Smooth measures and capacities associated with nonlocal parabolic operators.

We consider a family { L t , t ∈ [ 0 , T ] } of closed operators generated by a family of regular (non-symmetric) Dirichlet forms { (B (t) , V) , t ∈ [ 0 , T ] } on L 2 (E ; m) . We show that a bounded (signed) measure μ on (0 , T) × E is smooth, i.e. charges no set of zero parabolic capacity associated with ∂ ∂ t + L t , if and only if μ is of the form μ = f · m 1 + g 1 + ∂ t g 2 with f ∈ L 1 ((0 , T) × E ; d t ⊗ m) , g 1 ∈ L 2 (0 , T ; V ′) , g 2 ∈ L 2 (0 , T ; V) . We apply this decomposition to the study of the structure of additive functionals in the Revuz correspondence with smooth measures. As a by-product, we also give some existence and uniqueness results for solutions of semilinear equations involving the operator ∂ ∂ t + L t and a functional from the dual W ′ of the space W = { u ∈ L 2 (0 , T ; V) : ∂ t u ∈ L 2 (0 , T ; V ′) } on the right-hand side of the equation. [ABSTRACT FROM AUTHOR]