalous diffusion.\n\nWe present a general re
lation between the parameters of the equation and the distribution of any
stochastic process\, which provides a stochastic solution of Feynman-Kac
type. More precisely\, we derive a series representation in terms of the
time kernel $k$ and the symbol $-\\psi$ of the pseudodifferential operator
$L$ for the characteristic function of the one-dimensional marginals of a
ny stochastic solution. We explain how this series simplifies in the impor
tant case of homogeneous kernels which includes the kernel $k(t\,s)=(t-s)^
{\\beta-1}/\\Gamma(\\beta)$ for time-fractional evolution equations and\,
more generally\, kernels corresponding to Saigo-Maeda fractional diffinte
gration operators. The connection between Saigo-Maeda fractional diffinteg
ration operators and positive random variables with Laplace transform give
n by Prabhakar's three parameter generalization of the Mittag-Leffler func
tion is established. These results yield a stochastic representation for
(1) with a Saigo-Maeda kernel in terms of a randomly slowed down Lévy pr
ocess $(Y_{At^\\beta})_{t\\geq 0}$\, where $Y$ is a Lévy process with inf
initesimal generator $L$\, $A$ is an independent random variable with Lapl
ace transform given by the three-parameter Mittag-Leffler function\, and $
\\beta$ corresponds to the degree of homogeneity of the kernel. If $Y$ has
a stable distribution (e.g.\, in the case of a symmetric fractional Lapla
cian in space)\, the randomly slowed down Lévy process can be replaced by
a randomly scaled linear fractional stable motion\, providing a stochast
ic solution in terms of a self-similar process with stationary increments.
Similar results hold also in the case of equation (1) with more general
operator $L$ (it is enough to assume that $L$ generates a strongly continu
ous semigroup on some Banach space) and lead to Feynman-Kac formulae for s
uch equations (if $L:=L_0+V$ where $L_0$ generates a Markov process and $V
$ is a suitable potential). \n\n[1] Ch. Bender\, Ya.A. Butko. Stochastic
solutions of generalized time-fractional evolution equations// arXiv:2102.
00117 (2021)\n
LOCATION:https://researchseminars.org/talk/NonLocalOperators/46/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Grzegorz Karch (University of Wrocław)
DTSTART;VALUE=DATE-TIME:20210928T130000Z
DTEND;VALUE=DATE-TIME:20210928T140000Z
DTSTAMP;VALUE=DATE-TIME:20240715T174606Z
UID:NonLocalOperators/47
DESCRIPTION:Title: Concentration phenomena in a model of chemotaxis\nby Grzego
rz Karch (University of Wrocław) as part of Non-local operators\, probabi
lity and singularities\n\n\nAbstract\nIn this talk\, I shall discuss resul
ts obtained in the papers:\n\nBiler\, Piotr\; Karch\, Grzegorz\; Zienkiewi
cz\, Jacek\, Large\nglobal-in-time solutions to a nonlocal model of chemot
axis. Adv. Math.\n330 (2018)\, 834–875.\n\nBiler\, Piotr\; Karch\, Grzeg
orz\; Pilarczyk\, Dominika\, Global radial\nsolutions in classical Keller-
Segel model of chemotaxis. J.\nDifferential Equations 267 (2019)\, no. 11\
, 6352–6369.\n\nthe both on solutions to certain parabolic-elliptic mode
ls of\nchemotaxis. In these papers\, criteria for existence and nonexisten
ce\nof global-in-time solutions have been obtained.\n
LOCATION:https://researchseminars.org/talk/NonLocalOperators/47/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Mathav Murugan (University of British Columbia)
DTSTART;VALUE=DATE-TIME:20211026T140000Z
DTEND;VALUE=DATE-TIME:20211026T150000Z
DTSTAMP;VALUE=DATE-TIME:20240715T174606Z
UID:NonLocalOperators/48
DESCRIPTION:Title: On the comparison between jump processes and subordinated diffu
sions\nby Mathav Murugan (University of British Columbia) as part of N
on-local operators\, probability and singularities\n\n\nAbstract\nA well k
nown method to obtain heat kernel estimates and Harnack inequalities for j
ump processes is to compare the given jump process with a subordinated dif
fusion process. On any space that admits a diffusion which satisfies sub-G
aussian heat kernel bounds\, we show that a large family of jump processes
have a jump kernel comparable to that of a subordinated diffusion process
. If time permits\, I will also discuss another recent result that the par
abolic Harnack inequality implies that the jump kernel admits a density. T
his is joint work with Guanhua Liu (Tsinghua University).\n
LOCATION:https://researchseminars.org/talk/NonLocalOperators/48/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Carlo Marinelli (University College London)
DTSTART;VALUE=DATE-TIME:20211102T140000Z
DTEND;VALUE=DATE-TIME:20211102T150000Z
DTSTAMP;VALUE=DATE-TIME:20240715T174606Z
UID:NonLocalOperators/49
DESCRIPTION:Title: On the Malliavin differentiability of solutions to reaction-dif
fusion equations with multiplicative noise\nby Carlo Marinelli (Univer
sity College London) as part of Non-local operators\, probability and sing
ularities\n\n\nAbstract\nWe discuss some recent results about existence an
d regularity of the Malliavin derivative of the solution\, evaluated at fi
xed points in time and space\, to a parabolic dissipative stochastic PDE o
n $L^2(G)$\, where $G$ is an open bounded domain in $\\mathbb{R}^d$ with s
mooth boundary. The equation is driven by a multiplicative Wiener noise an
d the nonlinear drift term is the superposition operator associated to a r
eal function which is assumed to be monotone\, locally Lipschitz continuou
s\, and growing not faster than a polynomial. The arguments are based on t
he well-posedness theory in the mild sense for stochastic evolution equati
ons in Banach spaces\, as well as on monotonicity techniques and on a maxi
mum principle for stochastic evolution equations.\n\nPartly based on joint
work with Ll. Quer-Sardanyons.\n
LOCATION:https://researchseminars.org/talk/NonLocalOperators/49/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Simon Nowak (Bielefeld University)
DTSTART;VALUE=DATE-TIME:20211012T130000Z
DTEND;VALUE=DATE-TIME:20211012T140000Z
DTSTAMP;VALUE=DATE-TIME:20240715T174606Z
UID:NonLocalOperators/50
DESCRIPTION:Title: Regularity theory for nonlocal equations with VMO coefficients<
/a>\nby Simon Nowak (Bielefeld University) as part of Non-local operators\
, probability and singularities\n\n\nAbstract\nWe present some higher regu
larity results for nonlocal equations with possibly discontinuous coeffici
ents of VMO-type in fractional Sobolev spaces. While for corresponding loc
al elliptic equations with VMO coefficients it is only possible to obtain
higher integrability\, in our nonlocal setting we are able to also prove a
substantial amount of higher differentiability. Therefore\, our results a
re in some sense of purely nonlocal type\, following the recent trend of s
uch results in the literature.\n
LOCATION:https://researchseminars.org/talk/NonLocalOperators/50/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Michael Röckner (Bielefeld University)
DTSTART;VALUE=DATE-TIME:20211109T140000Z
DTEND;VALUE=DATE-TIME:20211109T150000Z
DTSTAMP;VALUE=DATE-TIME:20240715T174606Z
UID:NonLocalOperators/51
DESCRIPTION:Title: Strong dissipativity of generalized time-fractional derivatives
and quasi-linear (stochstic) partial differential equations\nby Micha
el Röckner (Bielefeld University) as part of Non-local operators\, probab
ility and singularities\n\nAbstract: TBA\n
LOCATION:https://researchseminars.org/talk/NonLocalOperators/51/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Peter Stollmann (Chemnitz University of Technology)
DTSTART;VALUE=DATE-TIME:20211019T130000Z
DTEND;VALUE=DATE-TIME:20211019T140000Z
DTSTAMP;VALUE=DATE-TIME:20240715T174606Z
UID:NonLocalOperators/52
DESCRIPTION:Title: On the decomposition principle and a Persson type theorem for g
eneral regular Dirichlet forms\nby Peter Stollmann (Chemnitz Universit
y of Technology) as part of Non-local operators\, probability and singular
ities\n\n\nAbstract\nWe present a decomposition principle for general regu
lar Dirichlet forms satisfying a spatial local compactness condition. We u
se the decomposition principle to derive a Persson type theorem for the\nc
orresponding Dirichlet forms. Our setting covers non-local forms as well
as local ones.\n
LOCATION:https://researchseminars.org/talk/NonLocalOperators/52/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Pascal Auscher (University of Paris-Saclay)
DTSTART;VALUE=DATE-TIME:20211116T140000Z
DTEND;VALUE=DATE-TIME:20211116T150000Z
DTSTAMP;VALUE=DATE-TIME:20240715T174606Z
UID:NonLocalOperators/53
DESCRIPTION:Title: On regularity of weak solutions to linear parabolic systems wit
h measurable coefficients\nby Pascal Auscher (University of Paris-Sacl
ay) as part of Non-local operators\, probability and singularities\n\n\nAb
stract\nWe discuss what can be seen as an old and seemingly closed problem
in regularity theory for parabolic systems. Still novel phenomenons show
up. Namely\, what is the regularity of weak solutions to equations or syst
ems of the form $\\partial_t u - div A\\\, grad\\\, u = f+ div F$ on a par
abolic cylinder assuming minimal conditions on A and (parabolic) scale in
variant integrability on f and F? Of course\, it depends on how one define
s a weak solution. We show\, and it seems that was not noticed under minim
al assumptions\, that local square integrability of u and its gradient im
