Walk-on-Spheres (WoS) is a particle-based technique used to describe physical phenomena as diverse as electrostatics, heat conduction, rendering, ion transport, and chemical reactions. It is fast, scalable, meshless, and applicable to geometrically complex domains, as long as the governing equations have constant coefficients. Despite these advantages, WoS has persistently struggled to deal with spatial heterogeneity, exhibiting significant loss of either accuracy or efficiency. We introduce a WoS method for parabolic and elliptic problems with spatially varying coefficients and demonstrate its superior performance through applications to heterogeneous transport phenomena. The method exhibits small bias and, where transport heterogeneity dominates the overall dynamics, a variance per sample is 1 to 2 orders of magnitude smaller than that of competing methods, including state-of-the-art WoS algorithms.

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• Brownian motion
• Classical transport
• Diffusion
• Escape problems
• Ionic transport
• Random walks