Université Paris 6Pierre et Marie Curie Université Paris 7Denis Diderot CNRS U.M.R. 7599 Probabilités et Modèles Aléatoires''

### A ratio limit theorem for recurrent diffusions on manifolds

Auteur(s):

Code(s) de Classification MSC:

• 35K10 General theory of second-order, parabolic equations
• 60Jxx Markov processes

Résumé: It is shown that the heat kernel $p_t$ on a recurrent Riemannian manifold $M$ has the following strong ratio limit property: for all $x,y,z \in M$ and all $s >0$, $p_{t+s}(x,y)/p_t(z,z) \to 1$ as $t \to +\infty$. This gives a positive answer to a conjecture of E. B. Davies. Some generalizations to other recurrent diffusions are also considered.

Mots Clés: Diffusion ; Laplace Beltrami operator ; heat kernel ; ratio limit theorem

Date: 1999-06-01

Prépublication numéro: PMA-505