Analysis of Stochastic Quantization for the Fractional Edwards Measure

Abstract

In [ 10 ] the existence of a diffusion process whose invariant measure is the fractional polymer or Edwards measure for fractional Brownian motion in dimension d ∈ ℕ with Hurst parameter H ∈ (0, 1) fulfilling dH Markov process is constructed via Dirichlet form techniques in infinite-dimensional (Gaussian) analysis. This article uses these results as starting point. In particular, we provide a Fukushima decomposition for the stochastic quantization of the fractional Edwards measure and prove that the constructed process solves a stochastic differential equation in infinite dimension for quasi-all starting points in a probabilistically weak sense. 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.