Stochastic Quantization for the Edwards Measure of Fractional Brownian Motion with $Hd=1$

TL;DR

This paper develops a Markov process with the fractional Edwards measure as its invariant, specifically for fractional Brownian motion with $Hd=1$, using Dirichlet form theory and quasi translation invariance.

Contribution

It introduces a novel construction of a Markov process for fractional Brownian motion with $Hd=1$, overcoming differentiability challenges via quasi translation invariance.

Findings

Constructed a Markov process with fractional Edwards measure as invariant

Proved closability of the Dirichlet form using quasi translation invariance

Extended the theory of stochastic quantization to fractional Brownian motion with $Hd=1$

Abstract

In this paper we construct a Markov process which has as invariant measure the fractional Edwards measure based on a $d$-dimensional fractional Brownian motion, with Hurst index $H$ in the case of $Hd=1$. We use the theory of classical Dirichlet forms. However since the corresponding self-intersection local time of fractional Brownian motion is not Meyer-Watanabe differentiable in this case, we show the closability of the form via quasi translation invariance of the fractional Edwards measure along shifts in the corresponding fractional Cameron-Martin space.