Stochastic search with space-dependent diffusivity

TL;DR

This work addresses stochastic search with a space-dependent diffusivity $D(x)$ and analyzes how the interpretation of multiplicative noise, encoded by $α∈[0,1]$, shapes first passage times and associated statistics. By combining forward/backward Fokker-Planck theory with spectral and perturbative methods, it derives general asymptotics for the full FPT distribution, all moments, mean residence times, and splitting probabilities for small or weakly reactive targets in general domains and dimensions, with validations via stochastic simulations. A key finding is the strong, sometimes counterintuitive dependence on the interpretation parameter: different $α$ can markedly alter MFPTs and the likelihood of discovering particular targets, even when $D(x)$ is fixed. The results extend the narrow-escape framework to heterogeneous media and provide practical insight into how spatial variations in diffusivity influence search efficiency in biological, chemical, and engineering contexts. The work connects to prior studies on space-dependent diffusion and demonstrates robust analytical predictions across 3d and 2d domains with both interior and boundary targets.

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 Itô-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.

Figures (5)

• Figure 1: A searcher diffuses (black path) with a space-dependent diffusivity (blue-green gradient) inside a general $d$-dimensional domain with small targets (red regions) on its boundary and its interior.
• Figure 2: Perfect targets in 3d for the Itô ($\alpha=0$), Stratonovich ($\alpha=1/2$), and kinetic interpretations ($\alpha=1$). The markers are computed from stochastic simulations, and the curves are the formulas in \ref{['eq:mfptperfect3d']} and \ref{['eq:splitperfect3d']}.
• Figure 3: Perfect targets in 2d for the Itô ($\alpha=0$), Stratonovich ($\alpha=1/2$), and kinetic interpretations ($\alpha=1$). The markers are computed from stochastic simulations, and the curves are the formulas in \ref{['eq:mfptperfect2d']} and \ref{['eq:splitperfect2d']}.
• Figure 4: Relative mean FPT error for perfect targets in 2d for the Itô ($\alpha=0$), Stratonovich ($\alpha=1/2$), and kinetic interpretations ($\alpha=1$)
• Figure 5: Imperfect targets in 2d. The markers are computed from stochastic simulations, and the curves are the formulas in \ref{['eq:mfptkappanarrow']} and \ref{['eq:splitkappanarrow']}.