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The diffusion equation is the primary tool to study the movement dynamics ofa free Brownian particle, but when spatial heterogeneities in the form ofpermeable interfaces are present, no fundamental equation has been derived.Here we obtain such an equation from a microscopic description using a latticerandom walk model. The sought after Fokker-Planck description and thecorresponding backward Kolmogorov equation are employed to investigatefirst-passage and local time statistics and gain new insights. Among them asurprising phenomenon, in the case of a semibounded domain, is the appearanceof a regime of dependence and independence on the location of the permeablebarrier in the mean first-passage time. The new formalism is completelygeneral: it allows to study the dynamics in the presence of multiple permeablebarriers as well as reactive heterogeneities in bounded or unbounded domainsand under the influence of external forces.

Diffusion through permeable interfaces: Fundamental equations and their application to first-passage and local time statistics