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

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 We follow in part a previous incomplete unpublished work of the first named author with M. Röckner and X.Y. Zhou. 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.