Stochastic quantization of the three-dimensional polymer measure via the Dirichlet form method

TL;DR

This paper constructs a diffusion process in three dimensions whose invariant measure is the polymer measure, using Dirichlet form techniques without relying on an integration by parts formula, advancing the mathematical understanding of polymer measures.

Contribution

It establishes the existence of a diffusion process with the three-dimensional polymer measure as invariant, employing Dirichlet form methods and proving closability without an integration by parts formula.

Findings

Existence of the diffusion process with the polymer measure as invariant

Proved closability of the gradient-type Dirichlet form in infinite dimensions

Established quasi-invariance of the polymer measure along Cameron-Martin vectors

Abstract

We prove that there exists a diffusion process whose invariant measure is the three dimensional polymer measure $\nu_\lambda$ for all $\lambda>0$. We follow in part a previous incomplete unpublished work of the first named author with M. R\"ockner and X.Y. Zhou. For the construction of $\nu_\lambda$ we rely on previous work by J. Westwater, E. Bolthausen and X.Y. Zhou. Using $\nu_\lambda$, the diffusion is constructed by means of the theory of Dirichlet forms on infinite-dimensional state spaces. The closability of the appropriate pre-Dirichlet form which is of gradient type is proven, by using a general closability result in [AR89a]. This result does not require an integration by parts formula (which does not even hold for the two-dimensional polymer measure $\nu_\lambda$) but requires the quasi-invariance of $\nu_\lambda$ along a basis of vectors in the classical Cameron-Martin space such that the Radon-Nikodym derivatives have versions which form a continuous process.