Smooth measures and capacities associated with nonlocal parabolic operators

TL;DR

This paper characterizes smooth measures related to nonlocal parabolic operators generated by time-dependent Dirichlet forms, providing a decomposition of measures, and studies solutions to associated semilinear equations.

Contribution

It introduces a precise measure decomposition linked to nonlocal parabolic operators and explores the structure of additive functionals and solutions to related semilinear equations.

Findings

Characterization of smooth measures via measure decomposition.

Application to the structure of additive functionals in Revuz correspondence.

Existence and uniqueness results for semilinear equations involving 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 $\mu=f\cdot m_1+g_1+\partial_tg_2$ with $f\in L^1((0,T)\times E;dt\otimes m)$, $g_1\in L^2(0,T;V')$, $g_2\in L^2(0,T;V)$. 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 $\mathcal{W}'$ of the space $\mathcal{W}=\{u\in L^2(0,T;V):\partial_t u\in L^2(0,T;V')\}$ on the right-hand side of the equation.