Probability and Stochastic Analysis Seminar  RSS

Tal Orenshtein 19/05/2021, 17:00 — 18:00 — Room P3.10, Mathematics Building Online
, Weierstrass Institute and TU Berlin

Rough walks in random environment

Random walks in random environment (RWRE) have been extensively studied in the last half-century. Functional central limit theorems (FCLT) hold in some prototypical classes such the reversible and the ballistic ones. The latter are treated using rather different techniques; Kipnis-Varadhan's theory for additive functionals of Markov processes is applicable in the reversible case whereas the main feature exploited in the ballistic class is a regeneration structure. Rough path theory is a deterministic theory which extends classical notions of integration to singular integrators in a continuous manner. It typically provides a framework for pathwise solutions of ordinary and partial stochastic differential equations driven by a singular noise. In the talk we shall discuss FCLT for additive functionals of Markov processes and regenerative processes lifted to the rough path space. The limiting rough path has two levels. The first one is the Brownian motion, whereas in the second we see a new feature: it is the iterated integral of the Brownian motion perturbed by a deterministic linear function called the area anomaly. The aforementioned classes of RWRE are covered as special cases. The results provide sharper information on the limiting path. In addition, the construction of new examples for SDE approximations is an immediate application.

Based on collaborations (some still in progress) with Johannes Bäumler, Noam Berger, Jean-Dominique Deuschel, Olga Lopusanschi, Nicolas Perkowski and Martin Slowik.

References

  1. Additive functionals as rough paths, with Jean-Dominique Deuschel and Nicolas Perkowski, Ann. Probab. 49(3): 1450-1479 (May 2021). DOI: 10.1214/20-AOP1488.
  2. Ballistic random walks in random environment as rough paths: convergence and area anomaly, with Olga Lopusanschi, ALEA, Lat. Am. J. Probab. Math. Stat. 18, 945–962 (April 2021) DOI: 10.30757/ALEA.v18-34.
  3. Rough invariance principle for delayed regenerative processes, arXiv:2101.05222.

link

Except for a few of the oldest sessions these are from the Seminário de Probabilidade e Mecânica Estatística at IMPA which is co-sponsored by several institutions, in particular Instituto Superior Técnico.