Strong uniqueness for both Dirichlet operators and stochastic dynamics to Gibbs measures on a path space with exponential interactions

Sergio Albeverio, Hiroshi Kawabi, Michael Röckner

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8 Citations (Scopus)


We prove Lp-uniqueness of Dirichlet operators for Gibbs measures on the path space C(R,Rd) associated with exponential type interactions in infinite volume by extending an SPDE approach presented in previous work by the last two named authors. We also give an SPDE characterization of the corresponding dynamics. In particular, for the first time, we prove existence and uniqueness of a strong solution for the SPDE, though the self-interaction potential is not assumed to be differentiable, hence the drift is possibly discontinuous. As examples, to which our results apply, we mention the stochastic quantization of P(φ)1-, exp(φ)1-, and trigonometric quantum fields in infinite volume. In particular, our results imply essential self-adjointness of the generator of the stochastic dynamics for these models. Finally, as an application of the strong uniqueness result for the SPDE, we prove some functional inequalities for diffusion semigroups generated by the above Dirichlet operators.

Original languageEnglish
Pages (from-to)602-638
Number of pages37
JournalJournal of Functional Analysis
Issue number2
Publication statusPublished - Jan 15 2012



  • Dirichlet operator
  • Essential self-adjointness
  • Exp(φ)-quantum fields
  • Gibbs measure
  • L-uniqueness
  • Logarithmic Sobolev inequality
  • Path space
  • SPDE
  • Strong uniqueness

ASJC Scopus subject areas

  • Analysis

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