Stochastic multiplicative processes with reset events

TL;DR

This paper investigates a stochastic multiplicative process with resets, revealing stationary power-law distributions, different regimes, and the effects of diffusion, supported by both numerical and analytical analysis.

Contribution

It introduces a detailed analysis of a stochastic process with resets, highlighting the impact of diffusion and identifying regimes with distinct behaviors.

Findings

Stationary power-law distribution with variable exponent

Two regimes: intermittent and regular behavior

Exponent of -2 at the boundary between regimes

Abstract

We study a stochastic multiplicative process with reset events. It is shown that the model develops a stationary power-law probability distribution for the relevant variable, whose exponent depends on the model parameters. Two qualitatively different regimes are observed, corresponding to intermittent and regular behaviour. In the boundary between them, the mean value of the relevant variable is time-independent, and the exponent of the stationary distribution equals -2. The addition of diffusion to the system modifies in a non-trivial way the profile of the stationary distribution. Numerical and analytical results are presented.