Stochastic Quantization of the Two-Dimensional Polymer Measure

Abstract.

We prove that there exists a diffusion process whose invariant measure is the two-dimensional polymer measure ν _g . The diffusion is constructed by means of the theory of Dirichlet forms on infinite-dimensional state spaces. We prove the closability of the appropriate pre-Dirichlet form which is of gradient type, using a general closability result by two of the authors. This result does not require an integration by parts formula (which does not hold for the two-dimensional polymer measure ν _g ) but requires the quasi-invariance of ν _g along a basis of vectors in the classical Cameron—Martin space such that the Radon—Nikodym derivatives (have versions which) form a continuous process. We also show the Dirichlet form to be irreducible or equivalently that the diffusion process is ergodic under time translations.