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**Heat kernel of a noncompact Riemannian manifold.**
*(English)*
Zbl 0829.58041

Cranston, Michael C. (ed.) et al., Stochastic analysis. Proceedings of the Summer Research Institute on stochastic analysis, held at Cornell University, Ithaca, NY, USA, July 11-30, 1993. Providence, RI: American Mathematical Society. Proc. Symp. Pure Math. 57, 239-263 (1995).

Let \(p(x, y, t)\) be a heat kernel of a noncompact Riemannian manifold. This survey paper discusses connections between the long time and long distance behaviour of the heat kernel and the geometry of the manifold.

From author’s introduction: “The following questions form the framework of the paper.

1. On which manifolds does the heat kernel satisfy the inequality \(p(x, y, t)\leq \text{const}/f(t)\) with a given increasing function \(f\), at least for large \(t\)? Necessary and sufficient conditions for this bound to hold are given in terms of the isoperimetric properties of the manifold.

2. How to describe the behaviour of the heat kernel and its derivatives for a large Riemannian distance \(r=d(x, y)\)? The main tool to answer this question is the detailed analysis of the integral \[ E_m (x, t)= \int_M |\nabla^m_y p|^2 (x, y, t)\exp (r^2/ Dt) dy \] which allows to produce pointwise estimates for the kernel. It turns out that the Gaussian factor in these estimates is not sensitive to the geometry of the manifold.

3) Harnack inequality and double sided estimates. Conditions are given for the Harnack inequality to hold in terms of the Poincaré inequality and the doubling volume property”.

For the entire collection see [Zbl 0814.00017].

From author’s introduction: “The following questions form the framework of the paper.

1. On which manifolds does the heat kernel satisfy the inequality \(p(x, y, t)\leq \text{const}/f(t)\) with a given increasing function \(f\), at least for large \(t\)? Necessary and sufficient conditions for this bound to hold are given in terms of the isoperimetric properties of the manifold.

2. How to describe the behaviour of the heat kernel and its derivatives for a large Riemannian distance \(r=d(x, y)\)? The main tool to answer this question is the detailed analysis of the integral \[ E_m (x, t)= \int_M |\nabla^m_y p|^2 (x, y, t)\exp (r^2/ Dt) dy \] which allows to produce pointwise estimates for the kernel. It turns out that the Gaussian factor in these estimates is not sensitive to the geometry of the manifold.

3) Harnack inequality and double sided estimates. Conditions are given for the Harnack inequality to hold in terms of the Poincaré inequality and the doubling volume property”.

For the entire collection see [Zbl 0814.00017].

Reviewer: B.Goldys (Kensington)

### MSC:

58J35 | Heat and other parabolic equation methods for PDEs on manifolds |

35B40 | Asymptotic behavior of solutions to PDEs |

35K05 | Heat equation |

58J60 | Relations of PDEs with special manifold structures (Riemannian, Finsler, etc.) |

58J65 | Diffusion processes and stochastic analysis on manifolds |

35K10 | Second-order parabolic equations |