On-diagonal lower bounds for heat kernels and Markov chains

Proceedings of the London Mathematical Society, 2007

We prove that in presence of L 2 Gaussian estimates, so-called Davies-Gaffney estimates, on-diagonal upper bounds imply precise off-diagonal Gaussian upper bounds for the kernels of analytic families of operators on metric measure spaces.

Journal de Mathématiques Pures et Appliquées, 2014

Let (X, d, µ) be a RCD * (K, N) space with K ∈ R and N ∈ [1, ∞). Suppose that (X, d) is connected, complete and separable, and supp µ = X. We prove that the Li-Yau inequality for the heat flow holds true on (X, d, µ) when K ≥ 0. A Baudoin-Garofalo inequality and Harnack inequalities for the heat flows are established on (X, d, µ) for general K ∈ R. Large time behaviors of heat kernels are also studied.

Bulletin of the Australian Mathematical Society, 2015

Given a domain ${\rm\Omega}$ of a complete Riemannian manifold ${\mathcal{M}}$, define ${\mathcal{A}}$ to be the Laplacian with Neumann boundary condition on ${\rm\Omega}$. We prove that, under appropriate conditions, the corresponding heat kernel satisfies the Gaussian upper bound $$\begin{eqnarray}h(t,x,y)\leq \frac{C}{[V_{{\rm\Omega}}(x,\sqrt{t})V_{{\rm\Omega}}(y,\sqrt{t})]^{1/2}}\biggl(1+\frac{d^{2}(x,y)}{4t}\biggr)^{{\it\delta}}e^{-d^{2}(x,y)/4t}\quad \text{for}~t>0,~x,y\in {\rm\Omega}.\end{eqnarray}$$ Here $d$ is the geodesic distance on ${\mathcal{M}}$, $V_{{\rm\Omega}}(x,r)$ is the Riemannian volume of $B(x,r)\cap {\rm\Omega}$, where $B(x,r)$ is the geodesic ball of centre $x$ and radius $r$, and ${\it\delta}$ is a constant related to the doubling property of ${\rm\Omega}$. As a consequence we obtain analyticity of the semigroup $e^{-t{\mathcal{A}}}$ on $L^{p}({\rm\Omega})$ for all $p\in [1,\infty )$ as well as a spectral multiplier result.

Journal of Functional Analysis, 2008

It is known that the couple formed by the two dimensional Brownian motion and its Lévy area leads to the heat kernel on the Heisenberg group, which is one of the simplest sub-Riemannian space. The associated diffusion operator is hypoelliptic but not elliptic, which makes difficult the derivation of functional inequalities for the heat kernel. However, Driver and Melcher and more recently H.-Q. Li have obtained useful gradient bounds for the heat kernel on the Heisenberg group. We provide in this paper simple proofs of these bounds, and explore their consequences in terms of functional inequalities, including Cheeger and Bobkov type isoperimetric inequalities for the heat kernel.

Bulletin of The Australian Mathematical Society, 2007

In a recent paper Miranda Jr., Pallara, Paronetto and Preunkert have shown that the classical De Giorgi's heat kernel characterisation of functions of bounded variation on Euclidean space extends to Riemannian manifolds with Ricci curvature bounded from below and which satisfy a uniform lower bound estimate on the volume of geodesic balls of fixed radius. We give a shorter proof of the same result assuming only the lower bound on the Ricci curvature.

Let Ω be a subset of a space of homogeneous type. Let A be the infinitesimal generator of a positive semigroup with Gaussian kernel bounds on L 2 (Ω). We then show Gaussian heat kernel bounds for operators of the type bA where b is a bounded, complex valued function.

Annals of Probability, 2011

We prove equivalent conditions for two-sided sub-Gaussian estimates of heat kernels on metric measure spaces.

2021

In this paper, we establish sharp two-sided estimates for transition densities of a large class of subordinate Markov processes. As applications, we show that the parabolic Harnack inequality and Hölder regularity hold for parabolic functions of such processes, and derive sharp two-sided Green function estimates. AMS 2020 Mathematics Subject Classification: Primary 60J35, 60J50, 60J76

The Quarterly Journal of Mathematics, 1996

2014

We prove a global Li-Yau inequality for a general Markov semigroup under a curvature-dimension condition. This inequality is stronger than all classical Li-Yau type inequalities known to us. On a Riemannian manifold, it is equivalent to a new parabolic Harnack inequality, both in negative and positive curvature, giving new subsequents bounds on the heat kernel of the semigroup. Under positive curvature we moreover reach ultracontractive bounds by a direct and robust method.