plies local $L^2$ bounds uniformly in time and more. Further\, with fur
ther integrability property on f and F\, we also obtain Hölder continuit
y in time with values in $L^p$ for some $p>2$\, which is new\, as well as
higher integrability for the gradient which was due to Giaquinta and Stru
we. This uses of half-order time derivatives with some non-local estimate
s. \n\nJoint work with S. Bortz\, M. Egert and O. Saari (JMPA\, 2019).\n
LOCATION:https://researchseminars.org/talk/NonLocalOperators/53/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Mark Veraar (Delft University of Technology)
DTSTART;VALUE=DATE-TIME:20211214T140000Z
DTEND;VALUE=DATE-TIME:20211214T150000Z
DTSTAMP;VALUE=DATE-TIME:20240715T174606Z
UID:NonLocalOperators/54
DESCRIPTION:Title: Stochastic Navier-Stokes equations with gradient noise in criti
cal spaces\nby Mark Veraar (Delft University of Technology) as part of
Non-local operators\, probability and singularities\n\n\nAbstract\nIn thi
s talk I will present some recent results on the stochastic Navier-Stokes
equations on the d-dimensional torus with gradient noise\, which arises in
the study of turbulent flows. Under very weak smoothness assumptions on t
he data one has local well-posedness. Moreover\, using a new bootstrap met
hod new regularization results for solution are obtained. New blow-up crit
eria are introduced and can be seen as stochastic versions of the Serrin b
low-up criteria. The latter is used to prove global well-posedness with hi
gh probability for small initial data in critical spaces in any dimensions
$d\\geq 2$. Moreover\, for d=2 we obtain new global well-posedness result
s and regularization phenomena\, which unify and extend several earlier re
sults.\n\nThe talk is based on joint work with Antonio Agresti https://arx
iv.org/abs/2107.03953\n
LOCATION:https://researchseminars.org/talk/NonLocalOperators/54/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Rupert Frank (University of Munich)
DTSTART;VALUE=DATE-TIME:20220118T140000Z
DTEND;VALUE=DATE-TIME:20220118T150000Z
DTSTAMP;VALUE=DATE-TIME:20240715T174606Z
UID:NonLocalOperators/55
DESCRIPTION:Title: Sobolev norms involving fractional Hardy operators\nby Rupe
rt Frank (University of Munich) as part of Non-local operators\, probabili
ty and singularities\n\n\nAbstract\nWe consider the fractional Schrödinge
r operator with Hardy potential and critical or subcritical coupling const
ant. This operator generates a natural scale of homogeneous Sobolev spaces
which we compare with the ordinary homogeneous Sobolev spaces. As a bypro
duct\, we obtain generalized and reversed Hardy inequalities for this oper
ator. Our results extend those obtained recently for ordinary (non-fractio
nal) Schrödinger operators and have an important application in the treat
ment of large relativistic atoms. \n\nThe talk is based on joint work with
K. Merz and H. Siedentop\n
LOCATION:https://researchseminars.org/talk/NonLocalOperators/55/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jeremy Quastel (University of Toronto)
DTSTART;VALUE=DATE-TIME:20220208T140000Z
DTEND;VALUE=DATE-TIME:20220208T150000Z
DTSTAMP;VALUE=DATE-TIME:20240715T174606Z
UID:NonLocalOperators/56
DESCRIPTION:Title: Integrable fluctuations in random growth\nby Jeremy Quastel
(University of Toronto) as part of Non-local operators\, probability and
singularities\n\n\nAbstract\nWe survey models in the KPZ universality clas
s and the integrable Markov process which governs their asymptotic fluctua
tions.\n
LOCATION:https://researchseminars.org/talk/NonLocalOperators/56/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Fritz Gesztesy (Baylor University)
DTSTART;VALUE=DATE-TIME:20211130T140000Z
DTEND;VALUE=DATE-TIME:20211130T150000Z
DTSTAMP;VALUE=DATE-TIME:20240715T174606Z
UID:NonLocalOperators/57
DESCRIPTION:Title: Continuity properties of the spectral shift function for massle
ss Dirac operators and an application to the Witten index\nby Fritz Ge
sztesy (Baylor University) as part of Non-local operators\, probability an
d singularities\n\n\nAbstract\nWe report on recent results regarding the l
imiting absorption principle for multi-dimensional\, massless Dirac-type o
perators (implying absence of singularly continuous spectrum) and continui
ty properties of the associated spectral shift function.\n\nWe will motiva
te our interest in this circle of ideas by briefly describing the connecti
on to index theory for non-Fredholm operators\, particularly\, to the noti
on of the Witten index.\n\nThis is based on various joint work with A. Car
ey\, J. Kaad\, G. Levitina\, R. Nichols\, D. Potapov\, F. Sukochev\, and D
. Zanin.\n
LOCATION:https://researchseminars.org/talk/NonLocalOperators/57/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jessica Lin (McGill University)
DTSTART;VALUE=DATE-TIME:20220308T140000Z
DTEND;VALUE=DATE-TIME:20220308T150000Z
DTSTAMP;VALUE=DATE-TIME:20240715T174606Z
UID:NonLocalOperators/58
DESCRIPTION:Title: Asymmetric and Symmetric Cooperative Motion\nby Jessica Lin
(McGill University) as part of Non-local operators\, probability and sing
ularities\n\n\nAbstract\nWe prove distributional convergence for a family
of random processes on $\\mathbb{Z}$\, which describe a type of random wal
k with dependent delay. The model generalizes the "hipster random walks" s
tudied by Addario-Berry et al [Probability Theory and Related Fields\, '20
]. We introduce a novel approach which relies on convergence results for f
inite difference schemes of certainly fully nonlinear PDEs. This talk is b
ased on joint work with Louigi Addario-Berry and Erin Beckman.\n
LOCATION:https://researchseminars.org/talk/NonLocalOperators/58/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Stéphane Menozzi (Université d'Évry Val d'Essonne)
DTSTART;VALUE=DATE-TIME:20220315T140000Z
DTEND;VALUE=DATE-TIME:20220315T150000Z
DTSTAMP;VALUE=DATE-TIME:20240715T174606Z
UID:NonLocalOperators/59
DESCRIPTION:Title: Heat kernel of supercritical SDEs with unbounded drifts\nby
Stéphane Menozzi (Université d'Évry Val d'Essonne) as part of Non-loca
l operators\, probability and singularities\n\n\nAbstract\nWe consider SDE
s driven by isotropic α-stable processes\, 0<α<2\, where:\n- the coeffic
ients are Hölder continuous in space\n- the "diffusion" coefficient is bo
unded and uniformly elliptic\n- the drift can be unbounded\n\nIf β is the
spatial Hölder regularity index of the coefficients\, we obtain under th
e condition α+β>1 existence of the density for such SDEs. Furthermore th
is density enjoys sharp two-sided estimates and we derive as well sharp bo
unds for its logarithmic derivative.\n\nImportantly\, we cover the whole s
upercritical range. The proof relies on ad hoc parametrix expansions and p
robabilistic techniques.\n
LOCATION:https://researchseminars.org/talk/NonLocalOperators/59/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Giorgio Metafune (Università del Salento)
DTSTART;VALUE=DATE-TIME:20220322T140000Z
DTEND;VALUE=DATE-TIME:20220322T150000Z
DTSTAMP;VALUE=DATE-TIME:20240715T174606Z
UID:NonLocalOperators/60
DESCRIPTION:Title: A unified approach to degenerate problems in the half-space
\nby Giorgio Metafune (Università del Salento) as part of Non-local opera
tors\, probability and singularities\n\n\nAbstract\nWe study elliptic and
parabolic problems governed by the singular elliptic operators\n\n$\n\\mat
hcal L =y^{\\alpha_1}\\Delta_{x} +y^{\\alpha_2}\\left(D_{yy}+\\frac{c}{y}D
_y -\\frac{b}{y^2}\\right)\,\\qquad\\alpha_1\, \\alpha_2 \\in\\R\n$\n\nin
the half-space $\\R^{N+1}_+=\\{(x\,y): x \\in \\R^N\, y>0\\}$. This is a j
oint paper with\n L. Negro and C. Spina.\n
LOCATION:https://researchseminars.org/talk/NonLocalOperators/60/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Martin Friesen (Dublin City University)
DTSTART;VALUE=DATE-TIME:20220405T130000Z
DTEND;VALUE=DATE-TIME:20220405T140000Z
DTSTAMP;VALUE=DATE-TIME:20240715T174606Z
UID:NonLocalOperators/61
DESCRIPTION:Title: Continuous affine Volterra processes\nby Martin Friesen (Du
blin City University) as part of Non-local operators\, probability and sin
gularities\n\n\nAbstract\nRecent empirical observations on intra-day stock
market data suggest that volatilities defined as short-term fluctuations
of asset prices exhibit a highly rough behavior on smaller time scales. Wh
ile such an effect is neither adequately captured nor predicted by Markovi
an models\, recent analysis has shown that their rough counterparts based
on the fractional Brownian motion or on Volterra processes perfectly captu
re these effects.\n\n \nIn this talk\, we focus on the particular class of
affine Volterra processes being characterized by the feature that their c
haracteristic function can be expressed in a semi-explicit form in terms o
f a solution of a Volterra Riccati equation.\n\nFor this equation\, we pro
vide a priori growth bounds\, Sobolev regularity in time\, continuous depe
ndence on parameters\, and differentiability in the initial state. Based o
n these findings\, we derive the existence of limiting distributions for a
large class of affine Volterra processes. To each of these distributions\
, we construct the associated stationary non-Markovian Volterra process an
d finally analyze the regularity of its law.\n\nThis is joint work with P.
