An asymptotic expansion of the fundamental solution to the heat equation and its applications

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The following three facts are shown through the construction of the fundamental solution to degenerate parabolic equations by means of the method of calculus of pseudo-differential operators :(1) Local index theorem holds for the Dirac Operators.(2) One can construct the fundamental solution in the exact form, which can be applied to obtain the eigenfunction expansions of the Fokker Planck operator.(3) The spectral zeta function of the degenerate operator, which is defined naturally on Nilmanilds, has the only one singularity.