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

Research output: Contribution to journalArticle

8 Citations (Scopus)

Abstract

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
Volume262
Issue number2
DOIs
Publication statusPublished - Jan 15 2012

Fingerprint

Path Space
Gibbs Measure
Stochastic Dynamics
Dirichlet
Uniqueness
Operator
Diffusion Semigroup
Interaction
Essential Self-adjointness
Functional Inequalities
Exponential Type
Quantum Fields
Strong Solution
Differentiable
Quantization
Existence and Uniqueness
Generator
Imply
Model

Keywords

  • 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

Cite this

Strong uniqueness for both Dirichlet operators and stochastic dynamics to Gibbs measures on a path space with exponential interactions. / Albeverio, Sergio; Kawabi, Hiroshi; Röckner, Michael.

In: Journal of Functional Analysis, Vol. 262, No. 2, 15.01.2012, p. 602-638.

Research output: Contribution to journalArticle

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