Asymptotic properties of heat kernels and their applications

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Ichihara has studied about some relationships between the spectrums of a second order elliptic partial differential operator and the asymptotic properties of the associated heat kernel. First, the exact decay order of the heat kernels in time has been given for a class of complete, simply-connected, negatively curved Riemannian manifolds with asymptotically constant negative sectional curvature. In particular, it has been clarified that there exists an intimate connection between the decay part of polynomial order of the heat kernel and a term of the perturbation of constant negative curvature. Furthermore large deviation principles of the Donsker-Varadhan type have been established for a class of diffusion processes with strong transience property. It is well known that the usual large deviation for a Markov process does not hold when the infimum of the spectrums for its infinitesimal generator ×(-1) is positive. Ichihara has shown that it is still possible to get a nice large deviation principle so far as the pinned process is concerned if an appropriate rate function is chosen.Osada has constructed a Gibbs measure on the path space under the existence of interaction potential and has proved its uniquness in a class of Gibbs measures. Making use of the theory of Dirichlet spaces, a diffusion process having the above Gibbs measure as an invariant probabilty measure has been constructed on the path space. Furthermore Osada has constructed diffusion processes on the Sierpinski Carpet and has given estimates for their transition densities.Chiyonobu has carried out precise estimates for integrations of functionals in an infinite dimensional space, which appear in statistical mechanics.