  This paper investigates sufficient conditions for a Feynman-Kac functional up
to an exit time to be the generalized viscosity solution of a Dirichlet
problem. The key ingredient is to find out the continuity of exit operator
under Skorokhod topology, which reveals the intrinsic connection between
overfitting Dirichlet boundary and fine topology. As an application, we
establish the sub and supersolutions for a class of non-stationary HJB
(Hamilton-Jacobi-Bellman) equations with fractional Laplacian operator via
Feynman-Kac functionals associated to $\alpha$-stable processes, which help
verify the solvability of the original HJB equation.
