First-passage distributions for the one-dimensional Fokker-Planck equation

TL;DR

This paper develops an analytical framework for understanding first-passage and first-return distributions in one-dimensional Fokker-Planck models, revealing how absorbing states influence these distributions and connecting them to known models like the voter model.

Contribution

It introduces a general theory for first-passage distributions in Fokker-Planck models with absorbing states, including new exponents and their dependence on model parameters.

Findings

Power-law decay with exponent -3/2 for Brownian motion in finite domains.

Near absorbing states, a different exponent, often -2, emerges.

The theory is validated with exact solutions and numerical simulations.

Abstract

We present an analytical framework to study the first-passage (FP) and first-return (FR) distributions for the broad family of models described by the one-dimensional Fokker-Planck equation in finite domains, identifying general properties of these distributions for different classes of models. When in the Fokker-Planck equation the diffusion coefficient is positive (nonzero) and the drift term is bounded, as in the case of a Brownian walker, both distributions may exhibit a power-law decay with exponent -3/2 for intermediate times. We discuss how the influence of an absorbing state changes this exponent. The absorbing state is characterized by a vanishing diffusion coefficient and/or a diverging drift term. Remarkably, the exponent of the Brownian walker class of models is still found, as long as the departure and arrival regions are far enough from the absorbing state, but the range of times where the power law is observed narrows. Close enough to the absorbing point, though, a new exponent may appear. The particular value of the exponent depends on the behavior of the diffusion and the drift terms of the Fokker-Planck equation. We focus on the case of a diffusion term vanishing linearly at the absorbing point. In this case, the FP and FR distributions are similar to those of the voter model, characterized by a power law with exponent -2. As an illustration of the general theory, we compare it with exact analytical solutions and extensive numerical simulations of a two-parameter voter-like family models. We study the behavior of the FP and FR distributions by tuning the importance of the absorbing points throughout changes of the parameters. Finally, the possibility of inferring relevant information about the steady-sate probability distribution of a model from the FP and FR distributions is addressed.