Marches aléatoires unipotentes dans les espaces homogènes // Unipotent random walks on homogeneous spaces

L'objectif de cette thèse est l'étude des propriétés de récurrence et d'équidistribution des marches aléatoires unipotentes dans les espaces homogènes de volume fini. Etant donné un groupe algébrique réel G (par exemple SL(d,R)) et un réseau L de G (par exemple SL(d,Z)), on cherchera donc à décrire le comportement asymptotique dans l'espace X= G/L des trajectoires de la chaine de Markov dont les incréments s'obtiennent en se déplaçant d'un point x de X à un point hx, où h est un élément aléatoire de G dont le support de la loi est contenu dans un sous-groupe unipotent H de G.
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The purpose of this PhD is the study of recurrence and equidistribution properties of unipotent random walks on finite volume homogeneous spaces. Given a real algebraic group G (for example SL(d,R)) and a lattice L in G (for example SL(d,Z)), we will therefore investigate the asymptotic behaviour in the space X=G/L of the trajectories of the Markov chain whose increments are obtained by moving from a point x in X to a point hx, where h is a random element in G whose law has support in a unipotent subgroup H of G.

The study of the asymptotic behaviour of the action of subgroups H of G in X=G/L, often called the theory of dynamics in homogenous spaces, finds its origins in questions from number theory. It grew in particular after the proof by Margulis of Oppenheim's conjecture in 1986. In the 90's, it witnessed a great development thanks to Ratner's theory which completely describes the closures of H-orbits when H is spanned by unipotent elements. Then, Shah conjectured that these results should be extendable to the case where the Zariski closure of H was spanned by unipotent elements. The case where this Zariski closure is semisimple is now well understood thanks to the work of Benoist and Quint. More recently, Eskin and Lindenstrauss described the stationary measures of random walks on X which are associated with a probability measure on G whose support spans a subgroup of G, with Zariski closure spanned by unipotent elements. The link between this metric result and the description of orbit closures remains to be established in general. The goal of this PhD is to develop a better understanding of the random walks which appear in this framework, in particular in case H has weak expansion properties.
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Début de la thèse : 01/10/2025

Contrat doctoral

Concours pour un contrat doctoral

Université de Montpellier

166 I2S - Information, Structures, Systèmes

Le candidat doit avoir des connaissances profondes en mathématiques fondamentales, en particulier en géométrie différentielle, théorie des groupes et théorie des probabilités.
The applicant needs to have a deep knowledge of fundamental mathematics, in particular in differential geometry, group theory and probability theory.