대한수학회지 (Journal of the Korean Mathematical Society)

  • 제34권3호
  • /
  • Pages.731-770
  • /
  • 1997
  • /
  • 0304-9914(pISSN)
  • /
  • 2234-3008(eISSN)
대한수학회 (Korean Mathematical Society)
권호 표지

DIRICHLET FORMS, DIRICHLET OPERATORS, AND LOG-SOBOLEV INEQUALITIES FOR GIBBS MEASURES OF CLASSICAL UNBOUNDED SPIN SYSTEM

  • Lim, Hye-Young (Department of Mathematics Sogang University ) ;
  • Park, Yong-Moon (Dept. of Math. and Institute for Math. Sciences Yonsei University ) ;
  • Yoo, Hyun-Jae (Institute for Mathematical Sciences Yonsei University )
  • 발행 : 1997.08.01
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초록

We study Diriclet forms and related subjects for the Gibbs measures of classical unbounded sping systems interacting via potentials which are superstable and regular. For any Gibbs measure $\mu$, we construct a Dirichlet form and the associated diffusion process on $L^2(\Omega, d\mu), where \Omega = (R^d)^Z^\nu$. Under appropriate conditions on the potential we show that the Dirichlet operator associated to a Gibbs measure $\mu$ 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, Hoegh-Krohn, Kondratiev, Rockner, and Kusuoka, etc, and also use the equilibrium condition and the regularity of Gibbs measures extensively.

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참고문헌

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