A CONCRETE APPROACH TO DIAGONAL SHORT TIME ASYMPTOTICS OF HEAT KERNELS ASSOCIATED WITH SUB-LAPLACIAN ON CR MANIFOLDS
Kyushu Journal of Mathematics
A diffusion process associated with the real sub-Laplacian b , the real part of the complex Kohn-Spencer Laplacian b , on a strictly pseudoconvex CR manifold is constructed by H. Kondo and S. Taniguchi [A construction of diffusion processes associated with sub-Laplacian on CR manifolds and its applications. J. Math. Soc. Japan 69(1) (2017), 111-125]. In this paper, we investigate the diagonal short time asymptotics of the heat kernel corresponding to the diffusion process by using Watanabe's
... mptotic expansion and give a representation for the asymptotic expansion of heat kernels which shows a relationship to the geometric structure. Introduction A diffusion process associated with the real sub-Laplacian b on a strictly pseudoconvex CR manifold M is constructed and the existence of a C ∞ heat kernel p(t, x, y), t ∈ [0, ∞), x, y ∈ M for b is studied in  . As a question arising naturally from this result, we consider the diagonal short asymptotics of the heat kernel, i.e. the asymptotic behavior of p (t, x, x) as t tends to zero. In  , the diffusion process on M is constructed via the Eells-Elworthy-Malliavin method. More precisely, a stochastic differential equation (SDE) on a complex unitary frame bundle is considered and its unique strong solution is projected onto M, and then the resulting process on M is the desired diffusion process generated by − b /2. Since the diffusion process is obtained as a strong solution of an SDE, we can utilize the asymptotic expansion theory of Ikeda and Watanabe (see  ) to investigate the asymptotic behavior of the heat kernel associated with the diffusion process. To be more precise, if the solution U ε t , t ∈ [0, ∞), of the SDE which is obtained by putting a parameter ε > 0 into the SDE defining the diffusion process has a suitable expansion in ε, then composing with the delta function and taking the generalized expectation gives an asymptotic expansion of p(ε 2 , x, x) in ε. When proceeding with this approach, the remaining task is to find the asymptotic expansion of U ε t . On this point, Takanobu  gives a quite general result taking advantage 2010 Mathematics Subject Classification: Primary 58J65; Secondary 60J60.