Spectral theory of diffusion in partially absorbing media

A probabilistic framework for studying single-particle diffusion in partially absorbing media has recently been developed terms of an encounter-based approach. The latter computes the joint probability density (generalized propagator) particle position $\X_t$ and a Brownian functional ${\mathcal U}_t$ that specifies amount time is contact with reactive component $\calM$. Absorption occurs as soon $\calU_t$ crosses randomly distributed threshold (stopping time). Laplace transforming propagator respect to leads classical boundary value problem (BVP) which constant rate absorption $z$, where $z$ corresponding variable. Hence, crucial step approach finding inverse transform. In case $\partial \calM$, this can be achieved by solving Robin BVP spectral decomposition Dirichlet-to-Neumann operator. paper we develop analogous construction substrate particular, show transformed computed pair operators. However, inverting transform more involved. We illustrate theory considering 1D example operators reduce scalars.