Energy functionals and killing transform
In the standard setting of potential theory for right Markov process due to Getoor and Sharpe, the author gives a probabilistic interpretation of energy functional w.r.t. the subprocess killed by a given exact multiplicative functional. The main theorem is stated as follows: Let \({\mathbf X}= (X_t, P^x)\) be a right Markov process and denote respectively by \(\text{Exc}({\mathbf X})\), \(\text{S}({\mathbf X})\) the classes of excessive measures and excessive functions associated with \({\mathbf X}\). The energy-functional \(L:\text{Exc}({\mathbf X})\times\text{S}({\mathbf X})\to\mathbb{R}\) (of \(X\)) is by definition \(L(m,u):= \sup\{\mu(u): \mu\geq 0\), \(\mu{\mathbf U}\leq m\}\), where \({\mathbf U}\) denotes the potential-operator. By the same way if \(M\) is a given exact multiplicative functional with positive life time, then the energy-functional of the \(M\)-subprocess will be denoted by \(L^M(m,u)\). Theorem. 1) \(L(m,u) = L^M(m-{\mathbf R}_Mm, u)\), where \({\mathbf R}_M\) denotes the balayage-operator for measures. 2) Furthermore, if \(m\) is dissipative, then \(L(m,u)= L^M(m,u- {\mathbf P}_Mu)\), where \({\mathbf P}_M\) denotes the balayage-operator for functions. On the way to show the above theorem the author gives an interpretation of the Feynman-Kac formula in terms of energy-functional.
scientific article; zbMATH DE number 1106390