From persistent random walks to the telegraph noise

Samuel Herrmann 1, 2 Pierre Vallois 1, 3
2 TOSCA
INRIA Lorraine, CRISAM - Inria Sophia Antipolis - Méditerranée , UHP - Université Henri Poincaré - Nancy 1, Université Nancy 2, INPL - Institut National Polytechnique de Lorraine, CNRS - Centre National de la Recherche Scientifique : UMR7502
3 BIGS - Biology, genetics and statistics
IECN - Institut Élie Cartan de Nancy, INRIA Lorraine
Abstract : We study a family of memory-based persistent random walks and we prove weak convergences after space-time rescaling. The limit processes are not only Brownian motions with drift. We have obtained a continuous but non-Markov process $(Z_t)$ which can be easely expressed in terms of a counting process $(N_t)$. In a particular case the counting process is a Poisson process, and $(Z_t)$ permits to represent the solution of the telegraph equation. We study in detail the Markov process $((Z_t,N_t); \ t\ge 0)$.
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  • HAL Id : hal-00326521, version 1
  • ARXIV : 0810.0650

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Samuel Herrmann, Pierre Vallois. From persistent random walks to the telegraph noise. Stochastics and Dynamics, World Scientific Publishing, 2010, 10 (2), pp.161-196. ⟨hal-00326521⟩

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