Communications in Mathematical Sciences

Volume 17 (2019)

Number 4

Long time behavior in locally activated random walks

Pages: 1071 – 1094



Nicolas Meunier (Laboratoire de Mathématiques et Modélisation d’Évry (LaMME), Université Évry Val d’Essonne, Évry, France)

Clément Mouhot (Department of Pure Mathematics and Mathematical Statistics (DPMMS), Centre for Mathematical Sciences, University of Cambridge, United Kingdom)

Raphaël Roux (Laboratoire de probabilités, statistique et modélisation (LPSM), Sorbonne Université, Paris, France)


We consider a $1$-dimensional Brownian motion whose diffusion coefficient varies when it crosses the origin. We study the long time behavior and we establish different regimes, depending on the variations of the diffusion coefficient: emergence of a non-Gaussian multipeaked probability distribution and a dynamical transition to an absorbing static state. We compute the generator and we study the partial differential equation which involves its adjoint. We discuss global existence and blow-up of the solution to this latter equation.


local time, random walk, dynamical transition, non-Gaussian probability distribution, blow-up

2010 Mathematics Subject Classification

35Q92, 60J65

C. Mouhot was partially supported by ERC grant MAFRAN.

Received 27 June 2017

Accepted 27 April 2019

Published 25 October 2019