Alexander Grigoryan’s method (GrigoryanGaussianBounds)for Gaussian upper bounds on the heat kernel translates on diagonal bounds into Gaussian upper bounds using riemannian distance function which was originally observed by Varadhan in 1966, that heat kernels should have . What happens when we replace the parabolic equation of the heat equation with a time-fractional equation? If we use the Caputo derivative

then there is a well-known relationship between solution of the Cauchy problem for the non-fractional standard Cauchy problem for the parabolic equation and the fractional one: if solves the nonfractional problem and with a second order differential equation such as the case of interest on riemannian manifolds the Laplace-Beltrami operator, and is the solution of the corresponding fractional equation then

First, let’s get some more information regarding the density function . It is the inverse stable subordinator density whose Laplace transform is . This has been precisely been analyzed by Mainardi-Luchko-Pagnini (fundamentalSolutionSTfrac):

Suppose then that is the parabolic kernel and is the fractional parabolic kernel. Assume as a convenient regularity condition and . Then

The last line follows from changing variables and integrating the Wright function using the (4.28) formula from the Mainardi-Luchko-Pagnini paper.