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 $$\nu _\lambda $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>ν</mml:mi> <mml:mi>λ</mml:mi> </mml:msub> </mml:math> for all $$\lambda &gt;0$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>λ</mml:mi> <mml:mo>&gt;</mml:mo> <mml:mn>0</mml:mn> </mml:mrow> </mml:math> . We follow in part a previous incomplete unpublished work of the first named author with M. Röckner and X. Y. Zhou (Stochastic quantization of the three-dimensional polymer measure, 1996).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 $$\nu _\lambda $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>ν</mml:mi> <mml:mi>λ</mml:mi> </mml:msub> </mml:math> we rely on previous work by J. Westwater, E. Bolthausen and X.Y. Zhou. Using $$\nu _\lambda $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>ν</mml:mi> <mml:mi>λ</mml:mi> </mml:msub> </mml:math> , 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 $$\nu _\lambda $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>ν</mml:mi> <mml:mi>λ</mml:mi> </mml:msub> </mml:math> ) but requires the quasi-invariance of $$\nu _\lambda $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>ν</mml:mi> <mml:mi>λ</mml:mi> </mml:msub> </mml:math> along a basis of vectors in the classical Cameron-Martin space such that the Radon-Nikodym derivatives have versions which form a continuous process.