Jin (BNU-HKBU)\n
LOCATION:https://researchseminars.org/talk/NonLocalOperators/61/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Peixue Wu (University of Illinois at Urbana-Champaign)
DTSTART;VALUE=DATE-TIME:20220412T130000Z
DTEND;VALUE=DATE-TIME:20220412T140000Z
DTSTAMP;VALUE=DATE-TIME:20240715T174606Z
UID:NonLocalOperators/62
DESCRIPTION:Title: Heat kernel estimates for non-local operators with multi-singul
ar killing potential.\nby Peixue Wu (University of Illinois at Urbana-
Champaign) as part of Non-local operators\, probability and singularities\
n\n\nAbstract\nWe study the heat kernel estimates for non-local operators
with multi-singular killing potential. To be specific\, given an open set
$D$ with boundary $\\partial D = \\cup_{k=1}^d \\cup_{j=1}^{m_k} \\Gamma_{
k\,j}$\, where for any $1\\leq k$\, $1 \\leq j \\leq m_k$\, $\\Gamma_{k\,j
}$ is a $C^{1\,\\beta}$ submanifold without boundary of codimension $1\\le
k\\le d$ and $\\{\\Gamma_{k\,j}\\}_{1\\le k\\le d\, 1\\le j \\le m_k}$ ar
e disjoint. We show that the heat kernel $p^D(t\,x\,y)$ of the following n
on-local operator with multi-singular critical killing potential \n\n\n$$\
n\\big( (\\Delta|_D)^{\\alpha/2} - \\kappa\\big)(f)(x):= p.v. {A}_{d\,-\\a
lpha} \\int_D \\frac{f(y)-f(x)}{|y-x|^{d+\\alpha}}dy - \\sum_{k=1}^d \\sum
_{j=1}^{m_k} \\lambda_{k\,j} \\delta_{\\Gamma_{k\,j}}(x)^{-\\alpha}\,\n$$\
n\nwhere $ \\lambda_{k\,j}>0\, \\alpha \\in (0\,2)$ has the following esti
mates: for any given $T>0$\, \n\n$$\np^D(t\,x\,y) \\asymp p(t\,x\,y) \\pro
d_{k=1}^d \\prod_{j=1}^{m_k} (\\frac{\\delta_{\\Gamma_{k\,j}}(x)}{t^{1/\\a
lpha}} \\wedge 1)^{p_{k\,j}}(\\frac{\\delta_{\\Gamma_{k\,j}}(y)}{t^{1/\\al
pha}} \\wedge 1)^{p_{k\,j}}\, \\quad \\forall t\\in (0\,T)\, x\,y\\in D\,\
n$$\n\nwhere $p(t\,x\,y)$ is the heat kernel of the $\\alpha$-stable proce
ss on $\\mathbb{R}^d$ and $p_{k\,j}$ and $\\lambda_{k\,j}$ are related thr
ough a strictly increasing function $\\lambda = C(k\,\\alpha\,p)$. Our met
hod is based on the result established in [Cho et al. Journal de Mathémat
iques Pures et Appliquées 143(2020): 208-256] and a detailed analysis of
$C^{1\,\\beta}$ manifolds. \n\nThis is joint work with Renming Song and Sh
ukun Wu: arXiv:2203.03891.\n
LOCATION:https://researchseminars.org/talk/NonLocalOperators/62/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Soobin Cho (Seoul National University)
DTSTART;VALUE=DATE-TIME:20220510T130000Z
DTEND;VALUE=DATE-TIME:20220510T140000Z
DTSTAMP;VALUE=DATE-TIME:20240715T174606Z
UID:NonLocalOperators/63
DESCRIPTION:Title: General law of iterated logarithm for Markov processes\nby
Soobin Cho (Seoul National University) as part of Non-local operators\, pr
obability and singularities\n\n\nAbstract\nIn this talk\, we discuss gener
al criteria and forms of both liminf and limsup laws of\niterated logarith
m (LIL) for continuous-time Markov processes. We establish LILs under loca
l assumptions near zero (near in finity\, respectively) on uniform bounds
of the first exit time from balls in terms of a function $\\phi$ and unif
orm bounds on the tails of the jumping measure in terms of a function $\\p
si$. One of the main results is that a simple ratio test in terms of the f
unctions $\\phi$ and $\\psi$ completely determines whether there exists a
positive nondecreasing function $R(t)$ such that $limsup|X_t|/R(t)$ is pos
itive and finite a.s.\, or not. Our results cover a large class of subordi
nate dffusions\, jump processes with mixed polynomial local growths and ra
ndom conductance models with long range jumps.\n
LOCATION:https://researchseminars.org/talk/NonLocalOperators/63/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Matteo Bonforte (University of Madrid)
DTSTART;VALUE=DATE-TIME:20220426T130000Z
DTEND;VALUE=DATE-TIME:20220426T140000Z
DTSTAMP;VALUE=DATE-TIME:20240715T174606Z
UID:NonLocalOperators/64
DESCRIPTION:Title: Nonlinear and Nonlocal Diffusions. Smoothing effects\, Green fu
nctions and functional inequalities\nby Matteo Bonforte (University of
Madrid) as part of Non-local operators\, probability and singularities\n\
n\nAbstract\nWe will consider the Cauchy problem for Nonlinear Diffusion e
quations of porous medium type $u_t=-\\mathcal{L} u^m$\, with $m>1$ and in
vestigate whether or not integrable data produce bounded solutions. The di
ffusion operator belongs to a quite general class of nonlocal operators\,
and we will see how different assumption on the operator imply (or not) sm
oothing properties. We will briefly compare the approach based on Moser it
eration and the approach through Green functions. On one hand\, we show th
at if the linear case ($m=1$) enjoys smoothing properties\, also the nonli
near will do. On the other hand\, we see that in some cases the nonlinear
diffusion enjoys the smoothing properties also when the linear counterpart
does not\, thanks to the convex nonlinearity.\n\nFollowing Nash' ideas\,
we see how smoothing properties are often equivalent to the validity of Ga
gliardo-Nirenberg-Sobolev (and Nash) inequalities: we explore these implic
ations also in the nonlinear and nonlocal context and the connection with
dual inequalities (Hardy-Littlewood-Sobolev) and Green function estimates.
\n\nThis is a work in progress with J. Endal (UAM\, Madrid).\n\nIf time
allows\, we will complete the panorama by showing related results on Eucli
dean bounded domains (joint works with Figalli\, Ros-Oton\, Sire\, Vazquez
) and/or on Riemannian Manifolds (joint works with Berchio\, Ganguly\, Gri
llo\, Muratori)\, together with a small detour on the Fast diffusion case
$m<1$ (joint work with Ibarrondo and Ispizua).\n
LOCATION:https://researchseminars.org/talk/NonLocalOperators/64/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Chiara Spina (University of Salento)
DTSTART;VALUE=DATE-TIME:20220531T130000Z
DTEND;VALUE=DATE-TIME:20220531T140000Z
DTSTAMP;VALUE=DATE-TIME:20240715T174606Z
UID:NonLocalOperators/65
DESCRIPTION:Title: $L^p$ estimates for a class of degenerate operators\nby
Chiara Spina (University of Salento) as part of Non-local operators\, prob
ability and singularities\n\n\nAbstract\nWe prove $L^p$-estimates for the
operator\n$$\\mathcal L=\\Delta_x+\\Delta_y +c\\frac{y}{|y|^2}\\cdot\\n
abla_y-\\frac{b}{|y|^{2}}=\\Delta_x+L_y\,$$\nwhere $L_y=\\Delta_y +c\\frac
{y}{|y|^2}\\cdot\\nabla_y-\\frac{b}{|y|^{2}}$. The parameters $b\,\\ c$
are constant real coefficients subject to the condition $ D:=b+\\left(\\fr
ac{M-2+c}{2}\\right)^2> 0$. \n\nWe work in the space $L^p_c:=L^p(\\R^{N+M}
\, |y|^c\\\, dxdy)$\, motivated by the fact that the weight $|y|^c$ makes
the operator symmetric in $L^2_c$ and we assume $M+c>0$\, so that the mea
sure $d\\mu=|y|^c\\\, dx\\\, dy$ is locally finite on $\\R^{N+M}$.\n\n\n\n
\nThe operators $\\Delta_x$\, $L_y$ commute and the whole operator $\\mat
hcal L$ satisfies the scaling property $I_s^{-1}\\mathcal L I_s=s^2\\mathc
al L$\, if $I_s u(x\,y)=u(sx\,sy)$. It is not difficult to see that $\\mat
hcal L$ generates a semigroup in $L^p_c$ if and only if $L_y$ generates in
$L^p(\\R^M\, |y|^c\\\, dy)$ and this is equivalent to $(M+c)\\\, \\left|\
\frac{1}{2}-\\frac 1 p\\right|<1+\\sqrt D$.\n\n\nWhen $M=1$ and $b=0$\, $
L_y$ is a Bessel operator and both $\\mathcal L=\\Delta_x+B_y$ and $D_t-\\
mathcal L$ play a major role in the investigation of the fractional powers
$(-\\Delta_x)^s$ and $(D_t-\\Delta_x)^s$\, $s=(1-c)/2$\, through the ``
extension procedure" of Caffarelli and Silvestre [1]. \n\n\nWhen $M=1$\, t
hat is in the half-space $\\R^{N+1}_+$\, all the results of this paper\,
and much more\, have been proved in [4] by taking advantage of sophistica
ted tools from operator valued harmonic analysis. More general\, non symm
etrizing weights $|y|^m\\\, dx\\\, dy$ are therein considered and both Dir
ichlet and Neumann boundary conditions. We refer the reader also to [2]\,
[3] for the case $b=0$ and with variable coefficients. \n\n\nHere we use
a different strategy and show that $L^p$-estimates for the pure $x$-deriv
atives\, that is the boundedness of the operators $D_{x_ix_j}\\mathcal L^{
-1}$\, follow from sub-solution estimates through an interpolation theorem
in absence of kernels in homogeneous spaces due to Z. Shen. Sub-solution
estimates\, that is improving of integrability for (sub) solutions of the
homogeneous equation $\\mathcal Lu=0$\, are proved by combining Cacciopoli
estimates\, weighted Sobolev embeddings and Moser iteration.\n\n[1] L. Ca
ffarelli\,L. Silvestre: An extension problem related to the fractional Lap
lacean\,\nComm. Partial Differential Equations\, 32 (2007)\, no. 7-9 1245-
1260.\n\n[2] H. Dong\, T. Phan: On parabolic and elliptic equations with s
ingular or degenerate coeff-\ncients\, arxiv: 2007.04385 2020\n\n[3] H. Do
ng\, T. Phan:Weighted mixed-norm Lp estimates for equations in non-diverge
nce form\nwith singular coeffcients: the Dirichlet problem arxiv: 2103.080
33 2021\n\n[4] G. Metafune\, L. Negro\, C. Spina: Lp estimates for the Caf
farelli-Silvestre extension\noperators\, Journal of Differential Equations
Volume 316\, (2022)\, Pages 290-345.\n
LOCATION:https://researchseminars.org/talk/NonLocalOperators/65/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Błażej Wróbel (University of Wroclaw)
DTSTART;VALUE=DATE-TIME:20220621T130000Z
DTEND;VALUE=DATE-TIME:20220621T140000Z
DTSTAMP;VALUE=DATE-TIME:20240715T174606Z
UID:NonLocalOperators/66
DESCRIPTION:Title: On Lp estimates for positivity-preserving Riesz transforms rela
ted to Schrödinger operators\nby Błażej Wróbel (University of Wroc
law) as part of Non-local operators\, probability and singularities\n\n\nA
bstract\nWe study $L^p$ boundedness properties of positivity preserving Ri
esz transforms related to a \nSchrödinger operator. Using interpolation t
echnique we establish $L^p$ boundedness for general non-negative potential
s. Then we present a counterexample showing that the $L^{\\infty}$ bounded
ness may fail. Next we give integral type conditions on the potential $V$
that guarantee the boundedness on the endpoints $p=1$ and $p=\\infty$. Our
conditions depend only on the global properties of V and are resistant to
small perturbations. In particular $V$ may have a power growth or an expo
nential growth. \n\nThe talk is based on a joint work with Maciej Kucharsk
i.\n
LOCATION:https://researchseminars.org/talk/NonLocalOperators/66/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jie Xiao (Memorial University)
DTSTART;VALUE=DATE-TIME:20220628T130000Z
DTEND;VALUE=DATE-TIME:20220628T140000Z
DTSTAMP;VALUE=DATE-TIME:20240715T174606Z
UID:NonLocalOperators/67
DESCRIPTION:Title: Energy formulae for fractional Schrodinger-Poisson system\n
by Jie Xiao (Memorial University) as part of Non-local operators\, probabi
lity and singularities\n\n\nAbstract\nThrough a geometric-capacity-theoret
ic approach\, this talk presents two new formulas for the fractional energ
y of a quantum particle arising from the fractional Schrodinger-Poisson s
ystem which models the behavior of a quantum particle within an unknown el
ectrostatic field.\n
LOCATION:https://researchseminars.org/talk/NonLocalOperators/67/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Xue-Mei Li (EPFL and Imperial College London)
DTSTART;VALUE=DATE-TIME:20221115T140000Z
DTEND;VALUE=DATE-TIME:20221115T150000Z
DTSTAMP;VALUE=DATE-TIME:20240715T174606Z
UID:NonLocalOperators/70
DESCRIPTION:Title: Hessian estimates of the logarithmic heat kernel\nby Xue-Me
i Li (EPFL and Imperial College London) as part of Non-local operators\, p
robability and singularities\n\n\nAbstract\nHessian estimates on the heat
kernel and its logarithmic estimates for important first steps toward unde
rstanding\ntoe Brownian bridge measure\, a natural measure on loop spaces.
