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Session S06 - Interacting Stochastic Systems

Tuesday, July 13, 13:25 ~ 14:00 UTC-3

From generalized Ray-Knight theorems to functional limit theorems for some models of self-interacting random walks on $\mathbb{Z}$

Elena Kosygina

Baruch College and the CUNY Graduate Center, United States   -   This email address is being protected from spambots. You need JavaScript enabled to view it.

Generalized Ray-Knight theorems for local times proved to be a very useful tool for studying the long time behavior of several models of self-interacting random walks on $\mathbb{Z}$. Examples include some classes of reinforced random walks, excited random walks, rotor walks with defects. An excited random walk on $\mathbb{Z}$ is a nearest-neighbor random walk whose probability $\omega_x(i)$ to jump to the right from site $x$ at time $n$ depends on $x$ and on the number of visits, $i$, to $x$ up to time $n$. The collection $\omega_x(i)$, where $x$ is an integer and $i$ is a positive integer, is sometimes called the ``cookie environment'' due to the following informal interpretation. On each site of $\mathbb{Z}$ there is an infinite stack of "cookies". Upon each visit to a site the walker eats a cookie from the bottom of the stack at that site and chooses the probability to jump to the right according to the ``flavor'' of the cookie eaten. We assume that the cookie stacks are i.i.d. and that the cookie ``flavors'' at each stack, $\omega_x(i),\, i\in\mathbb{N}$, follow a finite state Markov chain in $i$. Thus, the environment at each site is dynamic: it evolves according to the local time of the walk at each site rather than according to the random walk time. This talk will give an overview of some of the models of self-interacting random walks and then will discuss functional limit theorems for excited random walks with Markovian cookie stacks in the recurrent regime.

Joint work with Thomas Mountford (École polytechnique fédérale de Lausanne, Switzerland) and Jonathon Peterson (Purdue University, USA).

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