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Noyaux potentiels associés aux marches aléatoires sur les espaces homogènes. Quelques exemples clefs dont le groupe affine (Potential kernels associated to random walks on homogeneous spaces. Some key examples of the affine group). (French) Zbl 0571.60076
Théorie du potentiel, Proc. Colloq. J. Deny, Orsay/France 1983, Lect. Notes Math. 1096, 223-260 (1984).
[For the entire collection see Zbl 0543.00004.]
The paper is an attempt to study random walks on homogeneous spaces of non unimodular groups. With the aid of the ”affine group” of the real line, it is possible to exhibit new striking features as: - Harris recurrent Markov chains on \({\mathbb{R}}\) (viewed as \((ax+b)-\hom ogeneous\) space) admitting positive non constant harmonic functions. - New classes of invariant measures. It is worth noting that for the first type of example the author gets rid of any spread-out condition, making essential use of the Choquet-Deny theorem on \({\mathbb{R}}^*_+\).
Reviewer: O.Gebuhrer

MSC:
60G50 Sums of independent random variables; random walks
60J10 Markov chains (discrete-time Markov processes on discrete state spaces)
31C15 Potentials and capacities on other spaces
60B15 Probability measures on groups or semigroups, Fourier transforms, factorization