Stochastic quantization of the two-dimensional polymer measure

We prove that there exists a diiusion process whose invariant measure is the two-dimensional polymer measure g. The diiusion is constructed by means of the theory of Dirichlet forms on innnite-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 g) but requires the quasi-invariance of 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 diiusion process is ergodic under time translations.