As the Gaussian character of the heat kernel can be essentially captured by small time estimations, more accurate description of its behavior, particularly in the region of d(x,y)^2<t is still missing. This is related to the classic Gaussian bound estimates for heat kernels. In the case of complete Riemannian manifold, this requires a good estimate of the heat kernel in large time, which, essentially relies on a global lower bound of the curvature. Indeed, if we assume the Riemannian manifold M has nonnegative Ricci curvature, then its heat kernel satisfies the following Gaussian estimate.

\[

\frac{c(\epsilon)^{-1}}{\mathrm{Vol}(B(x,\sqrt{t}))}\exp{-\frac{d(x,y)^2}{(4-\epsilon)t}}\le p_t(x,y)\le \frac{c(\epsilon)}{\mathrm{Vol}(B(x,\sqrt{t}))}\exp{-\frac{d(x,y)^2}{(4+\epsilon)t}} , \quad\mbox{for all }x,y\in M, t>0, \epsilon>0

\]

where Vol(B(x,r)) is the volume of the geodesic ball centered at x and is of radius r. This is a consequence of the prominent Li-Yau inequality for global positive solution u(x,t) of the heat equation

\[

|\nabla\log u(x,t)|^2-\frac{\partial}{\partial t}\log u(x,t)\le \frac{n}{2t}, \quad \mbox{for all } x\in M, t>0.

\]

Based on the celebrated Bochner's formula, Bakry-Emery introduce the so-called Bakry-Emery criterion (also known as curvature-dimension inequality, which perfectly captures the notion of Ricci curvature lower bound. In their remarkable works, Bakry and Ledoux offered a new perspective on classical geometric theories and functional inequalities . In particular, they provided a more fundamental approach to Li-Yau type gradient estimates for positive solutions of the heat equation and associated Gaussian upper and lower bounds for the heat kernel.

In a series of works by Baudoin-Garofalo and their co-authors, a full generalization of the geometric analysis results were developed on sub-Riemannian manifolds with transverse symmetries, including Li-Yau type gradient estimates for positive solutions of the heat equation and associated Harnack inequalities.

\[

\frac{C(\epsilon)^{-1}}{\mathrm{Vol}(B(x,\sqrt

t))} \exp

\left(-\frac{D d(x,y)^2}{n(4-\epsilon)t}\right)\le p(x,y,t)\le \frac{C(\epsilon)}{\mathrm{Vol}(B(x,\sqrt

t))} \exp

\left(-\frac{d(x,y)^2}{(4+\epsilon)t}\right).

\]

In an joint work with Baudoin, we develop a family of curvature dimension inequalities that is suitable to a larger class of sub-Riemannian manifolds which may no longer be transversely symmetric. Typical examples are contact manifolds. We obtain

\[

\frac{c(\epsilon)^{-1}}{\mathrm{Vol}(B(x,\sqrt{t}))}\exp{-\frac{d(x,y)^2}{(4-\epsilon)t}}\le p_t(x,y)\le \frac{c(\epsilon)}{\mathrm{Vol}(B(x,\sqrt{t}))}\exp{-\frac{d(x,y)^2}{(4+\epsilon)t}} , \quad\mbox{for all }x,y\in M, t>0, \epsilon>0

\]

where Vol(B(x,r)) is the volume of the geodesic ball centered at x and is of radius r. This is a consequence of the prominent Li-Yau inequality for global positive solution u(x,t) of the heat equation

\[

|\nabla\log u(x,t)|^2-\frac{\partial}{\partial t}\log u(x,t)\le \frac{n}{2t}, \quad \mbox{for all } x\in M, t>0.

\]

Based on the celebrated Bochner's formula, Bakry-Emery introduce the so-called Bakry-Emery criterion (also known as curvature-dimension inequality, which perfectly captures the notion of Ricci curvature lower bound. In their remarkable works, Bakry and Ledoux offered a new perspective on classical geometric theories and functional inequalities . In particular, they provided a more fundamental approach to Li-Yau type gradient estimates for positive solutions of the heat equation and associated Gaussian upper and lower bounds for the heat kernel.

In a series of works by Baudoin-Garofalo and their co-authors, a full generalization of the geometric analysis results were developed on sub-Riemannian manifolds with transverse symmetries, including Li-Yau type gradient estimates for positive solutions of the heat equation and associated Harnack inequalities.

\[

\frac{C(\epsilon)^{-1}}{\mathrm{Vol}(B(x,\sqrt

t))} \exp

\left(-\frac{D d(x,y)^2}{n(4-\epsilon)t}\right)\le p(x,y,t)\le \frac{C(\epsilon)}{\mathrm{Vol}(B(x,\sqrt

t))} \exp

\left(-\frac{d(x,y)^2}{(4+\epsilon)t}\right).

\]

In an joint work with Baudoin, we develop a family of curvature dimension inequalities that is suitable to a larger class of sub-Riemannian manifolds which may no longer be transversely symmetric. Typical examples are contact manifolds. We obtain

- Stochastic completeness of the heat semigroup associated to the contact sub-Laplacian.
- Poincare (spectral gap) inequality and convergence of the associated diffusion process to equilibrium as time goes to infinity.
- Weak Bonnet-Myers type theorem and weak Bishop-Gromov comparison theorem.