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Renewal equations for single-particle diffusion in multi-layered media

In this paper we develop a probabilistic model of single-particle diffusion in 1D multi-layered media by constructing a multi-layered version of so-called snapping out Brownian motion (BM). The latter sews together successive rounds of reflected BM, each of which is restricted to a single layer. Each round of reflected BM is killed when the local time at one end of the layer exceeds an independent, exponentially distributed random variable. (The local time specifies the amount of time a reflected Brownian particle spends in a neighborhood of a boundary.) The particle then immediately resumes reflected BM in the same layer or the layer on the other side of the boundary with equal probability, and the process is iterated We proceed by constructing a last renewal equation for multi-layered snapping out BM that relates the full probability density to the probability densities of partially reflected BM in each layer. We then show how transfer matrices can be used to solve the Laplace transformed renewal equation, and prove that the renewal equation and corresponding multi-layer diffusion equation are equivalent. We illustrate the theory by analyzing the first passage time (FPT) problem for escape at the exterior boundaries of the domain. Finally, we use the renewal approach to incorporate a generalization of snapping out BM based on the encounter-based method for surface absorption; each round of reflected BM is now killed according to a non-exponential distribution for each local time threshold. This is achieved by considering a corresponding first renewal equation that relates the full probability density to the FPT densities for killing each round of reflected BM. We show that for certain configurations, non-exponential killing leads to an effective time-dependent permeability that is normalizable but heavy-tailed.