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## GRIGORYAN’S METHOD FOR GAUSSIAN UPPER BOUNDS TO TIME-FRACTIONAL DIFFUSIONS

Alexander Grigoryan’s method (GrigoryanGaussianBounds)for Gaussian upper bounds on the heat kernel translates on diagonal bounds $p(t,x,x) \le C/f(t)$ into Gaussian upper bounds using riemannian distance function $d(x,y)$ which was originally observed by Varadhan in 1966, that heat kernels should have $\lim_{t\rightarrow 0} 2t \log(p(t,x,y)) = -d^2(x,y)$.  What happens when we replace the parabolic equation of the heat equation with a time-fractional equation?  If we use $D^\beta_t$ the Caputo derivative

$D^\beta_t f = \int_0^t (t-s)^{-\beta} f'(s) ds$

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 $u(t,x)$ solves the nonfractional problem $\partial_t u = L u$ and $u(0,x) = \phi(x)$ with $L$ a second order differential equation such as the case of interest on riemannian manifolds the Laplace-Beltrami operator, and $v(t,x)$ is the solution of the corresponding fractional equation $D^\beta_t v = L v$ then

$v(t,x) = \int_0^{\infty} u( (t/s)^\beta, x) g_{\beta}(s) ds$

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

Suppose then that $p(t,x,x)$ is the parabolic kernel and $q(t,x,x)$ is the fractional parabolic kernel.  Assume $f( (t/s)^\beta ) \le A s^{-\beta} f(t^\beta)$ as a convenient regularity condition and $p(t,x,x) \le C/f(t)$.  Then

$q(t,x,x) = \int_0^{\infty} p(t,x,x) g_{\beta}(s) ds$

$\le C/f(t^\beta) \int_0^\infty A s^{-\beta-1} M_{\beta}(cs^{-\beta}) ds$

$\le C (1/\Gamma(1-\beta)) 1/f(t^\beta)$

The last line follows from changing variables $u=cs^{-\beta}$ and integrating the Wright function using the (4.28) formula from the Mainardi-Luchko-Pagnini paper.