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Stochastic search with space-dependent diffusivity
Hwai-Ray Tung1, Sean D Lawley1
1University of Utah, Department of Mathematics, Salt Lake City, Utah 84112, USA.
Physical Review. E
|July 24, 2026
Summary
Stochastic search with varying diffusivity depends on the interpretation of multiplicative noise. This study provides general formulas for search time and splitting probabilities, revealing counterintuitive dependencies.
Area of Science:
- Physics
- Applied Mathematics
- Physical Chemistry
Background:
- The canonical stochastic search problem models a searcher finding a target.
- Heterogeneous diffusion, where searcher diffusivity varies in space, is poorly understood.
- The Itô-Stratonovich dilemma arises from interpreting multiplicative noise in heterogeneous diffusion.
Purpose of the Study:
- To investigate the impact of multiplicative noise interpretation on stochastic search with space-dependent diffusivity.
- To derive general formulas for search time and splitting probabilities in heterogeneous diffusion scenarios.
- To analyze the dependence of stochastic search on noise interpretation in various dimensions and domain geometries.
Main Methods:
- Derivation of general asymptotic formulas for search time and splitting probabilities.
- Analysis of stochastic search with space-dependent diffusivity.
- Validation of theoretical results using stochastic simulations.
Main Results:
- General formulas for the probability distribution and moments of stochastic search time and splitting probabilities were obtained.
- Asymptotic results are valid for general space-dependent diffusivities, domains, dimensions, and target shapes/locations.
- Stochastic search outcomes can be strongly and counterintuitively influenced by the multiplicative noise interpretation.
Conclusions:
- The interpretation of multiplicative noise is crucial for understanding stochastic search with heterogeneous diffusion.
- The derived formulas provide a framework for analyzing complex search processes in diverse scientific and engineering applications.
- Future research can leverage these findings to explore more nuanced aspects of diffusion-limited processes.
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