Thierry Coulhon

Isopérimétrie pour les groupes et les variétés

Dans cet article, nous proposons une approche tres directe de differents inegalites isoperimetriques.

Gaussian heat kernel upper bounds via the Phragmén-Lindelöf theorem

We prove that in presence of

On-diagonal lower bounds for heat kernels and Markov chains

L^2

Gaussian lower bounds for random walks from elliptic regularity

Gaussian estimates, so-called Davies-Gaffney estimates, on-diagonal upper bounds imply precise off-diagonal Gaussian upper bounds for the kernels of analytic families of operators on metric measure spaces.

OFF-DIAGONAL HEAT KERNEL LOWER BOUNDS WITHOUT POINCARÉ

pt(x, y) = 1 (4πt)n/2 exp −|x− y| 4t  which shows that pt(x, y) behaves like t− n 2 for fixed x and y as t→ ∞. On other manifolds, its behaviour may be described by different functions of t, depending on the geometry of the manifold. The major question on complete non-compact manifolds is: What geometric terms are adequate to describe the long time behaviour of the heat kernel. Much has been known about upper bounds. The seminal works of Nash [N], Aronson [Ar], Varopoulos [V2], Carlen, Kusuoka, Stroock [CKS] and Davies [D1] have brought the understanding that the uniform upper bounds of the heat kernel are closely related to isoperimetric type inequalities including the Sobolev’s, Nash’s and the logarithmic Sobolev inequalities. More recent works [G2], [Carr], [C2] revealed the importance of a Faber-Krahn type inequality and of a generalized Nash inequality (see also the surveys [G4] and [C3]). The situation is quite different with lower bounds of the heat kernel. Until recently, only two methods were known: • a comparison type theorem ([DGM], [ChY]) which requires a pointwise restriction on the Ricci curvature;

NOYAU DE LA CHALEUR ET DISCRETISATION D'UNE VARIETE RIEMANNIENNE

Abstract We give a direct approach to off-diagonal lower bounds for reversible Markov chains on infinite graphs with regular volume growth and Poincare inequalities, by using ideas that go back to De Giorgi and Morrey.

Riesz transform and perturbation

On a manifold with polynomial volume growth satisfying Gaussian upper bounds of the heat kernel, a simple characterization of the matching lower bounds is given in terms of a certain Sobolev inequality. The method also works in the case of so-called sub-Gaussian or sub-diffusive heat kernels estimates, which are typical of fractals. Together with previously known results, this yields a new characterization of the full upper and lower Gaussian or sub-Gaussian heat kernel estimates.

Riesz transforms for p > 2

One considers a bounded geometry non-compact Riemannian manifold, and the graph obtained by discretizing this manifold. One shows that the uniform decay for large time of the heat kernel on the manifold and the decay of the standard random walk on the graph are the same, in the polynomial scale. As a consequence, such a large time behaviour of the heat kernel is invariant under rough isometries.

Pointwise Estimates for Transition Probabilities of Random Walks on Infinite Graphs

We prove that, on a complete noncompact Riemannian manifold with bounded geometry, the Lp boundedness of the Riesz transform, for p>2, is stable under a quasi-isometric and integrable change of metric. As an intermediate step, we treat the case of weighted divergence form operators in the Euclidean space.

Inégalités de Gagliardo-Nirenberg pour les semi-groupes d'opérateurs et applications

Abstract We show that, on a complete non-compact Riemannian manifold, the Riesz tranforms are bounded on L p for p>2 if suitable estimates for the heat kernel on 1 -forms hold true.