Diffusion with Partial Resetting

TL;DR

This paper introduces a generalized diffusion model with partial resetting, where a particle is returned only partway to its origin, revealing a transition in steady-state distributions from Laplace to Gaussian and providing a detailed probabilistic analysis.

Contribution

The study extends diffusion with resetting to partial resets, deriving the steady-state as an infinite sum of Laplace variables and analyzing the transition to Gaussian distribution.

Findings

Steady-state distribution transitions from Laplace to Gaussian as resetting varies.

Explicit expression for the time-dependent distribution in Fourier-Laplace space.

Extension of the model to drift-diffusion with partial resetting.

Abstract

Inspired by many examples in nature, stochastic resetting of random processes has been studied extensively in the past decade. In particular, various models of stochastic particle motion were considered where upon resetting the particle is returned to its initial position. Here we generalize the model of diffusion with resetting to account for situations where a particle is returned only a fraction of its distance to the origin, e.g., half way. We show that this model always attains a steady-state distribution which can be written as an infinite sum of independent, but not identical, Laplace random variables. As a result, we find that the steady-state transitions from the known Laplace form which is obtained in the limit of full resetting to a Gaussian form which is obtained close to the limit of no resetting. A similar transition is shown to be displayed by drift-diffusion whose steady-state can also be expressed as an infinite sum of independent random variables. Finally, we extend our analysis to capture the temporal evolution of drift-diffusion with partial resetting, providing a bottom-up probabilistic construction that yields a closed form solution for the time dependent distribution of this process in Fourier-Laplace space. Possible extensions and applications of diffusion with partial resetting are discussed.