Mini-cours

Mini-cours d'Amine Asselah

Titre : Folding a transient random walk.

Résumé : The motivation for the material presented in this course comes from the phenomenon of polymer folding, which we simply think of as (the many ways of) forcing a random walk, say on the three-dimensional lattice, to localize in a small region. Thus, we aim at presenting simple and nonetheless useful tools to study the simple random walk. Our plan is

(i) to estimate the probability a random walk visits uniformly some domains, using simple, though original arguments.

(ii) to explain how the Newtonian capacity enters the game and highlight its central role, and properties.

(iii) to analyze some folding occurrences.

We should assume no prerequisite, and should provide all interesting proofs.

Mini-cours de Jean Bertoin

Titre: Noise reinforcement for Lévy processes

Résumé: In a step reinforced random walk, at each integer time and with a fixed probability p in (0,1), the walker repeats one of his previous steps chosen uniformly at random, and with complementary probability 1-p, the walker makes an independent new step with a given distribution.

Examples in the literature include the so-called elephant random walk and the shark random swim. We consider here a continuous time analog, when the random walk is replaced by a Lévy process. For sub-critical (or admissible) memory parameters p < p_c, where p_c is related to the Blumenthal-Getoor index of the Lévy process, we construct a noise reinforced Lévy process. Our main result shows that the step-reinforced random walks corresponding to discrete time skeletons of the Lévy process, converge weakly to the noise reinforced Lévy process as the time-mesh goes to 0.

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