Many physical processes depend on the time it takes a diffusing particle to find a target. Though this classical quantity is now well understood in various scenarios, little is known when the diffusivity depends on the location of the particle. For such heterogeneous diffusion, the multiplicative noise requires an interpretation (e.g., Itô, Stratonovich, or kinetic). Here, we analytically determine the mean escape time and splitting probabilities for an arbitrary heterogeneous diffusion in an arbitrary three-dimensional domain with small targets that can be perfectly or imperfectly absorbing. Our analysis reveals general principles for how search depends on heterogeneous diffusion and its interpretation. An intricate picture emerges in which, for instance, increasing the diffusivity can decrease, have no effect, or even increase the escape time. Our results could be used to determine the appropriate interpretation for specific physical systems.

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• Brownian motion
• Diffusion
• Escape problems
• Stochastic processes
• Fokker–Planck equation
• Markovian processes
• Stochastic analysis
• Stochastic differential equations