Stochastic search with space-dependent diffusivity

TL;DR

This paper explores how stochastic search efficiency and behavior are affected by space-dependent diffusivity and the interpretation of multiplicative noise, providing general formulas and simulations for various conditions.

Contribution

It develops a comprehensive theoretical framework for stochastic search with space-dependent diffusivity, including formulas for search time distributions and moments under different interpretations.

Findings

Search time distribution depends strongly on noise interpretation.

General formulas derived for heterogeneous diffusion in arbitrary domains.

Simulations confirm theoretical predictions and reveal counterintuitive effects.

Abstract

The canonical model of stochastic search tracks a randomly diffusing "searcher" until it finds a "target." Owing to its many applications across science and engineering, this perennially popular problem has been thoroughly investigated in a variety of models. However, aside from some exactly solvable one-dimensional examples, very little is known if the searcher diffusivity varies in space. For such space-dependent or "heterogeneous" diffusion, one must specify the interpretation of the multiplicative noise, which is termed the It\^{o}-Stratonovich dilemma. In this paper, we investigate how stochastic search with space-dependent diffusivity depends on this interpretation. We obtain general formulas for the probability distribution and all the moments of the stochastic search time and the so-called splitting probabilities assuming that the targets are small or weakly reactive. These asymptotic results are valid for general space-dependent diffusivities in general domains in any space dimension with targets of general shape which may be in the interior or on the boundary of the domain. We illustrate our theory with stochastic simulations. Our analysis predicts that stochastic search can depend strongly and counterintuitively on the multiplicative noise interpretation.