## 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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