Ifan Johnston

In 1967 Aronson showed that the fundamental solution of the heat equation for a second order uniformly elliptic operator in divergence form satisfies two-sided Gaussian estimates. Using this celebrated result, we investigate two-sided estimates for the fundamental solution of the fractional analogues of the heat equation, where one replaces the time derivative with a Caputo fractional derivative of order \(\beta \in (0, 1)\) and also replace the second order elliptic operator with a homogeneous pseudo-differential operator. The starting point for these estimates is given by a formula, which is due to Zolotarev, that links Mittag-Leffler functions with \(\beta\)-stable densities via the Laplace transform. Probabilistically speaking, the solution of such fractional evolution equations is typically some time-changed Brownian motion, or time-changed stable process. This is joint work with Vassili Kolokoltsov.