Efficiency of random search with space-dependent diffusivity

TL;DR

This paper investigates how space-dependent diffusion affects random search efficiency, deriving new formulas under various stochastic interpretations and revealing that heterogeneity generally reduces efficiency, with implications for optimizing search strategies.

Contribution

It provides the first explicit expressions for search efficiency in environments with space-dependent diffusivity, highlighting the influence of stochastic interpretation on results.

Findings

Stratonovich interpretation yields a closed-form efficiency expression.

Heterogeneous diffusivity profiles decrease search efficiency.

Efficiency depends on diffusivity distribution, not spatial arrangement.

Abstract

We address the problem of random search for a target in an environment with space-dependent diffusion coefficient $D(x)$. From a general form of the diffusion differential operator that includes It\^o, Stratonovich, and H\"anggi-Klimontovich interpretations of the associated stochastic process, we obtain the first-passage time distribution and the search efficiency $\mathcal{E}=\langle 1/t \rangle$. For the paradigmatic power-law diffusion coefficient $D(x) = D_0|x|^{\alpha}$, with $\alpha<2$, which controls whether the mobility increases or decreases with the distance from a target at the origin, we show the impact of the different interpretations. For the Stratonovich framework, we obtain a closed expression of the search efficiency, valid for arbitrary diffusion coefficient $D(x)$. We show that a heterogeneous diffusivity profile leads to lower efficiency than the homogeneous average level, and the efficiency depends only on the distribution of diffusivity values and not on its spatial organization, features that breakdown under other interpretations.