Feynman-Kac propagators and viscosity solutions

The aim of this paper is to study viscosity solutions to the following terminal value problem on [0, t] × E: where E is a locally compact second countable Hausdorff topological space equipped with a reference measure m, f  L(m), and V satisfies a Kato type condition. It is assumed that a transition probability density p is given, and the family of operators A() is defined by where Y denotes the free backward propagator associated with p. It is shown in the paper that under some restrictions on p, V , ₀  [0,t), and x₀  E, the backward Feynman-Kac propagator Y_V associated with p and V generates a viscosity solution to the terminal value problem above at the point (₀, x₀). Similar result holds in the case where the function V is replaced by a time-dependent family  of Borel measures on E.