Feynman-Kac propagators and viscosity solutions

Abstract

The aim of this paper is to study viscosity solutions to the following terminal value problem on [0, t] × E: $$ \left\{ {\begin{array}{*{20}l} {\frac{{\partial u}} {{\partial t}}(\tau ,x) + [A(\tau )u(\tau )](x) - V(\tau ,x)u(\tau ,x) = 0} \hfill \\ {u(t,x) = f(x),} \hfill \\ \end{array} } \right. $$ 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(τ) is defined by $$ A(\tau )h(x) = \mathop {\lim }\limits_{^{^{\epsilon \to 0 + } } } \frac{{Y(\tau + \epsilon ,\tau )h(x) - h(x)}} {\epsilon }, $$ where Y denotes the free backward propagator associated with p. It is shown in the paper that under some restrictions on p, V , τ0 ∈ [0,t), and x0 ∈ E, the backward Feynman-Kac propagator YV associated with p and V generates a viscosity solution to the terminal value problem above at the point (τ0, x0). Similar result holds in the case where the function V is replaced by a time-dependent family μ of Borel measures on E.