Communications in Mathematical Sciences
Volume 17 (2019)
Long time behavior in locally activated random walks
Pages: 1071 – 1094
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
C. Mouhot was partially supported by ERC grant MAFRAN.
Received 27 June 2017
Accepted 27 April 2019
Published 25 October 2019