Journal of the
Korean Mathematical Society
JKMS

ISSN(Print) 0304-9914 ISSN(Online) 2234-3008

Article

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J. Korean Math. Soc. 1997; 34(3): 731-770

Printed September 1, 1997

Copyright © The Korean Mathematical Society.

Dirichlet forms, Dirichlet operators, and log-Sobolev inequalities for Gibbs measures of classical unbounded spin system

Hye Young Lim, Yong Moon Park, and Hyun Jae Yoo

Sogang University, Yonsei University and Yonsei University

Abstract

We study Dirichlet forms and related subjects for the Gibbs measures of classical unbounded spin systems interacting via potentials which are superstable and regular. For any Gibbs measure $\m$, we construct a Dirichlet form and the associated diffusion process on $L^2(\Om, d\m)$, where $\Om=(\BR^d)^{\BZ^\n}$. Under appropriate conditions on the potential we show that the Dirichlet operator associated to a Gibbs measure $\m$ is essentially self-adjoint on the space of smooth bounded cylinder functions. Under the condition of uniform log-concavity, the Gibbs measure exists uniquely and there exists a mass gap in the lower end of the spectrum of the Dirichlet operator. We also show that under the condition of uniform log-concavity, the unique Gibbs measure satisfies the log-Sobolev inequality. We utilize the general scheme of the previous works on the theory in infinite dimensional spaces developed by e.g., Albeverio, Antonjuk, H\o egh-Krohn, Kondratiev, R\"ockner, and Kusuoka, etc, and also use the equilibrium condition and the regularity of Gibbs measures extensively.

Keywords: classical unbounded spin systems, equilibrium condition, Gibbs measures, regularity of measures, Dirichlet forms, diffusion processes, Dirichlet operators, log-Sobolev inequality

MSC numbers: 47D07, 47N55, 60J60