%0 Journal Article %1 Deng2019-jx %A Deng, Chang-Song %A Schilling, René L %D 2019 %I Springer Science and Business Media LLC %J Fract. Calc. Appl. Anal. %K %N 4 %P 968--989 %T Exact asymptotic formulas for the heat kernels of space and time-fractional equations %V 22 %X 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\\\ \frac\\partial^\beta\\t^\beta\u(t,x) = -(-\Delta_x)^u(t,x), \beta,\gammaın(0,1). \end\array\$

@article{Deng2019-jx, abstract = {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 $\begin\{array\}\{\} \displaystyle \frac\{\partial^\beta\}\{\partial t^\beta\}u(t,x) = -(-\Delta_x)^\gamma u(t,x), \quad \beta,\gamma\in(0,1). \end\{array\}$}, added-at = {2024-09-10T11:56:37.000+0200}, author = {Deng, Chang-Song and Schilling, Ren{\'e} L}, biburl = {https://puma.scadsai.uni-leipzig.de/bibtex/248f691c9585f795bc8b8e382ac8c996b/scadsfct}, interhash = {154d0a19f444b9ae53896bac9d4c2a54}, intrahash = {48f691c9585f795bc8b8e382ac8c996b}, journal = {Fract. Calc. Appl. Anal.}, keywords = {}, language = {en}, month = aug, number = 4, pages = {968--989}, publisher = {Springer Science and Business Media LLC}, timestamp = {2024-09-10T15:15:57.000+0200}, title = {Exact asymptotic formulas for the heat kernels of space and time-fractional equations}, volume = 22, year = 2019 }