Feynman-Kac semigroup with discontinuous additive functionals

Abstract

LetX be a symmetric stable process of index α, 0<α<2, inRd, let μ be a (signed) Radon measure onRd belonging to the Kato classKd, α and letF be a Borel function onRd×Rd satisfying certain conditions. Suppose thatA t μ is the continuous additive functional with μ as its Revuz measure and $$\begin{gathered} A_{t } = A_t^\mu + \Sigma F(X_{s - } , X_s ) \hfill \\ 0< S \leqslant t \hfill \\ \end{gathered} $$ Then the defined semigroup $$T_t f(x) = E^x \{ e^{A_1 } f(X_1 )\} $$ is called the Feynman-Kac semigroup. In this paper we study the Feynman-Kac semigroup (T t ) t>0 and identify the bilinear form corresponding to it.