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Exact Asymptotic Formulas for the Heat Kernels of Space and Time-Fractional Equations

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Abstract

This paper aims to study the asymptotic behaviour of the fundamental solutions (heat kernels) of non-local (partial and pseudo differential) equations with fractional operators in time and space. In particular, we obtain exact asymptotic formulas for the heat kernels of time-changed Brownian motions and Cauchy processes. As an application, we obtain exact asymptotic formulas for the fundamental solutions to the n-dimensional fractional heat equations in both time and space

$$\begin{array}{*{20}{c}} {\frac{{{\partial ^\beta }}}{{\partial {t^\beta }}}u\left( {t,x} \right) = - {{\left( { - {\Delta _x}} \right)}^\gamma }u\left( {t,x} \right),} {\beta,\gamma \in \left( {0,1} \right)} \\ \end{array}.$$

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Authors and Affiliations

  1. School of Mathematics and Statistics, Wuhan University, Wuhan, 430072, China

    Chang-Song Deng

  2. Fakultät Mathematik Institut für Mathematische Stochastik, TU Dresden, 01062, Dresden, Germany

    René L. Schilling

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  1. Chang-Song Deng
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  2. René L. Schilling
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Correspondence to Chang-Song Deng.

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Deng, CS., Schilling, R.L. Exact Asymptotic Formulas for the Heat Kernels of Space and Time-Fractional Equations. FCAA 22, 968–989 (2019). https://doi.org/10.1515/fca-2019-0052

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