Conditional gauge theorem for non-local Feynman-Kac transforms
Let \(X=(X_t,P_x)\) be a transient Borel standard process on a Lusin space \(E\). Further assume that, for an excessive measure \(m\) on \(E\), \(X\) is \(m\)-irreducible having Green function \(G(x,y)\). The main purpose of this paper is to prove the gauge and the conditional gauge theorem related to discontinuous additive functionals. For two Borel functions \(q\) on \(E\) and \(F\) on \(E\times E\) vanishing on the diagonal, let \[ A_{q+F} (t)=\int^t_0 q(X_s)ds+ \sum_{s\leq t}F(X_{s-},X_s) \quad\text{and}\quad e_{q+F} (t)=\exp (A_{q+F} (t)). \] The function \(q\) is said to be of class \({\mathbf K}_1\) if \(\beta(q)= \sup_{B \subset K,m(B)< \delta}\sup_x G(|q|I_{K^c\cup B}) <1\) for some \(K\) and \(\delta >0\). Also \(F\) is said a function of class \({\mathcal I}_\infty\) if \(F\) is bounded and \(N|F|(x) \mu_H(dx)\) is a Kato type measure, where \((N,H)\) is the Lévy system and \(\mu_H\) is the Revuz measure of \(H\). The gauge theorem in this paper says that, for \(q\in{\mathcal K}_1\) and \(F\in{\mathcal I}_\infty\), \(E_x (e_{g+F}(\zeta))\) is either bounded or identically infinite. When the dual process exists, the corresponding classes \({\mathbf S}_1\) and \({\mathbf A}_\infty\) related to the \(h\)-process \(X^z= (X_t,P^z_x)\) for \(h=G( \cdot,z)\) are introduced. For \(q\in {\mathbf S}_1\) and \(F\in {\mathbf A}_\infty\), it is proved that \(E^z_x(e_{q+F} (\zeta^z))\) is bounded if it is finite at some \((x_0,z_0)\). Furthermore, an expression of the bilinear form of the process determined by a Feynman-Kac transformation corresponding to \(e_{q+F}(t)\) is given.
scientific article; zbMATH DE number 1903768