Heat Kernel Bounds on Metric Measure Spaces and Some Applications

Let $(X,d,\mu)$ be a $RCD^\ast(K, N)$ space with $K\in \mathbb{R}$ and $N\in [1,\infty]$. For $N\in [1,\infty)$, we derive the upper and lower bounds of the heat kernel on $(X,d,\mu)$ by applying the parabolic Harnack inequality and the comparison principle, and then sharp bounds for its gradient, which are also sharp in time. When $N=\infty$, we also establish a sharp upper bound of the heat kernel by using the dimension free Harnack inequality. For applications, we study the large time behavior of the heat kernel, the stability of solutions to the heat equation, and show the $L^p$ boundedness of (local) Riesz transforms.Comment: 27pp,Section 6 was removed, to appear in Potential Ana

Harnack Inequalities for SDEs with Multiplicative Noise and Non-regular Drift

The log-Harnack inequality and Harnack inequality with powers for semigroups associated to SDEs with non-degenerate diffusion coefficient and non-regular time-dependent drift coefficient are established, based on the recent papers \cite{Flandoli, Zhang11}. We consider two cases in this work: (1) the drift fulfills the LPS-type integrability, and (2) the drift is uniformly H\"older continuous with respect to the spatial variable. Finally, by using explicit heat kernel estimates for the stable process with drift, the Harnack inequality for the stochastic differential equation driven by symmetric stable process is also proved.Comment: All comments are welcom