An advection-diffusion process with proportional resetting

TL;DR

This paper introduces a new stochastic diffusion model with a proportional resetting mechanism, providing exact solutions for its distribution and moments, useful for systems experiencing sudden proportional reductions.

Contribution

It develops a novel advection-diffusion process with proportional resetting, described by a pantograph equation, and derives exact statistical properties.

Findings

Distribution interpolates between exponential and Gaussian forms.

Exact steady-state distribution and moments are obtained.

Model applicable to systems with proportional random reductions.

Abstract

This paper presents a diffusion process with a novel resetting mechanism in which the amplitude of the process is instantaneously converted to a proportion of its value at random times. This model is described by a Langevin equation with both additive Gaussian white noise and multiplicative Poisson shot noise terms. The distribution function obeys a pantograph equation, a functional partial differential equation evaluated at two amplitudes simultaneously. From this equation the exact statistical moments and steady-state distribution of the process are calculated. The distribution interpolates between exponential and Gaussian extremes depending on the proportion of the amplitude lost in each reset. These results will be useful for applications in which stochastic quantities are suddenly reduced in proportion to their values due to random events.