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