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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 is sensitive to noise interpretation. Our formulas predict search time and probabilities for heterogeneous diffusion in complex systems.
Area of Science:
- Physics
- Physical Chemistry
- Applied Mathematics
Background:
- The canonical stochastic search model assumes constant diffusivity.
- Heterogeneous diffusion, where diffusivity varies in space, presents challenges due to the Itô-Stratonovich dilemma.
- Existing models are limited, especially in higher dimensions and complex geometries.
Purpose of the Study:
- To investigate the impact of multiplicative noise interpretation on stochastic search with space-dependent diffusivity.
- To develop general formulas for search time and splitting probabilities in heterogeneous diffusion scenarios.
- To provide a theoretical framework applicable to diverse domains and target configurations.
Main Methods:
- Derivation of general formulas for probability distributions and moments of stochastic search time.
- Analysis of splitting probabilities for small or weakly reactive targets.
- Asymptotic analysis valid for general space-dependent diffusivities, domains, and target shapes.
- Validation through stochastic simulations.
Main Results:
- General formulas for stochastic search time and splitting probabilities were obtained.
- Results are applicable to various dimensions, domains, and target configurations (interior/boundary).
- Stochastic search outcomes can be strongly and counterintuitively influenced by the noise interpretation.
Conclusions:
- The interpretation of multiplicative noise is crucial for understanding heterogeneous stochastic search.
- The derived formulas offer a powerful tool for analyzing complex diffusion-limited processes.
- This work advances the understanding of search processes in non-uniform environments across scientific disciplines.
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