  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 $\X_t$ and a Brownian functional ${\mathcal
U}_t$ that specifies the amount of time the particle is in contact with a
reactive component $\calM$. Absorption occurs as soon as $\calU_t$ crosses a
randomly distributed threshold (stopping time). Laplace transforming the
propagator with respect to $\calU_t$ 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 $\partial \calM$, this can be achieved by solving a
classical Robin BVP in terms of the spectral decomposition of a
Dirichlet-to-Neumann operator. In this paper we develop the analogous
construction in the case of a reactive substrate $\calM$. In particular, we
show that the Laplace transformed propagator can be computed in terms of the
spectral decomposition of a pair of Dirichlet-to-Neumann operators. However,
inverting the Laplace transform with respect to $z$ is more involved. We
illustrate the theory by considering a 1D example where the
Dirichlet-to-Neumann operators reduce to scalars.
