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Transition operators on co-compact G-spaces.

Laurent Saloff-CosteWolfgang Woess — 2006

Revista Matemática Iberoamericana

We develop methods for studying transition operators on metric spaces that are invariant under a co-compact group which acts properly. A basic requirement is a decomposition of such operators with respect to the group orbits. We then introduce reduced transition operators on the compact factor space whose norms and spectral radii are upper bounds for the Lp-norms and spectral radii of the original operator. If the group is amenable then the spectral radii of the original and...

On the relation between elliptic and parabolic Harnack inequalities

Waldemar HebischLaurent Saloff-Coste — 2001

Annales de l’institut Fourier

We show that, if a certain Sobolev inequality holds, then a scale-invariant elliptic Harnack inequality suffices to imply its a priori stronger parabolic counterpart. Neither the relative Sobolev inequality nor the elliptic Harnack inequality alone suffices to imply the parabolic Harnack inequality in question; both are necessary conditions. As an application, we show the equivalence between parabolic Harnack inequality for Δ on M , (i.e., for t + Δ ) and elliptic Harnack inequality for - t 2 + Δ on × M .

Stability results for Harnack inequalities

Alexander Grigor'yanLaurent Saloff-Coste — 2005

Annales de l’institut Fourier

We develop new techniques for proving uniform elliptic and parabolic Harnack inequalities on weighted Riemannian manifolds. In particular, we prove the stability of the Harnack inequalities under certain non-uniform changes of the weight. We also prove necessary and sufficient conditions for the Harnack inequalities to hold on complete non-compact manifolds having non-negative Ricci curvature outside a compact set and a finite first Betti number or just having asymptotically...

Random walks on finite rank solvable groups

Ch. PittetLaurent Saloff-Coste — 2003

Journal of the European Mathematical Society

We establish the lower bound p 2 t ( e , e ) exp ( t 1 / 3 ) , for the large times asymptotic behaviours of the probabilities p 2 t ( e , e ) of return to the origin at even times 2 t , for random walks associated with finite symmetric generating sets of solvable groups of finite Prüfer rank. (A group has finite Prüfer rank if there is an integer r , such that any of its finitely generated subgroup admits a generating set of cardinality less or equal to r .)

The logarithmic Sobolev constant of some finite Markov chains

Guan-Yu ChenWai-Wai LiuLaurent Saloff-Coste — 2008

Annales de la faculté des sciences de Toulouse Mathématiques

The logarithmic Sobolev constant is always bounded above by half the spectral gap. It is natural to ask when this inequality is an equality. We consider this question in the context of reversible Markov chains on small finite state spaces. In particular, we prove that equality holds for simple random walk on the five cycle and we discuss assorted families of chains on three and four points.

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