For the operator (−∆)^ν, the heat kernel is a generalized exponential (Fox-Wright) function E_{ν,d/2}(−x^2/4τ^{1/ν}), with power-law asymptotics for noninteger ν and oscillatory exponential asymptotics for integer ν.
Heat kernel estimates for pseudodifferential operators, fractional Laplacians and Dirichlet-to-Neumann operators
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abstract
The purpose of this article is to establish upper and lower estimates for the integral kernel of the semigroup exp(-tP) associated to a classical, strongly elliptic pseudodifferential operator P of positive order on a closed manifold. The Poissonian bounds generalize those obtained for perturbations of fractional powers of the Laplacian. In the selfadjoint case, extensions to t in C_+ are studied. In particular, our results apply to the Dirichlet-to-Neumann semigroup.
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Heat kernel for higher-order differential operators and generalized exponential functions
For the operator (−∆)^ν, the heat kernel is a generalized exponential (Fox-Wright) function E_{ν,d/2}(−x^2/4τ^{1/ν}), with power-law asymptotics for noninteger ν and oscillatory exponential asymptotics for integer ν.