Modul: MAT971 Stochastische Prozesse

## The Monkey walk: a Markov process with random reinforced relocations

Dr. Cécile Mailler talk

Speaker invited by: Prof. Dr. Jean Bertoin

**Date:** 17.10.18 **Time:** 17.15 - 18.15 **Room:** ETH HG G 43

In this joint work with Gerónimo Uribe-Bravo, we prove and extend results from the physics literature about a random walk with random reinforced relocations. The ``walker’’ evolves in $\mathbb Z^d$ or $\mathbb R^d$ according to a Markov process, except at some random jump-times, where it chooses a time uniformly at random in its past, and instatnly jumps to the position it was at that random time. This walk is by definition non-Markovian, since the walker needs to remember all it’s past.

We prove that, under moment conditions on the inter-jump-times, and provided that the underlying Markov process verifies a distributional limit theorem, we show a distributional limit theorem for the position of the walker at large time. The proof relies on exploiting the branching structure of this random walk with random relocations; we are able to extend the model further by allowing the memory of the walker to decay with time.