Random search with resetting in heterogeneous environments

TL;DR

This paper provides analytical solutions for the mean first-passage time of a stochastic search with resetting in a heterogeneous 1D environment, revealing how space-dependent diffusivity influences search efficiency.

Contribution

It derives exact and asymptotic expressions for search times in heterogeneous environments under various stochastic interpretations, advancing understanding of resetting in complex media.

Findings

Exact closed-form for Stratonovich case

Asymptotic approximations for small and large reset rates

Analysis of heterogeneity effects on search efficiency

Abstract

We investigate random searches under stochastic position resetting at rate $r$, in a bounded 1D environment with space-dependent diffusivity $D(x)$. For arbitrary shapes of $D(x)$ and prescriptions of the associated multiplicative stochastic process, we obtain analytical expressions for the average time $T$ for reaching the target (mean first-passage time), given the initial and reset positions, in good agreement with stochastic simulations. For arbitrary $D(x)$, we obtain an exact closed-form expression for $T$, within Stratonovich scenario, while for other prescriptions, like It\^o and anti-It\^o, we derive asymptotic approximations for small and large rates $r$. Exact results are also obtained for particular forms of $D(x)$, such as the linear one, with arbitrary prescriptions, allowing to outline and discuss the main effects introduced by diffusive heterogeneity on a random search with resetting. We explore how the effectiveness of resetting varies with different types of heterogeneity.