Stochastic dynamics with multiplicative dichotomic noise: heterogeneous telegrapher's equation, anomalous crossovers and resetting

TL;DR

This paper explores complex diffusion behaviors in heterogeneous media using a generalized telegrapher's equation, revealing various anomalous regimes, crossover dynamics, and the effects of stochastic resetting leading to non-equilibrium steady states.

Contribution

It introduces a heterogeneous telegrapher's equation framework capturing diverse diffusion regimes and analyzes the impact of stochastic resetting on system dynamics.

Findings

Identification of multiple diffusion regimes including hyperdiffusion and subdiffusion.

Demonstration of anomalous diffusion exponents and crossover behaviors.

Establishment of non-equilibrium stationary states under stochastic resetting.

Abstract

We analyze diffusion processes with finite propagation speed in a non-homogeneous medium in terms of the heterogeneous telegrapher's equation. In the diffusion limit of infinite-velocity propagation we recover the results for the heterogeneous diffusion process. The heterogeneous telegrapher's process exhibits a rich variety of diffusion regimes including hyperdiffusion, ballistic motion, superdiffusion, normal diffusion and subdiffusion, and different crossover dynamics characteristic for complex systems in which anomalous diffusion is observed. The anomalous diffusion exponent in the short time limit is twice the exponent in the long time limit, in accordance to the crossover dynamics from ballistic diffusion to normal diffusion in the standard telegrapher's process. We also analyze the finite-velocity heterogeneous diffusion process in presence of stochastic Poissonian resetting. We show that the system reaches a non-equilibrium stationary state. The transition to this non-equilibrium steady state is analysed in terms of the large deviation function.