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

TL;DR

This paper establishes strong uniqueness and existence of solutions for SPDEs associated with Gibbs measures on path spaces with exponential interactions, enabling analysis of complex quantum field models.

Contribution

It extends SPDE methods to prove $L^{p}$-uniqueness and strong solutions for non-differentiable potentials in infinite-volume Gibbs measures.

Findings

Proves $L^{p}$-uniqueness of Dirichlet operators for exponential interactions.

Establishes existence and uniqueness of strong solutions for the associated SPDEs.

Derives functional inequalities for diffusion semigroups related to these models.

Abstract

We prove $L^{p}$-uniqueness of Dirichlet operators for Gibbs measures on the path space $C(\mathbb R, \mathbb R^{d})$ 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, 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 $P(\phi)_{1}$-, ${\rm exp}(\phi)_{1}$-, 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. These inequalities are improvements of previous work by the second named author.