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In the first part of the paper we prove various results on regularity ofFeynman-Kac functionals of Hunt processes associated with time dependentsemi-Dirichlet forms. In the second part we study the Cauchy problem forsemilinear parabolic equations with measure data involving operators associatedwith time-dependent forms. Model examples are non-symmetric divergence formoperators and fractional laplacians with possibly variable exponents. We firstintroduce a definition of a solution resembling Stampacchia's definition in thesense of duality and then, using the results of the first part, we prove theexistence, uniqueness and regularity of solutions of the problem under mildassumptions on the data.

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