Dirichlet forms perturbated by additive functionals of extended Kato class
Let \(({\mathcal E},{\mathcal F})\) be a quasi-regular Dirichlet form on \(L^2(E,m)\) and \(X= (X_t, P^x)\) be the associated Borel right process. For a \([0,\infty]\)-valued, not necessarily continuous, additive functional \(A_t\), let \(\nu_A\) be the bivariate Revuz measure defined by \(\nu_A(f)= \lim_{t\downarrow 0} {1\over t} P^m\left(\int_{]0,t]} f(X_{s-},X_s)dA_s\right)\). Denote by \((\text{Exp }A)_t\) the Stieltjes exponent of \(A\) given as the unique solution of the equation \(Z_t= 1+ \int_{]0,t]} Z_{s-} dA_s\). The main purpose of this paper is to investigate the property of the perturbed semigroup \(P^{-A}_t f(x)= P^x((\text{Exp }A)_t f(X_t))\). Introduce the extended Kato class \({\mathcal P}_K\) defined as the family of additive functionals \(A_t\) such that \(\nu_A\) is symmetric and \(\inf_{t>0} \inf_{\text{Cap}(N)= 0} \sup_{x\not\in N} P^x(A_t)< 1\). Then it is shown, for \(A_t\in{\mathcal P}_K\), that \(P^{-A}_t\) is a symmetric strongly continuous bounded operator on \(L^2(E,m)\). Further, let \(({\mathcal E}^{-A},{\mathcal D}^{-A})\) be the perturbed bilinear form defined by \({\mathcal D}^{-A}={\mathcal D}\cap L^2(\rho_A)\) and \({\mathcal E}^{-A}(u,v)={\mathcal E}(u,v)- \nu_A(u\otimes v)\), where \(\rho_A\) is the Revuz measure of \(A\). Then it is also shown that \(({\mathcal E}^{-A},{\mathcal D}^{-A})\) is closable on \(L^2(m)\) and \(P^{-A}_t\) is the semigroup naturally associated with the closure.
scientific article; zbMATH DE number 1132997