J. Korean Math. Soc. 1997; 34(3): 731-770
Printed September 1, 1997
Copyright © The Korean Mathematical Society.
Hye Young Lim, Yong Moon Park, and Hyun Jae Yoo
Sogang University, Yonsei University and Yonsei University
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
1996; 33(4): 823-855
2023; 60(4): 779-797
2002; 39(6): 931-951
2011; 48(1): 147-158
© 2022. The Korean Mathematical Society. Powered by INFOrang Co., Ltd