It also find applications in studying the existence of Lipschitz\ntranspo
rt maps. I shall discuss some new techniques in these studies and results.
This is joint work with Xin Chen and Bo Wu.\n
LOCATION:https://researchseminars.org/talk/NonLocalOperators/70/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Longjie Xie (Jiangsu Normal University)
DTSTART;VALUE=DATE-TIME:20221011T130000Z
DTEND;VALUE=DATE-TIME:20221011T140000Z
DTSTAMP;VALUE=DATE-TIME:20240715T174606Z
UID:NonLocalOperators/71
DESCRIPTION:Title: Weak and strong well-posedness of critical and supercritical SD
Es with singular coefficients\nby Longjie Xie (Jiangsu Normal Universi
ty) as part of Non-local operators\, probability and singularities\n\n\nAb
stract\nConsider the following time-dependent stable-like operator with dr
ift:\n$$\n\\mathscr{L}_t\\varphi(x)=\\int_{\\mathbb{R}^d}\\big[\\varphi(x+
z)-\\varphi(x)-z^{(\\alpha)}\\cdot\\nabla\\varphi(x)\\big]\\sigma(t\,x\,z)
\\nu_\\alpha(d z)+b(t\,x)\\cdot\\nabla \\varphi(x)\,\n$$\nwhere $d\\geq 1$
\, $\\nu_\\alpha$ is an $\\alpha$-stable type Lévy measure with $\\alpha\
\in(0\,1]$ and $z^{(\\alpha)}=1_{\\alpha=1}1_{|z|\\leq1}z$\, $\\sigma$ is
a real-valued Borel function on $\\mathbb{R}_+\\times\\mathbb{R}^d\\times\
\mathbb{R}^d$ and $b$ is an $\\mathbb{R}^d$-valued Borel function\non $\\m
athbb{R}_+\\times\\mathbb{R}^d$. By using the Littlewood-Paley theory\, we
establish the well-posedness for the martingale problem associated with $
\\mathscr{L}_t$ under the sharp balance condition $\\alpha+\\beta\\geq1$\,
where $\\beta$ is the Hölder index of $b$ with respect to $x$.\nMoreover
\, we also study a class of stochastic differential equations driven by Ma
rkov processes with generators of the form $\\mathscr{L}_t$.\nWe prove the
pathwise uniqueness of strong solutions for such equations when the coeff
icients are in certain Besov spaces.\n
LOCATION:https://researchseminars.org/talk/NonLocalOperators/71/
END:VEVENT
BEGIN:VEVENT
SUMMARY:The Anh Bui (Macquarie University)
DTSTART;VALUE=DATE-TIME:20221213T140000Z
DTEND;VALUE=DATE-TIME:20221213T150000Z
DTSTAMP;VALUE=DATE-TIME:20240715T174606Z
UID:NonLocalOperators/72
DESCRIPTION:Title: On Sobolev norms involving generalized Hardy operators\nby
The Anh Bui (Macquarie University) as part of Non-local operators\, probab
ility and singularities\n\n\nAbstract\nConsider the operator on $L^{2}(\\m
athbb{R}^d)\, d\\ge 1$\n $$\n \\mathcal L_a = (-\\Delta)^{\\alpha/2}+a|x
|^{-\\alpha} \\quad \\text{with} \\quad 0<\\alpha<\\min\\{2\, d\\}.\n $$\
n Under the condition $a\\ge -\\frac{2^\\alpha\\Gamma((d+\\alpha)/4)^2}{\
\Gamma((d-\\alpha)/4)^2}$ the operator is non negative and self-adjoint.\n
We prove that fractional powers {$\\mathcal{L}^{s/2}_a$}\n for $s\\in(0
\,2]$ satisfy the estimates\n $$\n \\|\\mathcal{L}_{a}^{s/2}f\\|_{L^{p
}}\n \\lesssim\\|(-\\Delta)^{\\alpha s/4}f\\|_{L^{p}}\,\n \\qquad\
n \\|(-\\Delta)^{s/2}f\\|_{L^{p}}\n \\lesssim \\|\\mathcal{L}_{a}^
{\\alpha s/4}f\\|_{L^{p}}\n $$\n for suitable ranges of $p$. Our result
fills the remaining gap\n in earlier results. The method of proof is base
d on square function estimates for operators whose heat kernel has a weak
decay. This talk is based on joint work with P. D'Ancona.\n
LOCATION:https://researchseminars.org/talk/NonLocalOperators/72/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Fabrice Baudoin (University of Connecticut)
DTSTART;VALUE=DATE-TIME:20221206T140000Z
DTEND;VALUE=DATE-TIME:20221206T150000Z
DTSTAMP;VALUE=DATE-TIME:20240715T174606Z
UID:NonLocalOperators/73
DESCRIPTION:Title: Asymptotic windings of the unitary Brownian motion\nby Fabr
ice Baudoin (University of Connecticut) as part of Non-local operators\, p
robability and singularities\n\n\nAbstract\nWe study several matrix diffus
ion processes constructed from a unitary Brownian motion. In particular\,
we use the Stiefel fibration to lift the Brownian motion of the complex Gr
assmannian to the complex Stiefel manifold and deduce a skew-product decom
position of the Stiefel Brownian motion. As an application\, we prove asym
ptotic laws for the determinants of the block entries of the unitary Brown
ian motion. This is a joint work with Jing Wang (Purdue University).\n
LOCATION:https://researchseminars.org/talk/NonLocalOperators/73/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Sergio Polidoro (University of Modena and Reggio Emilia)
DTSTART;VALUE=DATE-TIME:20221220T140000Z
DTEND;VALUE=DATE-TIME:20221220T150000Z
DTSTAMP;VALUE=DATE-TIME:20240715T174606Z
UID:NonLocalOperators/74
DESCRIPTION:Title: Asymptotic bounds for the relativistic Fokker-Planck operator\nby Sergio Polidoro (University of Modena and Reggio Emilia) as part of
Non-local operators\, probability and singularities\n\n\nAbstract\nWe con
sider a class of second order degenerate kinetic operators L in the framew
ork of special relativity. We first describe L as an Hörmander operator w
hich is invariant with respect to Lorentz transformations. Then we prove a
Lorentz-invariant Harnack type inequality\, and we derive accurate asympt
otic lower bounds for positive solutions to Lf=0. As a consequence we obta
in upper and lower bounds for the density of the relativistic stochastic
process associated to L.\n\nThis is a joint work with Francesca Anceschi a
nd Annalaura Rebucci.\n
LOCATION:https://researchseminars.org/talk/NonLocalOperators/74/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Abdelaziz Rhandi (University of Salerno)
DTSTART;VALUE=DATE-TIME:20221108T130000Z
DTEND;VALUE=DATE-TIME:20221108T140000Z
DTSTAMP;VALUE=DATE-TIME:20240715T174606Z
UID:NonLocalOperators/75
DESCRIPTION:Title: Bounds for the gradient of the transition kernel for elliptic o
perators with unbounded diffusion\, drift and potential terms\nby Abde
laziz Rhandi (University of Salerno) as part of Non-local operators\, prob
ability and singularities\n\n\nAbstract\nWe prove global Sobolev regularit
y and pointwise upper bounds for the gradient of transition densities asso
ciated with second order differential operators in R^d with unbounded diff
usion\, drift and potential terms.\n\nThis is joint work with Markus Kunze
and Marianna Porfido.\n
LOCATION:https://researchseminars.org/talk/NonLocalOperators/75/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Kaj Nyström (Uppsala University)
DTSTART;VALUE=DATE-TIME:20221025T130000Z
DTEND;VALUE=DATE-TIME:20221025T140000Z
DTSTAMP;VALUE=DATE-TIME:20240715T174606Z
UID:NonLocalOperators/76
DESCRIPTION:Title: Parabolic uniform rectifiability and caloric measure I: $A_\\in
fty$ implies parabolic uniform rectifiability of a parabolic Lipschitz gra
ph\nby Kaj Nyström (Uppsala University) as part of Non-local operator
s\, probability and singularities\n\n\nAbstract\nWe prove that if a parabo
lic Lipschitz graph domain has the property that its caloric measure is a
parabolic $A_\\infty$ weight with respect to surface measure\, then the fu
nction defining the graph has a half-order time derivative in the space of
(parabolic) bounded mean oscillation. Equivalently\, we prove that the $A
_\\infty$ property of caloric measure implies that the boundary is parabol
ic uniformly rectifiable. Consequently\, by combining our result with the
work of Lewis and Murray we resolve\, in the setting of parabolic Lipschit
z graph domains\, a longstanding open problem in the field by proving that
the $L^p$ solvability (for some $p > 1$) of the Dirichlet problem for the
heat equation is equivalent to parabolic uniformly rectifiability. This
is joint work with S. Bortz\, S. Hofmann\, and J.M. Martell.\n
LOCATION:https://researchseminars.org/talk/NonLocalOperators/76/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jean-François Jabir (Higher School of Economics)
DTSTART;VALUE=DATE-TIME:20230117T140000Z
DTEND;VALUE=DATE-TIME:20230117T150000Z
DTSTAMP;VALUE=DATE-TIME:20240715T174606Z
UID:NonLocalOperators/77
DESCRIPTION:Title: Multidimensional stable-driven McKean-Vlasov SDEs with distribu
tional interaction kernel\nby Jean-François Jabir (Higher School of E
conomics) as part of Non-local operators\, probability and singularities\n
\n\nAbstract\nThis talk will be focused on presenting existence and unique
ness results\, in a weak and a strong sense\, for McKean-Vlasov models dri
ven by alpha-stable Lévy processes and an interaction kernel lying in a B
esov space with non-positive exponent. In this specific setting\, we exhi
bit how\, quantitatively\, the McKean non-linearity\, together with the no
ise\, provides a regularisation effect allowing to pass beyond classical -
or more recently established - characteristic thresholds ensuring the wel
lposedness of linear SDEs with singular derive. Application to physically-
based McKean-Vlasov models and the propagation of chaos property of some r
elated particle approximation will be also discussed. \n\n\n(This work is
a joint collaboration with P-E. Chaudru de Raynal\, Laboratoire de Mathé
matiques Jean Leray\, Nantes\, and S. Menozzi\, LaMME\, Univ. Evry Val d
’Essonne - Paris Saclay.)\n
LOCATION:https://researchseminars.org/talk/NonLocalOperators/77/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Xicheng Zhang (Beijing Institute of Technology)
DTSTART;VALUE=DATE-TIME:20230131T140000Z
DTEND;VALUE=DATE-TIME:20230131T150000Z
DTSTAMP;VALUE=DATE-TIME:20240715T174606Z
UID:NonLocalOperators/78
DESCRIPTION:Title: Second order fractional mean-field SDEs with singular kernels
and measure initial data\nby Xicheng Zhang (Beijing Institute of Tech
nology) as part of Non-local operators\, probability and singularities\n\n
\nAbstract\nIn this work we establish the local and global well-posedness
of weak and strong solutions to second order fractional mean-field SDEs wi
th singular/distribution interaction kernels and measure initial value\, w
here the kernel can be\nNewton or Coulomb potential\, Riesz potential\, Bi
ot-Savart law\, etc. Moreover\, we also show the stability\, smoothness an
d the short time singularity and large time decay estimates of the density
.\nOur results reveal a phenomenon that for nonlinear mean-field equations
