First-passage time statistics for non-linear diffusion

TL;DR

This paper investigates the first-passage time statistics for non-linear diffusion processes with power-law dependent diffusivity, providing exact and approximate solutions for survival probabilities and mean first-passage times in different potential scenarios.

Contribution

It introduces analysis of non-linear diffusion equations for first-passage times, filling a gap in understanding non-linear diffusive processes compared to linear models.

Findings

Exact and approximate expressions for survival probability and first-passage time distribution.

Divergent mean first-passage time for free diffusion, finite in harmonic potential.

Derived exact formula for mean first-passage time in harmonic potential.

Abstract

Evaluating the completion time of a random algorithm or a running stochastic process is a valuable tip not only from a purely theoretical, but also pragmatic point of view. In the formal sense, this kind of a task is specified in terms of the first-passage time statistics. Although first-passage properties of diffusive processes, usually modeled by different types of the linear differential equations, are permanently explored with unflagging intensity, there still exists noticeable niche in this subject concerning the study of the non-linear diffusive processes. Therefore, the objective of the present paper is to fill this gap, at least to some extent. Here, we consider the non-linear diffusion equation in which a diffusivity is power-law dependent on the concentration/probability density, and analyse its properties from the viewpoint of the first-passage time statistics. Depending on the value of the power-law exponent, we demonstrate the exact and approximate expressions for the survival probability and the first-passage time distribution along with its asymptotic representation. These results refer to the freely and harmonically trapped diffusing particle. While in the former case the mean first-passage time is divergent, even though the first-passage time distribution is normalized to unity, it is finite in the latter. To support this result, we derive the exact formula for the mean first-passage time to the target prescribed in the minimum of the harmonic potential.