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 for flat generalized Laplacians with anisotropic scaling
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abstract
We calculate the closed analytic form of the solution of heat kernel equation for the anisotropic generalizations of flat Laplacian. We consider a UV as well as UV/IR interpolating generalizations. In all cases, the result can be expressed in terms of Fox-Wright psi-functions. We perform different consistency checks, analytically reproducing some of the previous numerical or qualitative results, such as spectral dimension flow. Our study should be considered as a first step towards the construction of a heat kernel for curved Ho\v{r}ava-Lifshitz geometries, which is an essential ingredient in the spectral action approach to the construction of the Ho\v{r}ava-Lifshitz gravity.
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2019 1verdicts
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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.