MARK KAC SEMINAR

In this talk, we discuss the stochastic homogenisation process of the Gaussian Free Field (GFF) and the bi-Laplacian field defined in random environments. Both can be characterized in terms of random, non-homogeneous elliptic operators. Under standard assumptions of stochastic homogenisation, we identify the limit fields, respective, as the usual GFF and bi-Laplacian field, up to multiplicative constants. The results hold both as the scaling limit of a discrete and continuous settings for dimension larger than 2. Background in both fractional Gaussian fields and stochastic homogenisation will be presented during the lecture and the viewer is not expected to be familiar with such subjects. This is a joint work with W. Ruszel.

In this talk, we consider one-dimensional continuous-time random walks driven by dynamic random environemts. We require the environment to have good spatial mixing properties, meaning that, if one observes space time boxes that are sufficiently far away in the spatial dimension, the configurations inside these boxes are almost independent. We prove that, provided the random walk is ballistic with a large enough speed, it satisfies a law of large numbers. As an example of application, we prove that ballistic random walks evolving on top of the zero-range process and the assymetric exclusion process satisfy law of large numbers. The proofs are based on multiscale renormalization and might be of independent interest. Joint work with Weberson Arcanjo, Marcelo Hilário, and Renato S. dos Santos.