Long-Time Asymptotics for Subordinated Fractional Diffusion Equations
Mohamed Majdoub, Ezzedine Mliki
TL;DR
This work addresses the long-time dynamics of linear fractional diffusion equations under random time changes induced by subordinators, focusing on a mixed local-nonlocal generator $\mathscr{L}_{a,b}$ and Caputo time derivatives. The authors develop a unified framework using subordination with density $G_t$ and Cesàro means to derive universal time-asymptotics for the subordinated solutions, facilitated by Laplace-transform methods and the Feller–Karamata Tauberian theorem. The main result shows that $\mathbf{M}_t(v^{E}(x,t))$ scales as $\|v(x,\cdot)\|_1 \Gamma(\varrho+1)^{-1} t^{\varrho-1} L(t)$, with explicit decay rates for kernel classes ${\bf C}_1$--${\bf C}_5$ and for specific parameter regimes $(\alpha,\gamma,a,b)$, thereby linking stochastic time-changes to deterministic PDE asymptotics. This provides a rigorous bridge between stochastic subordinators and memory-bearing diffusion processes, offering insights for modeling long-time dynamics in contexts such as population dynamics and ecological systems.
Abstract
We study the long-time behavior of solutions to a class of evolution equations arising from random-time changes driven by subordinators. Our focus is on fractional diffusion equations involving mixed local and nonlocal operators. By combining techniques from probability theory, asymptotic analysis, and partial differential equations (PDEs), we characterize the dynamics of the subordinated solutions. This approach extends classical fractional dynamics and establishes a deeper connection between stochastic processes and deterministic PDEs.