Question

Let $(M,g^{TM})$ a Riemannian manifold of dimension $n$ and $\Delta$ the Laplace–Beltrami operator. I would like to find a reference (analytic or probabilistic) for the following classic result.

If $p_t(x,y)$ is the kernel of the semigroup $e^{t\Delta}$, then there exist $C,c>0$, such that $$p_{t}(x,y)\leq \frac{C}{t^{n/2}}e^{-cd(x,y)^2/t}.$$

Answer 1

A probabilistic proof is given in the book: "Stochastic Analysis on Manifolds" by E. Hsu. See Theorem 5.3.4, which also gives the lower bound.

Answer 2

I'm pretty sure this can be found in Davies, Heat kernels and spectral theory. I'll check when I get to my office in an hour or so.

Answer 3

You may also be interested in the following paper of Grigoryan, Gaussian upper bounds for the heat kernel on arbitrary manifolds, which establishes the desired Gaussian bounds whenever one can show that there exists $C>0$ such that for all $x\in M$ and $t>0$, $p_t(x,x) \leq Ct^{-n/2}$. The latter estimate may be obtained via a Sobolev inequality or a Nash inequality or through other means (and I'm sure it's discussed in the Davies and Saloff-Coste books already mentioned). This paper also establishes Gaussian upper bounds even when the function appearing in the 'on-diagonal bound' is not of the form $t^{-n/2}$, or if one only has control of $p_t(x,x)$ at two points $x_1$ and $x_2$.