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SUMMARY:Carsten Wiuf (University of Copenhagen)
DTSTART:20260423T153000Z
DTEND:20260423T160000Z
DTSTAMP:20260421T123619Z
UID:MoRN/145
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/MoRN/145/">Q
 SDs for one-species reaction networks</a>\nby Carsten Wiuf (University of 
 Copenhagen) as part of Seminar on the Mathematics of Reaction Networks\n\n
 \nAbstract\nThis talk is concerned with quasi-stationary distributions (QS
 Ds) on $\\mathbb{N}_0$ for one-species stochastic reaction networks concei
 ved as continuous time Markov chains (CTMCs).  A QSD describes the long ti
 me behaviour of the CTMC before absorption. \n Examples  arise  in ecology
 \, epidemiology\, cellular biology and chemistry.  \n\n Let  $(X_t)_{t\\ge
  0}$\, $X_t\\in \\mathbb{N}_0$\, describe the species count of a one-speci
 es reaction network that eventually is absorbed into a trapping set $A\\su
 bseteq\\mathbb{N}_0$. For example\, $S\\to 2S$ and $S\\to 0$\, here $A=\\{
 0\\}$. A  probability distribution $\\nu$   with support on  $A^c=\\mathbb
 {N}_0\\!\\setminus\\! A$ is a QSD\, if  the following holds:\n\n$\\bullet$
  If the initial count $X_0$  is chosen from $\\nu$\, then $X_t$\, $t>0$\, 
 remains distributed as  $\\nu$\, provided the chain is not absorbed.\n\n\n
 Formulated in math terms:\n$$\\mathbb{P}_\\nu(X_t\\in B\\\, |\\\, t<\\tau_
 A)=\\nu(B)\,\\quad B\\subseteq A^c\,\\quad\\text{for all}\\quad t\\ge 0\,$
 $\n where $ \\tau_A=\\inf\\{t\\ge0 \\colon X_t\\in A\\}$ is the time until
  absorption\, and $\\mathbb{P}_{\\nu}$ is the distribution of the process 
 before absorption  with initial distribution $\\nu$. \n\nWe start by looki
 ng at some examples to shape the intuition. Then\, we characterise all QSD
 s $\\nu$   in terms of a finite number of generating terms $\\nu(J)=(\\nu(
 i_1)\,\\ldots\,\\nu(i_d))$\, $J=\\{i_1\,\\ldots\,i_d\\}$ and a correspondi
 ng eigenvalue $\\theta\\in\\R$. This is to say\, $\\nu(n)=\\sum_{j\\in J}R
 _j(n\,\\theta)\\nu(j)$\, where  \nthe coefficients $R_j(n\,\\theta)$ are g
 iven recursively in $n$. Based on this and relying on Perron-Frobenius the
 ory for infinite matrices\, we prove existence of an extremal QSD for King
 man's parameter $\\theta_K>0$\, provided the CTMC is ultimately absorbed. 
 Furthermore\, we show the existence of a series of polynomials  of increas
 ing degree\, $ \\rho_n(\\theta)$\, such that $\\theta(n)\\downarrow\\theta
 _K$\, where $\\theta(n)$ is the smallest real root  of $ \\rho_n(\\theta)$
 .   These results mimic results for birth-death processes  (seminal work b
 y Karlin and McGregor). \n\nWe further discuss numerial means to find  the
  generator  and to determine Kingman's parameter.  The results are illustr
 ated with numerical examples  from stochastic reaction network theory.\n\n
 This is joint work with Chuang Xu (University of Hawaii) and Mads Chr Hans
 en (formerly at University of Copenhagen).\n
LOCATION:https://researchseminars.org/talk/MoRN/145/
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