Smooth measures and capacities associated with nonlocal parabolic operators

Abstract

We consider a family $\{L_t,t\in [0,T]\}$ of closed operators generated by a family of regular (non-symmetric) Dirichlet forms $\{(B^{(t)},V),t\in [0,T]\}$ on $L^2(E;m)$. We show that a bounded (signed) measure $\mu$ on $(0,T)\times E$ is smooth, i.e. charges no set of zero parabolic capacity associated with $\frac{\partial }{\partial t}+L_t$, if and only if $\mu$ is of the form $\mu =f\cdot m_1+g_1+{\partial }_t{g}_2$ with $f\in L^1((0,T)\times E;dt\otimes m)$, $g_1\in L^2(0,T;V^{\prime })$, $g_2\in L^2(0,T;V)$. 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 $\frac{\partial }{\partial t}+L_t$ and a functional from the dual ${\mathcal{W}}^{\prime }$ of the space $\mathcal{W}=\{u\in L^2(0,T;V):{\partial }_{t}u\in L^2(0,T;V^{\prime })\}$ on the right-hand side of the equation.