Long time behavior in locally activated random walks

TL;DR

This paper investigates the long-term behavior of a 1D Brownian motion with a position-dependent diffusion coefficient, revealing regimes with non-Gaussian distributions and phase transitions, through PDE analysis.

Contribution

It introduces a detailed analysis of how varying diffusion at the origin affects long-term dynamics, including regimes and phase transitions, using PDE methods.

Findings

Emergence of non-Gaussian, multi-peaked distributions

Identification of a dynamical transition to an absorbing state

Analysis of PDE solutions showing global existence and blow-up

Abstract

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 mul-tipeaked 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.