\, the regularity of the initial distribution could balance the singularit
y of the kernel. The precise relationship between the singularity of kerne
ls and the regularity of\ninitial value are calculated\, which belongs to
the subcritical regime in scaling sense. In particular\, our results prov
ide microscopic probability explanation and establish a unified treatment
for\nmany physical models such as fractional Vlasov-Poisson-Fokker-Planck
system\, the vorticity formulation of 2D-fractal Navier-Stokes equations\,
surface quasi-geostrophic models\, fractional porous medium equation with
viscosity\, etc.\n(This is a joint work with Zimo Ham and Michael Rockner
.)\n
LOCATION:https://researchseminars.org/talk/NonLocalOperators/78/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Lorenzo Marino (Polish Academy of Sciences)
DTSTART;VALUE=DATE-TIME:20230124T140000Z
DTEND;VALUE=DATE-TIME:20230124T150000Z
DTSTAMP;VALUE=DATE-TIME:20240715T174606Z
UID:NonLocalOperators/79
DESCRIPTION:Title: Weak regularisation by degenerate Lévy noise\nby Lorenzo M
arino (Polish Academy of Sciences) as part of Non-local operators\, probab
ility and singularities\n\n\nAbstract\nThe current talk presents some new
results about the regularisation by noise phenomena for multidimensional O
DEs\, where the random disturbance stands as a Lévy process (with suitabl
e properties) and it perturbs the dynamics only on some components. In par
ticular\, we aim to exhibit the minimal Hölder regularity on the determin
istic drift ensuring the well-posedness\, in a weak probabilistic sense\,
of the associated SDE. Due to the noise degeneracy\, an hypoelliptic-type
framework is implemented assuming a weak Hörmander condition on the drift
. As a by-product of our method of proof\, Krylov-type estimates for the c
anonical solution process are also established. In conclusion\, we also sh
ow through suitable counter-examples that there exists indeed an (almost)
sharp threshold on the Hölder regularity exponents of the drift ensuring
the weak well-posedness for the SDE.\n\n\nThe work this talk is based upon
is a joint collaboration with S. Menozzi (LaMME\, Université d’Evry Va
l d’Essonne).\n
LOCATION:https://researchseminars.org/talk/NonLocalOperators/79/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Renming Song (University of Illinois Urbana-Champaign)
DTSTART;VALUE=DATE-TIME:20230207T140000Z
DTEND;VALUE=DATE-TIME:20230207T150000Z
DTSTAMP;VALUE=DATE-TIME:20240715T174606Z
UID:NonLocalOperators/80
DESCRIPTION:Title: Potential theory of Dirichlet forms with jump kernels blowing u
p at the boundary\nby Renming Song (University of Illinois Urbana-Cham
paign) as part of Non-local operators\, probability and singularities\n\n\
nAbstract\nIn this talk\, I will present some recent results on potential
theory of Dirichlet forms on the half-space $\\R^d_+$ defined by the jump
kernel $J(x\,y)=|x-y|^{-d-\\alpha}\\mathcal{B}(x\,y)$\, where $\\alpha\\in
(0\,2)$ and $\\mathcal{B}(x\,y)$ can blow up to infinity at the boundary.
The main results include boundary Harnack principle and sharp two-sided G
reen function estimates.\n\nThis talk is based on a joint paper with Panki
Kim and Zoran Vondracek.\n
LOCATION:https://researchseminars.org/talk/NonLocalOperators/80/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Krzysztof Bogdan (Wroclaw University of Science and Technology)
DTSTART;VALUE=DATE-TIME:20230314T140000Z
DTEND;VALUE=DATE-TIME:20230314T150000Z
DTSTAMP;VALUE=DATE-TIME:20240715T174606Z
UID:NonLocalOperators/81
DESCRIPTION:Title: The fractional Laplacian with reflections\nby Krzysztof Bog
dan (Wroclaw University of Science and Technology) as part of Non-local op
erators\, probability and singularities\n\n\nAbstract\nMotivated by the no
tion of isotropic $\\alpha$-stable Lévy process confined\, by ''reflectio
ns''\, to a bounded open Lipschitz set\, we study related analytical objec
ts. In particular\, we construct the corresponding transition semigroup\,
and prove the exponential speed of convergence of the semigroup to a uniqu
e stationary distribution over a long time. This is a joint work with Mark
us Kunze.\n
LOCATION:https://researchseminars.org/talk/NonLocalOperators/81/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Oleg Butkovsky (Technische Universität Berlin)
DTSTART;VALUE=DATE-TIME:20230404T130000Z
DTEND;VALUE=DATE-TIME:20230404T140000Z
DTSTAMP;VALUE=DATE-TIME:20240715T174606Z
UID:NonLocalOperators/82
DESCRIPTION:Title: Stochastic equations with singular drift driven by fractional B
rownian motion\nby Oleg Butkovsky (Technische Universität Berlin) as
part of Non-local operators\, probability and singularities\n\n\nAbstract\
nJoint work with Khoa Le and Leonid Mytnik [1]. We consider stochastic dif
ferential equation\n$$\nd X_t=b(X_t) dt +d W_t^H\,\n$$\nwhere the drift $b
$ is either a measure or an integrable function\, and $W^H$ is a $d$-dimen
sional fractional Brownian motion with Hurst parameter $H\\in(0\,1)$\, $d\
\in\n$. For the case where $b\\in L_p(\\R^d)$\, $p\\in[1\,\\infty]$ we sho
w weak existence of solutions to this equation under the condition\n$$\n
\\frac{d}p<\\frac1H-1\,\n$$\nwhich is an extension of the Krylov-Rockner c
ondition (2005) to the fractional case. We construct a counter-example sh
owing optimality of this condition. If $b$ is a Radon measure\, particular
ly the delta measure\, we prove weak existence of solutions to this equati
on under the optimal condition $H<\\frac1{d+1}$. We also show strong wel
l-posedness of solutions to this equation under certain conditions.\nTo es
tablish these results\, we utilize the stochastic sewing technique and dev
elop a new version of the stochastic sewing lemma.\n\n[1] Butkovsky\, O.\,
Lê\, K.\, & Mytnik\, L. (2023). Stochastic equations with singular drift
driven by fractional Brownian motion. arXiv preprint arXiv:2302.11937.\n
LOCATION:https://researchseminars.org/talk/NonLocalOperators/82/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Vincenzo Ambrosio (Università Politecnica delle Marche)
DTSTART;VALUE=DATE-TIME:20230627T130000Z
DTEND;VALUE=DATE-TIME:20230627T140000Z
DTSTAMP;VALUE=DATE-TIME:20240715T174606Z
UID:NonLocalOperators/83
DESCRIPTION:Title: The nonlinear fractional relativistic Schrödinger equation
\nby Vincenzo Ambrosio (Università Politecnica delle Marche) as part of N
on-local operators\, probability and singularities\n\n\nAbstract\nWe consi
der the following class of fractional relativistic Schrödinger equations:
\n\\[\n\\begin{cases}\n(-\\Delta+m^{2})^{s}u + V(\\varepsilon x) u= f(u)\,
\\quad {\\rm in} \\quad \\mathbb{R}^{N}\,\\\\\nu \\in H^{s}(\\mathbb{R}^{N
})\, \\quad u>0 \\quad {\\rm in} \\quad \\mathbb{R}^{N}\,\n\\end{cases}\n\
\]\nwhere $\\varepsilon>0$ is a small parameter\, $s\\in (0\, 1)$\, $m>0$\
, $N> 2s$\, $(-\\Delta+m^{2})^{s}$ is the fractional relativistic Schrödi
nger operator\, $V:\\mathbb{R}^{N}\\rightarrow \\mathbb{R}$ is a continuo
us potential satisfying a local condition\, and $f:\\mathbb{R}\\rightarrow
\\mathbb{R}$ is a continuous subcritical nonlinearity. We first show that
\, for $\\varepsilon>0$ small enough\, the above problem has a weak soluti
on $u_{\\varepsilon}$ (with exponential decay at infinity) which concentra
tes around a local minimum point of $V$ as $\\varepsilon\\rightarrow 0$. W
e also relate the number of positive solutions with the topology of the se
t where the potential $V$ attains its minimum value.\n\nThe main results w
ill be established by using a penalization technique\, the generalized Neh
ari manifold method and Ljusternik-Schnirelman theory.\n
LOCATION:https://researchseminars.org/talk/NonLocalOperators/83/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Marvin Weidner (Universitat de Barcelona)
DTSTART;VALUE=DATE-TIME:20230516T130000Z
DTEND;VALUE=DATE-TIME:20230516T140000Z
DTSTAMP;VALUE=DATE-TIME:20240715T174606Z
UID:NonLocalOperators/84
DESCRIPTION:Title: The nonlocal Bernstein technique and the nonlocal obstacle prob
lem\nby Marvin Weidner (Universitat de Barcelona) as part of Non-local
operators\, probability and singularities\n\n\nAbstract\nThe Bernstein te
chnique is an elementary but powerful tool in the regularity theory for el
liptic and parabolic equations. It is based on the insight that\, if deriv
atives of a solution are also subsolutions to an equation\, then the maxim
um principle can be used in order to obtain regularity estimates for these
solutions.\nIn the first part of this talk\, we explain how the Bernstein
technique can be extended to a large class of integro-differential equati
ons driven by nonlocal operators that are comparable to the fractional Lap
lacian. In the second part\, we discuss several applications of this techn
ique to the regularity theory for the nonlocal obstacle problem in a bound
ed domain.\nThis talk is based on a joint work with Xavier Ros-Oton and Da
mià Torres-Latorre.\n
LOCATION:https://researchseminars.org/talk/NonLocalOperators/84/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Nikolay Krylov (University of Minnesota)
DTSTART;VALUE=DATE-TIME:20230523T140000Z
DTEND;VALUE=DATE-TIME:20230523T150000Z
DTSTAMP;VALUE=DATE-TIME:20240715T174606Z
UID:NonLocalOperators/85
DESCRIPTION:Title: On weak solutions of time inhomogeneous Ito's equations with VM
O diffusion and Morrey drift\nby Nikolay Krylov (University of Minneso
ta) as part of Non-local operators\, probability and singularities\n\n\nAb
stract\nWe prove the existence and weak uniqueness of weak solutions of It
o's stochastic time dependent equations with irregular diffusion and drift
terms of Morrey class with mixed norms.\n
LOCATION:https://researchseminars.org/talk/NonLocalOperators/85/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Alexei Kulik (Wrocław University of Science and Technology)
DTSTART;VALUE=DATE-TIME:20230606T130000Z
DTEND;VALUE=DATE-TIME:20230606T140000Z
DTSTAMP;VALUE=DATE-TIME:20240715T174606Z
UID:NonLocalOperators/86
DESCRIPTION:Title: Drift reduction and parametrix construction for stochastic diff
erential equations driven by cylindrical Lévy noises.\nby Alexei Kuli
