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Self-adjointness of some infinite-dimensional elliptic operators and application to stochastic quantization

Abstract

We consider an operator ϕ = Lϕ−: <CDU(x), Dϕ> in a Hilbert space H, where L is an Ornstein–Uhlenbeck operator, UW 1,4(H, μ) and μ is the invariant measure associated with L. We show that is essentially self-adjoint in the space L 2(H, ν) where ν is the “Gibbs” measure ν(dx) = Z −:1 e −:2U(x) dx. An application to Stochastic quantization is given.

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Da Prato, G., Tubaro, L. Self-adjointness of some infinite-dimensional elliptic operators and application to stochastic quantization. Probab Theory Relat Fields 118, 131–145 (2000). https://doi.org/10.1007/s004400000073

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  • DOI: https://doi.org/10.1007/s004400000073

Mathematics Subject Classification (2000)

  • 47B25
  • 60H15
  • 81S20

Key words

  • Essential self-adjointness
  • Stochastic partial differential equations
  • Stochastic quantization