Reference for estimation gaussian of the heat kernel - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-24T16:33:08Z http://mathoverflow.net/feeds/question/88433 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/88433/reference-for-estimation-gaussian-of-the-heat-kernel Reference for estimation gaussian of the heat kernel shu 2012-02-14T14:23:02Z 2012-02-15T20:01:58Z <p>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.</p> <blockquote> <p>If $p_t(x,y)$ is the kernel of the semigroup $e^{t\Delta}$, then there exist $C,c>0$, such that $$p_{t}(x,y)\leq \frac{C}{t^{n/2}}e^{-cd(x,y)^2/t}.$$</p> </blockquote> http://mathoverflow.net/questions/88433/reference-for-estimation-gaussian-of-the-heat-kernel/88434#88434 Answer by Nate Eldredge for Reference for estimation gaussian of the heat kernel Nate Eldredge 2012-02-14T15:16:19Z 2012-02-14T15:16:19Z <p>I'm pretty sure this can be found in Davies, <em>Heat kernels and spectral theory</em>. I'll check when I get to my office in an hour or so.</p> http://mathoverflow.net/questions/88433/reference-for-estimation-gaussian-of-the-heat-kernel/88436#88436 Answer by Hans for Reference for estimation gaussian of the heat kernel Hans 2012-02-14T15:21:30Z 2012-02-14T15:21:30Z <p>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.</p> http://mathoverflow.net/questions/88433/reference-for-estimation-gaussian-of-the-heat-kernel/88551#88551 Answer by mfolz for Reference for estimation gaussian of the heat kernel mfolz 2012-02-15T20:01:58Z 2012-02-15T20:01:58Z <p>You may also be interested in the following paper of Grigoryan, <a href="http://www.math.uni-bielefeld.de/~grigor/super.pdf" rel="nofollow">Gaussian upper bounds for the heat kernel on arbitrary manifolds</a>, 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 $x_2$. </p>