k (Wrocław University of Science and Technology) as part of Non-local ope
rators\, probability and singularities\n\n\nAbstract\nIn the talk\, we wil
l present an analytic construction for the unique weak solution \nof an SD
E driven by a cylindrical Lévy noise which are spatially inhomogeneous in
the sense that different coordinates of the driving Lévy process may hav
e different scaling properties. We will discuss how the classical paramet
rix method for constructing fundamental solutions to parabolic equations
should be adapted in order to handle numerous difficulties which arise in
this non-local setting\, including essential singularity\, lack of scalin
g\, and presence of the drift (gradient) term which may be not order-domin
ated by the noise.\n
LOCATION:https://researchseminars.org/talk/NonLocalOperators/86/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Guohuan Zhao (Chinese Academy of Sciences)
DTSTART;VALUE=DATE-TIME:20230711T130000Z
DTEND;VALUE=DATE-TIME:20230711T140000Z
DTSTAMP;VALUE=DATE-TIME:20240715T174606Z
UID:NonLocalOperators/87
DESCRIPTION:Title: Lévy-type operators with low singularity kernels: regularity e
stimates and martingale problem\nby Guohuan Zhao (Chinese Academy of S
ciences) as part of Non-local operators\, probability and singularities\n\
n\nAbstract\nThe main focus of this talk is on the linear non-local operat
or $L$ defined by\n$$\nL u (x) = \\int_{\\mathbb{R}^d} (u(x+z)-u(x)) a(x\,
z)J(z)~d z.\n$$\nHere $J$ is the jumping kernel of a L\\'evy process\, whi
ch exhibits only a low-order singularity near the origin and does not perm
it standard scaling. To analyze elliptic equations associated with $L$\, I
will introduce generalized Orlicz-Besov spaces that are specifically tail
ored for this purpose. Moreover\, I will establish certain regularity prop
erties of the solutions to such equations in these spaces. Additionally\,
I intend to introduce the martingale problem associated with $L$. By explo
iting analytic results\, we demonstrate the well-posedness of the martinga
le problem under mild conditions\, and establish a new Krylov-type estimat
e for the corresponding Markov processes. This is based on joint work with
Eryan Hu from Tianjin University.\n
LOCATION:https://researchseminars.org/talk/NonLocalOperators/87/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Phan Thành Nam (Ludwig-Maximilians-Universität München)
DTSTART;VALUE=DATE-TIME:20231024T130000Z
DTEND;VALUE=DATE-TIME:20231024T140000Z
DTSTAMP;VALUE=DATE-TIME:20240715T174606Z
UID:NonLocalOperators/88
DESCRIPTION:Title: Cwikel-Lieb-Rozenblum type estimates for Schrödinger operators
with Hardy potential\nby Phan Thành Nam (Ludwig-Maximilians-Universi
tät München) as part of Non-local operators\, probability and singularit
ies\n\n\nAbstract\nThe celebrated Cwikel-Lieb-Rozenblum (CLR) inequality s
tates that the number of negative eigenvalues of the Schrödinger operator
$-\\Delta-V(x)$ in $L^2(R^d)$ is bounded from above by the integration of
|V|^{d/2}. Up to a universal constant factor\, this bound is optimal for
a wide range of fermionic systems\, from one-body systems where it is equi
valent to the standard Sobolev inequality\, to large systems where it is c
onsistent with Weyl's semiclassical approximation. I will discuss extensio
ns of the CLR bound when $V(x)$ may be as singular as the Hardy potential
$(d/2-1)^2 |x|^{-2}$. The critical singularity requires a logarithmic corr
ection which has been noticed for the one-body case but seems unknown for
the general case. The talk is based on joint work with Giao Ky Duong\, Thi
Minh Thao Le\, and Phuoc Tai Nguyen.\n
LOCATION:https://researchseminars.org/talk/NonLocalOperators/88/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Nguyen H. Lam (Memorial University)
DTSTART;VALUE=DATE-TIME:20231121T140000Z
DTEND;VALUE=DATE-TIME:20231121T150000Z
DTSTAMP;VALUE=DATE-TIME:20240715T174606Z
UID:NonLocalOperators/89
DESCRIPTION:Title: Hardy-Rellich type inequalities: A new approach and symmetrizat
ion principle\nby Nguyen H. Lam (Memorial University) as part of Non-l
ocal operators\, probability and singularities\n\n\nAbstract\nWe present a
new way to use the notion of Bessel pair to establish the optimal Hardy-R
ellich type inequalities. We also talk about necessary and sufficient cond
itions on the weights for the Hardy-Rellich inequalities to hold. Symmetry
properties of the Rellich type and Hardy-Rellich type inequalities will a
lso be discussed. The talk is based on joint work with Anh Do\, Guozhen Lu
\, and Lu Zhang.\n
LOCATION:https://researchseminars.org/talk/NonLocalOperators/89/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Francesco Russo (ENSTA Paris)
DTSTART;VALUE=DATE-TIME:20231205T140000Z
DTEND;VALUE=DATE-TIME:20231205T150000Z
DTSTAMP;VALUE=DATE-TIME:20240715T174606Z
UID:NonLocalOperators/90
DESCRIPTION:Title: Weak Dirichlet processes with jumps and applications\nby Fr
ancesco Russo (ENSTA Paris) as part of Non-local operators\, probability a
nd singularities\n\n\nAbstract\nIn this talk we will revisit the notion of
weak Dirichlet process\nwhich is the natural extension of semimartingale
with jumps.\nIf $X$ is such a process\, then it is the sum of a local mart
ingale $M$ and a\nmartingale ortogonal process $A$ in the sense\nthat $[A\
,N] = 0$ for every continuous local martingale $N$.\nWe remark that if $[A
] = 0$ then $X$ is a Dirichlet process.\nThe notion of Dirichlet process i
s not very suitable in the\njump case since in this case $A$ is forced to
be continuous.\n\nThe talk will discuss the following points.\n\n- To prov
ide a (unique) decomposition which\n is also significant for semimartinga
les with jumps.\n\n- To introduce the notion of characteristics\n (simila
rly to the case of semimartingales)\n in equivalence with some It\\^o typ
e chain rules.\n\n- To discuss various examples of such processes\n arisi
ng from path-dependent martingale problems.\n This includes path-dependen
t stochastic differential equations\n with involving a distributional dri
ft and with jumps. \n\n\nThe talk is based on a joint paper with E. Bandi
ni (Bologna).\n\nhttp://uma.ensta-paristech.fr/$\\sim$russo\n
LOCATION:https://researchseminars.org/talk/NonLocalOperators/90/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Elena Issoglio (University of Torino)
DTSTART;VALUE=DATE-TIME:20231003T130000Z
DTEND;VALUE=DATE-TIME:20231003T140000Z
DTSTAMP;VALUE=DATE-TIME:20240715T174606Z
UID:NonLocalOperators/91
DESCRIPTION:Title: McKean SDEs with singular coefficients\nby Elena Issoglio (
University of Torino) as part of Non-local operators\, probability and sin
gularities\n\n\nAbstract\nIn this talk we consider a class of SDEs with dr
ift depending on the law density of the solution\, known as McKean SDEs. T
he novelty here is that the drift is singular in the sense that it is `mul
tiplied' by a generalised function (element of a negative fractional Sobol
ev space). Those equations are interpreted in the sense of a suitable sing
ular martingale problem\, thus a key tool is the study of the correspondin
g singular Fokker-Planck equation. We define the notion of solution to the
singular McKean equation and show its existence and uniqueness. This is b
ased on a joint work with F. Russo (ENSTA).\n
LOCATION:https://researchseminars.org/talk/NonLocalOperators/91/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Zhen-Qing Chen (University of Washington)
DTSTART;VALUE=DATE-TIME:20231114T140000Z
DTEND;VALUE=DATE-TIME:20231114T150000Z
DTSTAMP;VALUE=DATE-TIME:20240715T174606Z
UID:NonLocalOperators/92
DESCRIPTION:Title: Boundary Harnack principle for non-local operators\nby Zhen
-Qing Chen (University of Washington) as part of Non-local operators\, pro
bability and singularities\n\n\nAbstract\nThe classical boundary Harnack p
rinciple asserts that two positive harmonic\nfunctions that vanish on a po
rtion of the boundary of a smooth domain decay\nat the same rate. It is we
ll known that scale invariant boundary Harnack\ninequality holds for Lapla
cian \\Delta on uniform domains and holds for\nfractional Laplacians \\Del
ta^s on any open sets. It has been an open\nproblem whether the scale-inva
riant boundary Harnack inequality holds on\nbounded Lipschitz domains for
Levy processes with Gaussian components such\nas the independent sum of a
Brownian motion and an isotropic stable process\n(which corresponds to \\D
elta + \\Delta^s).\n \nIn this talk\, I will present a necessary and suffi
cient\ncondition for the scale-invariant boundary Harnack inequality to ho
ld for a\nclass of non-local operators on metric measure spaces through a\
nprobabilistic consideration. This result will then be applied to give a\n
sufficient geometric condition for the scale-invariant boundary Harnack\ni
nequality to hold for subordinate Brownian motions having Gaussian compone
nts \non bounded Lipschitz domains in Euclidean spaces. This condition is
almost optimal and \na counterexample will be given showing that the scal
e-invariant BHP may fail\non some bounded Lipschitz domains with large Lip
schitz constants.\n\nBased on joint work with Jieming Wang.\n
LOCATION:https://researchseminars.org/talk/NonLocalOperators/92/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Franco Flandoli (Scuola Normale Superiore di Pisa)
DTSTART;VALUE=DATE-TIME:20240305T140000Z
DTEND;VALUE=DATE-TIME:20240305T150000Z
DTSTAMP;VALUE=DATE-TIME:20240715T174606Z
UID:NonLocalOperators/93
DESCRIPTION:Title: Stochasticity into fluids\nby Franco Flandoli (Scuola Norma
le Superiore di Pisa) as part of Non-local operators\, probability and sin
gularities\n\n\nAbstract\nFrom the beginning of the advent of Stochastic P
artial Differential Equations (SPDEs)\, classes of equations related to fl
uid dynamics were considered. Even earlier\, Landau and Lischitz wrote a N
avier-Stokes equation perturbed by additive noise\, in their volume on flu
id mechanics. However\, deciding the form of stochasticity realistic or in
teresting for fluid dynamics remains one of the most important questions s
till debated and stochastic analysis helps a lot to address this problem.
