1. Stochastic quantization of the three-dimensional polymer measure via the Dirichlet form method
- Author
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Albeverio, Sergio, Kusuoka, Seiichiro, Liang, Song, and Nakashima, Makoto
- Subjects
Mathematics - Probability ,Mathematical Physics ,81S20, 60J65, 60J46, 60H30 - Abstract
We prove that there exists a diffusion process whose invariant measure is the three dimensional polymer measure $\nu_\lambda$ for small $\lambda>0$. We follow in part a previous incomplete unpublished work of the first named author with M. R\"ockner and X.Y. Zhou.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., Comment: 87 pages, 8 figures
- Published
- 2023