Appl Math Optim 40:341–354 (1999)
1999 Springer-Verlag New York Inc.
Stochastic Quantization of the Two-Dimensional Polymer Measure
and X. Y. Zhou
Institut f¨ur Angewandte Mathematik, Universit¨at Bonn,
D-53115 Bonn, Germany
Department of Mathematics, University of Kansas,
Lawrence, KS 66045, USA
Institute of Mathematical Sciences, Academia Sinica,
Wuhan 430071, China
Fakult¨at f¨ur Mathematik, Universit¨at Bielefeld,
33501 Bielefeld, Germany
Institute of Mathematics, Beijing Normal University,
Beijing 100875, People’s Republic of China
Institut f¨ur Mathematik, Ruhr-Universit¨at Bochum,
D-44780 Bochum, Germany
Abstract. We prove that there exists a diffusion process whose invariant measure
is the two-dimensional polymer measure ν
. 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 poly-
mer measure ν
) but requires the quasi-invariance of ν
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
Key Words. Two-dimensional polymer measure, Closability, Dirichlet forms,
Diffusion processes, Ergodicity, Quasi-invariance.
AMS Classiﬁcation. Primary 60J65, Secondary 60H30.
Thisresearchwasﬁnanciallysupported bySFB343 Bielefeldand SFB237Bochum–D¨usseldorf–Essen.
Deceased April 22, 1996.