I will try to review part of the present understanding of this modeling is
sue and its consequences.\n
LOCATION:https://researchseminars.org/talk/NonLocalOperators/93/
END:VEVENT
BEGIN:VEVENT
SUMMARY:José Luis Pérez Garmendia (CIMAT)
DTSTART;VALUE=DATE-TIME:20231219T140000Z
DTEND;VALUE=DATE-TIME:20231219T150000Z
DTSTAMP;VALUE=DATE-TIME:20240715T174606Z
UID:NonLocalOperators/94
DESCRIPTION:Title: Multi-armed Lévy bandits with periodic decision opportunities<
/a>\nby José Luis Pérez Garmendia (CIMAT) as part of Non-local operators
\, probability and singularities\n\n\nAbstract\nWe consider a version of t
he continuous-time multi-armed bandit problem where decision\nopportunitie
s arrive at Poisson arrival times and study its Gittins index policy. When
driven by a Lévy process\, we will show that the Gittins index can be ex
pressed in terms of a Wiener-Hopf factorization of the Lévy process obser
ved at the arrival times of an independent Poisson process.\n
LOCATION:https://researchseminars.org/talk/NonLocalOperators/94/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Vassili Kolokoltsov (Moscow State University\, Higher School of Ec
onomics and the University of Warwick)
DTSTART;VALUE=DATE-TIME:20240123T140000Z
DTEND;VALUE=DATE-TIME:20240123T150000Z
DTSTAMP;VALUE=DATE-TIME:20240715T174606Z
UID:NonLocalOperators/95
DESCRIPTION:Title: Convergence rates for functional central limit theorems with st
able laws and domains of quasi-attraction\nby Vassili Kolokoltsov (Mos
cow State University\, Higher School of Economics and the University of Wa
rwick) as part of Non-local operators\, probability and singularities\n\n\
nAbstract\nThe talk will be devoted to the three new directions of researc
h:\n\n1) Rates of convergence in the functional CLT with stable limits\; 2
) Domains of quasi-attraction\n\nas distributions\, whose normalised sums
of $n$ i.i.d terms approach stable laws for large\,\n\nbut not too large $
n$ (full quantitative and qualitative description of this effect in a func
tional setting)\;\n\n3) Rates of convergence of CTRWs (continuous time\n\n
random walks) to fractional evolutions. The ideas of the talk are taken fr
om the recent author's papers\n\n(1) The Rates of Convergence for Function
al Limit Theorems with\n\nStable Subordinators and for CTRW Approximations
to\n\nFractional Evolutions. Fractal Fract. (2023)\, 7\, 335.\n\nhttps://
doi.org/10.3390/fractalfract7040335\n\n(2) Domains of Quasi Attraction: Wh
y Stable Processes Are Observed in Reality?\n\nFractal Fract. (2023)\, 7\,
752.\n\nhttps://doi.org/10.3390/fractalfract7100752\n
LOCATION:https://researchseminars.org/talk/NonLocalOperators/95/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Krzysztof Burdzy (University of Washington)
DTSTART;VALUE=DATE-TIME:20240416T150000Z
DTEND;VALUE=DATE-TIME:20240416T160000Z
DTSTAMP;VALUE=DATE-TIME:20240715T174606Z
UID:NonLocalOperators/96
DESCRIPTION:Title: Simple nonlinear PDEs inspired by billiards\nby Krzysztof B
urdzy (University of Washington) as part of Non-local operators\, probabil
ity and singularities\n\n\nAbstract\nHow many times can $n$ billiard balls
collide\nin the open $d$-dimensional space? I will provide some\nestimate
s. I will explain how the above question leads\nto a ``pinned billiard bal
ls'' model. On a large scale\,\nthe model seems to have a hydrodynamic lim
it.\nThe parameters of the conjectured limit should satisfy\nsimple nonlin
ear PDEs. While the existence and properties\nof the conjectured hydrodyna
mic limit are open questions\,\nI will provide a quite complete analysis o
f the conjectured PDEs.\n
LOCATION:https://researchseminars.org/talk/NonLocalOperators/96/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Alberto Saldaña (National Autonomous University of Mexico)
DTSTART;VALUE=DATE-TIME:20240319T140000Z
DTEND;VALUE=DATE-TIME:20240319T150000Z
DTSTAMP;VALUE=DATE-TIME:20240715T174606Z
UID:NonLocalOperators/97
DESCRIPTION:Title: The logarithmic Laplacian: a new tool in the analysis of PDEs w
ith fractional diffusions\nby Alberto Saldaña (National Autonomous Un
iversity of Mexico) as part of Non-local operators\, probability and singu
larities\n\n\nAbstract\nIn this talk\, I will give an introduction to the
logarithmic Laplacian\, which is a relatively new pseudodifferential opera
tor that has shown to be a powerful tool in the study of linear and nonlin
ear fractional PDEs. I will describe some of its main properties\, applic
ations\, and some recent regularity results.\n
LOCATION:https://researchseminars.org/talk/NonLocalOperators/97/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Nikita Simonov (Sorbonne Université)
DTSTART;VALUE=DATE-TIME:20240409T130000Z
DTEND;VALUE=DATE-TIME:20240409T140000Z
DTSTAMP;VALUE=DATE-TIME:20240715T174606Z
UID:NonLocalOperators/98
DESCRIPTION:Title: Fast diffusion equations\, tails and convergence rates\nby
Nikita Simonov (Sorbonne Université) as part of Non-local operators\, pro
bability and singularities\n\n\nAbstract\nUnderstanding the intermediate a
symptotic and computing convergence rates towards equilibria are among the
major problems in the study of parabolic equations. Convergence rates dep
end on the tail behaviour of solutions. This observation raised the follow
ing question: how can we understand the tail behaviour of solutions from t
he tail behaviour of the initial datum?\n\nIn this talk\, I will discuss t
he asymptotic behaviour of solutions to the fast diffusion equation. It is
well known that non-negative solutions behave for large times as the Bare
nblatt (or fundamental) solution\, which has an explicit expression. In th
is setting\, I will introduce the Global Harnack Principle (GHP)\, precise
global pointwise upper and lower estimates of non-negative solutions in t
erms of the Barenblatt profile. I will characterize the maximal (hence opt
imal) class of initial data such that the GHP holds by means of an integra
l tail condition. As a consequence\, I will provide rates of convergence t
owards the Barenblatt profile in entropy and in stronger norms such as the
uniform relative error.\n
LOCATION:https://researchseminars.org/talk/NonLocalOperators/98/
END:VEVENT
BEGIN:VEVENT
SUMMARY:René Schilling (Technische Universität Dresden)
DTSTART;VALUE=DATE-TIME:20240507T130000Z
DTEND;VALUE=DATE-TIME:20240507T140000Z
DTSTAMP;VALUE=DATE-TIME:20240715T174606Z
UID:NonLocalOperators/99
DESCRIPTION:Title: The Liouville Theorem for Lévy Generators (and beyond) and the
Unique Continuation Property\nby René Schilling (Technische Universi
tät Dresden) as part of Non-local operators\, probability and singulariti
es\n\n\nAbstract\nWe discuss necessary and sufficient criteria for certain
Fourier\nmultiplication operators to satisfy the Liouville property (boun
ded\nharmonic functions are a.s.\\ constant) and the local continuation\np
roperty (bounded functions\, that are harmonic and identically zero on a\n
domain\, are a.s. zero on the whole space). Since the operators generate\n
stochastic processes\, there is also a probabilistic interpretation of\nth
ese findings.\n\nThis is a joint work with David Berger (TU Dresden)\,\nEu
gene Shargorodsky (King's College\, London) and Teo Sharia (Royal\nHollowa
y\, London).\n
LOCATION:https://researchseminars.org/talk/NonLocalOperators/99/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Zimo Hao (Universität Bielefeld)
DTSTART;VALUE=DATE-TIME:20240312T140000Z
DTEND;VALUE=DATE-TIME:20240312T150000Z
DTSTAMP;VALUE=DATE-TIME:20240715T174606Z
UID:NonLocalOperators/100
DESCRIPTION:Title: SDEs with supercritical distributional drifts\nby Zimo Hao
(Universität Bielefeld) as part of Non-local operators\, probability and
singularities\n\n\nAbstract\nLet $d\\geq 2$. In this talk\, we investiga
te the following stochastic\ndifferential equation (SDE) in ${\\mathbb R}
^d$ driven by Brownian motion\n$$\n{\\rm d} X_t=b(t\,X_t){\\rm d} t+\\sqrt
{2}{\\rm d} W_t\,\n$$\nwhere $b$ belongs to the space ${\\mathbb L}_T^q \\
mathbf{H}_p^\\alpha$\nwith $\\alpha \\in [-1\, 0]$ and $p\,q\\in[2\, \\inf
ty]$\, which is a\ndistribution-valued and divergence-free vector field.\n
In the subcritical case $\\frac dp+\\frac 2q<1+\\alpha$\, we establish the
\nexistence and uniqueness of a weak solution to the integral equation:\n$
$\nX_t=X_0+\\lim_{n\\to\\infty}\\int^t_0b_n(s\,X_s){\\rm d} s+\\sqrt{2} W_
t.\n$$\nHere\, $b_n:=b*\\phi_n$ represents the mollifying approximation\,
and the\nlimit is taken in the $L^2$-sense.\nIn the critical and supercrit
