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

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{gather*} \frac{\partial^\beta}{\partial t^\beta}u(t,x) = -(-\Delta_x)^\gamma u(t,x), \quad \beta,\gamma\in(0,1). \end{gather*}