Analysis of Stochastic Quantization for the fractional Edwards Measure

Abstract

We analyse a diffusion process whose invariant measure is the fractional polymer or Edwards measure for fractional Brownian motion in dimension $d\in\mathbb{N}$ with Hurst parameter $H\in(0,1)$ fulfilling $dH < 1$. We make use of a construction of the diffusion via Dirichlet form techniques in infinite dimensional (Gaussian) analysis. By providing a Fukushima decomposition for the stochastic quantization of the fractional Edwards measure we prove that the constructed process solves weakly a stochastic differential equation in infinite dimension for quasi-all starting points. Moreover, the solution process is driven by an Ornstein--Uhlenbeck process taking values in an infinite dimensional distribution space and is unique, in the sense that the underlying Dirichlet form is Markov unique. The equilibrium measure, which is by construction the fractional Edwards measure, is specified to be an extremal Gibbs state and therefore, the constructed stochastic dynamics is time ergodic. The studied stochastic differential equation provides in the language of polymer physics the dynamics of the bonds, i.e. stochastically spoken the noise of the process. An integration leads then to polymer paths. We show that if one starts with a continuous polymer configuration the integrated process stays almost surely continuous during the time evolution.