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# Stochastic Deformations of Sample Paths of Random Walks and Exclusion Models

Abstract : This study in centered on models accounting for stochastic deformations of sample paths of random walks, embedded either in ^$\mathbb{Z}^2$ or in $\mathbb{Z}^3$. These models are immersed in multi-type particle systems with exclusion. Starting from examples, we give necessary and sufficient conditions for the underlying Markov processes to be reversible, in which case their invariant measure has a Gibbs form. Letting the size of the sample path increase, we find the convenient scalings bringing to light phase transition phenomena. Stable and metastable configurations are bound to time-periods of limiting deterministic trajectories which are solution of nonlinear differential systems: in the example of the ABC model, a system of Lotka-Volterra class is obtained, and the periods involve elliptic, hyper-elliptic or more general functions. Lastly, we discuss briefly the contour of a general approach allowing to tackle the transient regime via differential equations of Burgers' type.
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https://hal.inria.fr/inria-00071476
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Submitted on : Tuesday, May 23, 2006 - 5:40:47 PM
Last modification on : Thursday, February 3, 2022 - 11:18:13 AM
Long-term archiving on: : Sunday, April 4, 2010 - 10:17:52 PM

### Identifiers

• HAL Id : inria-00071476, version 1

### Citation

Guy Fayolle, Cyril Furtlehner. Stochastic Deformations of Sample Paths of Random Walks and Exclusion Models. [Research Report] RR-5106, INRIA. 2004. ⟨inria-00071476⟩

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