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SUMMARY:Zach Selk (Queen’s University)
DTSTART:20230928T161500Z
DTEND:20230928T173000Z
DTSTAMP:20260423T022922Z
UID:NEDNT/60
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/NEDNT/60/">S
 tochastic Calculus for the Theta Process</a>\nby Zach Selk (Queen’s Univ
 ersity) as part of New England Dynamics and Number Theory Seminar\n\nLectu
 re held in Online.\n\nAbstract\nThe Theta process\, $X(t)$\, is a complex 
 valued stochastic process of number theoretical origin arising as a scalin
 g limit of quadratic Weyl sums $$\\sum_{n=1}^N e^{2\\pi i \\left(\\frac{1}
 {2}(n^2+\\beta)x+\\alpha n\\right)}\,$$ where $(\\alpha\,\\beta)\\in \\mat
 hbb R^2 \\setminus \\mathbb Q^2$ and $x\\in \\mathbb R$ is chosen at rando
 m according to any law absolutely continuous with respect to Lebesgue meas
 ure. The Theta process can be explicitly represented as $X(t)=\\sqrt{t} \\
 Theta(\\Gamma g \\Phi^{2 \\log t})$ where $\\Theta$ is an automorphic func
 tion defined on Lie group $G$\, invariant under left multiplication under 
 lattice $\\Gamma$. Additionally\, $g\\in \\Gamma \\setminus G$ is chosen H
 aar uniformly at random and $\\Phi$ is the geodesic flow on $\\Gamma \\set
 minus G$. The Theta process shares several similar properties with the Bro
 wnian motion. In particular\, both lack differentiability and have the sam
 e $p$ variation and H\\”older properties.\nSimilarly to Brownian motion\
 , standard calculus and even Young/Riemann-Stieltjes calculus techniques d
 o not work. However\, Brownian motion is what is known as a martingale all
 owing for a classical theory of It\\^o calculus which makes use of cancell
 ations “on average”. The It\\^o calculus can be used to prove several 
 properties of Brownian motion such as its conformal invariance\, bounds on
  its running maximum in terms of its quadratic variation\, absolutely cont
 inuous changes in measure and much more. \nUnfortunately\, we show that th
 e Theta process $X$ is not a (semi)martingale\, therefore It\\^o technique
 s don’t work. However\, a new theory introduced in 1998 by Terry Lyons c
 alled rough paths theory handles processes with the same analytic regulari
 ty as $X$. The key idea in rough paths theory is that constructing stochas
 tic calculus for a signal can be reduced to constructing the “iterated i
 ntegrals” of the signal. In this talk\, we will show the construction of
  the iterated integrals – the “rough path” – above the process $X$
 . Joint with Francesco Cellarosi.\n
LOCATION:https://researchseminars.org/talk/NEDNT/60/
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