ical case $1+\\alpha\\leq\\frac dp+\\frac\n2q<2+\\alpha$\, assuming the in
itial distribution has an $L^2$-density\, we\nshow the existence of weak s
olutions and associated Markov processes.\nMoreover\, under the additional
assumption that $b=b_1+b_2+\\div a$\, where\n$b_1\\in {\\mathbb L}^\\inft
y_T{\\mathbf B}^{-1}_{\\infty\,2}$\, $b_2\\in\n{\\mathbb L}^2_TL^2$\,\nand
$a$ is a bounded antisymmetric matrix-valued function\, we establish\nthe
convergence of mollifying approximation solutions without the need\nto su
btract a subsequence.\nTo illustrate our results\, we provide examples of
Gaussian random fields\nand singular interacting particle systems\, includ
ing the two-dimensional\nvortex models.\n
LOCATION:https://researchseminars.org/talk/NonLocalOperators/100/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Milica Tomasevic (École polytechnique)
DTSTART;VALUE=DATE-TIME:20240528T130000Z
DTEND;VALUE=DATE-TIME:20240528T140000Z
DTSTAMP;VALUE=DATE-TIME:20240715T174606Z
UID:NonLocalOperators/101
DESCRIPTION:Title: Particle approximation of the doubly parabolic Keller-Segel eq
uation in the plane\nby Milica Tomasevic (École polytechnique) as par
t of Non-local operators\, probability and singularities\n\n\nAbstract\nIn
this talk\, we study a stochastic system of $N$ particles associated with
the parabolic-parabolic Keller-Segel system in the plane. This particle s
ystem is singular and non Markovian in that its drift term depends on the
past of the particles. When the sensitivity parameter is sufficiently smal
l\, we show that this particle system indeed exists for any $N \\geq 2$\,
we show tightness in $N$ of its empirical measure\, and that any weak limi
t point of this empirical measure\, as $N\\to \\infty$\, solves some nonli
near martingale problem\, which in particular implies that its family of t
ime-marginals solves the parabolic-parabolic Keller-Segel system in some w
eak sense. The main argument of the proof consists of a Markovianization o
f the interaction kernel: We show that\, in some loose sense\, the two-by-
two path-dependant interaction can be controlled by a two-by-two Coulomb i
nteraction\, as in the parabolic-elliptic case. This is a joint work with
N. Fournier (Sorbonne Université).\n
LOCATION:https://researchseminars.org/talk/NonLocalOperators/101/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Huyuan Chen (Jiangxi Normal University)
DTSTART;VALUE=DATE-TIME:20240521T130000Z
DTEND;VALUE=DATE-TIME:20240521T140000Z
DTSTAMP;VALUE=DATE-TIME:20240715T174606Z
UID:NonLocalOperators/102
DESCRIPTION:Title: The Cauchy problem associated to the logarithmic Laplacian
\nby Huyuan Chen (Jiangxi Normal University) as part of Non-local operator
s\, probability and singularities\n\n\nAbstract\nIn this talk\, we study
the Cauchy problem \n$$\\partial_tu+ \\mathcal{L}_{\\Delta} u=0 \\ \\ {\\
rm in}\\ \\\, (0\,\\frac N2) \\times \\R^N\,\\quad\\quad u(0\,\\cdot)=0\
\ \\ {\\rm in}\\ \\\, \\R^N\\setminus \\{0\\}.$$\nwhere $L_\\Delta$ is th
e logarithmic Laplacian operator\, a singular integral operator with symb
ol $2\\log |\\zeta|$. We apply our results to give a classification of the
solutions of\n$$\n\\begin{cases}\n \\partial_t u+\\mathcal{L}_{\\Delta} u
=0 \\quad \\ &{\\rm in}\\ \\ (0\,T)\\times \\R^N \\\\\n \\phantom{ \\
\\\, }\n\\displaystyle u(0\,\\cdot)=f\\quad \\ &{\\rm{in}}\\ \\ \\R
^N\n\\end{cases}\n$$\nand obtain an expression of the fundamental solutio
n of the associated stationary equation in $\\R^N$\,\nand of the fundament
al solution in a bounded domain\, i.e.\n$$\\mathcal{L}_{\\Delta} u=k\\del
ta_0\\quad {\\rm in}\\ \\ \\mathcal{D}'(\\Omega)\\quad {\\rm such\\ that
\n }\\quad u=0\\quad {\\rm in}\\ \\ \\R^N\\setminus\\Omega. $$\n\nThis is
a joint work with Laurent Véron.\n
LOCATION:https://researchseminars.org/talk/NonLocalOperators/102/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Ivan Biočić (University of Zagreb)
DTSTART;VALUE=DATE-TIME:20240618T130000Z
DTEND;VALUE=DATE-TIME:20240618T140000Z
DTSTAMP;VALUE=DATE-TIME:20240715T174606Z
UID:NonLocalOperators/103
DESCRIPTION:Title: Semilinear equations for subordinate spectral Laplacian: moder
ate and large solutions\nby Ivan Biočić (University of Zagreb) as pa
rt of Non-local operators\, probability and singularities\n\n\nAbstract\nI
n this talk\, we solve semilinear problems in bounded $C^{1\,1}$ domains f
or non-local operators with a non-homogeneous Dirichlet boundary condition
\, based on the work [1] and a joint work in progress with Vanja Wagner. T
he operators cover and extend the case of the spectral fractional Laplacia
n\, and are modelled using the process called subordinate killed Brownian
motion. Our focus will be on the potential-probabilistic approach to these
problems with an emphasis on methods\, intuition\, and calculations. This
approach is a consequence of recent developments in [2\,3].\n\nWe present
an integral representation of harmonic functions for such non-local opera
tors and give sharp boundary behaviour of Green and Poisson potentials. H\
\"older regularity of distributional solutions is given as well as a versi
on of Kato's inequality. We explore moderate (i.e. harmonically bounded) s
olutions and large (i.e. harmonically unbounded) solutions to the semiline
ar problem. Large solutions are obtained by using a Keller-Osserman-type c
ondition\, by an approximation method.\n\n\n[1] I. Biočić\, Semilinear D
irichlet problem for subordinate spectral Laplacian\, Communications on Pu
re and Applied Analysis\, 22 (2023)\, 851-898.\n\n[2] I. Biočić\, Z. Vo
ndraček\, V. Wagner\, Semilinear equations for non-local operators: Beyon
d the fractional Laplacian\, Nonlinear Analysis\, 207 (2021)\, 112303.\n\n
[3] P. Kim\, R. Song\, Z. Vondraček\, Potential theory of subordinate kil
led Brownian motion\, Transactions of the American mathematical society\,
371 (2019)\, 3917-3969.\n
LOCATION:https://researchseminars.org/talk/NonLocalOperators/103/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Aleksandar Mijatović (University of Warwick)
DTSTART;VALUE=DATE-TIME:20240702T130000Z
DTEND;VALUE=DATE-TIME:20240702T140000Z
DTSTAMP;VALUE=DATE-TIME:20240715T174606Z
UID:NonLocalOperators/105
DESCRIPTION:Title: Subexponential lower bounds for $f$-ergodic Markov processes\nby Aleksandar Mijatović (University of Warwick) as part of Non-local
operators\, probability and singularities\n\n\nAbstract\nIn this talk I wi
ll describe a criterion for establishing lower bounds on the rate of conv
ergence in $f$-variation of a continuous-time ergodic Markov process to it
s invariant measure. The criterion consists of novel super- and submarting
ale conditions for certain functionals of the Markov process. It provides
a general approach for proving lower bounds on the tails of the invariant
measure and the rate of convergence in $f$-variation of a Markov process\,
analogous to the widely used Lyapunov drift conditions for upper bounds.
Our key innovation\, which will be discussed in the talk\, produces lower
bounds on the tails of the heights and durations of the excursions from b
ounded sets of a continuous-time Markov process using path-wise arguments.
\n\nI will present applications of our theory to elliptic diffusions and
Levy-driven stochastic differential equations with known polynomial/stre
tched exponential upper bounds on their rates of convergence. Our lower bo
unds match asymptotically the known upper bounds for these classes of mode
ls\, thus establishing their rate of convergence to stationarity. The gene
rality of our approach suggests that\, analogous to the Lyapunov drift con
ditions for upper bounds\, our methods can be expected to find applicatio
ns in many other settings. This is joint work with Miha Brešar at Warwick
. The paper is available on https://arxiv.org/abs/2403.14826\n
LOCATION:https://researchseminars.org/talk/NonLocalOperators/105/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Sergio Andraus (Tsukuba Gakuin University)
DTSTART;VALUE=DATE-TIME:20240924T130000Z
DTEND;VALUE=DATE-TIME:20240924T140000Z
DTSTAMP;VALUE=DATE-TIME:20240715T174606Z
UID:NonLocalOperators/106
DESCRIPTION:by Sergio Andraus (Tsukuba Gakuin University) as part of Non-l
ocal operators\, probability and singularities\n\nInteractive livestream:
https://ulaval.zoom.us/j/65411768971?pwd=MzBmb0VIcTd6OUZkNEJzc2tPRUxwQT09\
nAbstract: TBA\n
LOCATION:https://researchseminars.org/talk/NonLocalOperators/106/
URL:https://ulaval.zoom.us/j/65411768971?pwd=MzBmb0VIcTd6OUZkNEJzc2tPRUxwQ
T09
END:VEVENT
END:VCALENDAR