Stochastic Quantization of the Three-Dimensional Polymer Measure via Dirichlet Form Method

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 We follow in part a previous incomplete unpublished work of the first named author with M. Röckner and X. Y. Zhou (Stochastic quantization of the three-dimensional polymer measure, 1996). 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 by the first named author and Röckner (Probab Theory Related Fields 83(3):405–434, 1989). 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.