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Strong uniqueness for certain infinite-dimensional Dirichlet operators and applications to stochastic quantization. (English) Zbl 0953.60056

Suppose that we are given a classical Dirichlet form of gradient type on a rigged Hilbert space. Initially, this form is defined on some domain \(D\). The operator associated with the Dirichlet form generates a Markov semigroup on the corresponding \(L^2\) space. If \(D\) is contained in the domain of the generator, one can ask the question whether the extension in \(L^2\) of the generator restricted to \(D\) is unique. If there is only one lower bounded self-adjoint extension in \(L^2\) of the generator originally defined on \(D\), we say the Dirichlet operator is strongly unique. This paper studies the strong uniqueness of Dirichlet operators. The authors also study the extension of the problem in the \(L^p\) setting. The main results are used to study stochastic quantization of field theory in finite volume.

MSC:

60H99 Stochastic analysis
31C25 Dirichlet forms
47B25 Linear symmetric and selfadjoint operators (unbounded)
47D07 Markov semigroups and applications to diffusion processes
81S99 General quantum mechanics and problems of quantization
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