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Extinction time for a random walk in a random environment

Research paper by Anna De Masi, Errico Presutti, Dimitrios Tsagkarogiannis, Maria Eulalia Vares

Indexed on: 28 Jul '15Published on: 28 Jul '15Published in: Mathematics - Probability



Abstract

We consider a random walk with death in $[-N,N]$ moving in a time dependent environment. The environment is a system of particles which describes a current flux from $N$ to $-N$. Its evolution is influenced by the presence of the random walk and in turn it affects the jump rates of the random walk in a neighborhood of the endpoints, determining also the rate for the random walk to die. We prove an upper bound (uniform in $N$) for the survival probability up to time $t$ which goes as $c\exp\{-bN^{-2}t\}$, with $c$ and $b$ positive constants.