Exact Asymptotic Formulas for the Heat Kernels of Space and Time-Fractional Equations
arXiv:1803.11435
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 -dimensional fractional heat equations in both time and space \begin{gather*} \frac{\partial^β}{\partial t^β}u(t,x) = -(-Δ_x)^γu(t,x), \quad β,γ\in(0,1). \end{gather*}
arXiv commentThis paper is now published (in revised form) in Fract. Calc. Appl. Anal., Vol. 22, No 4 (2019), pp. 968--989, DOI: 10.1515/fca-2019-0052 , and is available online at http://www.degruyter.com/view/j/fca