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Perturbation of Dirichlet forms — lower semiboundedness, closability, and form cores. (English) Zbl 0743.60071

Summary: We study perturbations \({\mathcal E}^ \mu:={\mathcal E}+Q_ \mu\) of Dirichlet forms \(\mathcal E\) on some \(L^ 2\) space \(L^ 2(m)\) given by quadratic forms \(Q_ \mu(f,g)=\int fg d\mu\) with \(\mu\) a signed Borel measure whose positive and negative parts are smooth measures with respect to the given Dirichlet form. We give necessary and sufficient conditions for \({\mathcal E}^ \mu\) to be a lower semibounded closed quadratic form with a unique associated continuous semigroup \({\mathcal P}^ \mu\) on \(L^ 2(m)\). We also study the associated resolvent semigroup, as well as the relation between \({\mathcal E}^ \mu\) and and the largest closed quadratic form \(\tilde{\mathcal E}^ \mu\) which is smaller than \({\mathcal E}^ \mu\). In particular we exhibit form cores.

MSC:

60J35 Transition functions, generators and resolvents
31C25 Dirichlet forms
60J45 Probabilistic potential theory
47D07 Markov semigroups and applications to diffusion processes
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