Abstract:
Random walk in random environment (RWRE) is a model to describe propagation of heat or diffusion of matter through a highly irregular medium. The latter is expressed locally in the model in terms of a random environment according to which the process evolves randomly in time. In a few fundamental classes the phenomenon of homogenization of the media takes place. One way this can be expressed is in the fact that on large scales, the RWRE fluctuates as a Brownian motion with a deterministic covariance matrix given in terms the (law of the) environment.
Rough path theory enables the construction of solutions to SDEs so that the solution map is continuous with respect to the noise. One important application guarantees that if the approximation converges to the noise in the rough path topology, the SDEs driven by the noise approximations converge, in an appropriate sense, to a well-defined SDE different than the original one, so that the correction term is explicit in terms of the noise approximation.
In this talk we shall present our current program, in which one lifts RWRE in various classes to the rough path space and shows a convergence to an enhanced Brownian motion in the rough path topology. Interestingly, the limiting second level of the lifted RWRE may have a linear correction, called area anomaly, which we identify. Except for the immediate application to approximations of SDEs, and potentially to SPDEs, this adds some new information on the RWRE limiting path. Time permitted, we shall elaborate on the tools to tackle these problems. Based on joint works with Olga Lopusanschi, with Jean-Dominique Deuschel and Nicolas Perkowski and with Johaness Bäumler, Noam Berger and Martin Slowik.
Biography:
Tal Orenshtein obtained a Master's Degree in Infinite Combinatorial Topology in 2009 at the Weizmann Institute in Rehovot - Israel, and he obtained a PhD in Probability from the same institution in 2014. Between 2012-2014, he was also a research member of the Probability Group at TU Munich. In 2014-2016 he was a postdoctoral researcher at the Insitut Camille Jordan - Université Lyon 1, and since 2016 he is a postdoctoral researcher in Berlin, working at Humboldt University Berlin, TU Berlin and more recently also at the Weierstrass Institute Berlin.
Seminar by the NYU-ECNU Institute of Mathematical Sciences at NYU Shanghai