Semi-Dirichlet forms, Feynman–Kac functionals and the Cauchy problem for semilinear parabolic equations

In the first part of the paper we prove various results on regularity of Feynman–Kac functionals of Hunt processes associated with time-dependent semi-Dirichlet forms. In the second part we study the Cauchy problem for semilinear parabolic equations with measure data involving operators associated with time-dependent forms. Model examples are non-symmetric divergence form operators and fractional laplacians with possibly variable exponents. We first introduce a definition of a solution resembling Stampacchia's definition in the sense of duality and then, using the results of the first part, we prove the existence, uniqueness and regularity of solutions of the problem under mild assumptions on the data.