Abstract

In this talk, I present a partial exclusion process in random environment, a system of random walks where the random environment is obtained by assigning a maximal occupancy to each site of the Euclidean lattice. This maximal occupancy is allowed to randomly vary among sites, and partial exclusion occurs. Under the assumption of ergodicity under translation and uniform ellipticity of the environment, we prove that the quenched hydrodynamic limit is a heat equation with a homogenized diffusion matrix. To this purpose, we exploit the stochastic self-duality property to transfer a homogenization result concerning random walks in the same environment with arbitrary starting points to the particle system. The first part of the talk is based on a joint work with Frank Redig (TU Delft) and Federico Sau (IST Austria). Finally, I will discuss some recent progresses in the understanding of what happens when removing the uniform ellipticity assumption. After recalling some results on the Bouchaud’s trap model, I will show that, when assuming that the maximal occupancies are heavy tailed and i.i.d., the hydrodynamic limit is the fractional-kinetics equation. The second part of the talk is based on an ongoing project with Alberto Chiarini (University of Padova) and Frank Redig (TU Delft).