Smooth measures and capacities associated with nonlocal parabolic operators

Abstract

We consider a family {L_t,, tin [0,T]} of closed operators generated by a family of regular (non-symmetric) Dirichlet forms {(B^{(t)},V),tin [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 =fcdot m_1+g_1+partial _tg_2 with fin L^1((0,T)times E;mathrm{d}totimes m), g_1in L^2(0,T;V'), g_2in 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}}}={uin L^2(0,T;V):partial _t uin L^2(0,T;V')} on the right-hand side of the equation.