Appears in collection : 2017 - T2 - Stochastic Dynamics out of Equilibrium

In the first lecture I will discuss the notion of hydrodynamic limit and as example we will present the proof for the symmetric simple exclusion process in contact with slow or fast reservoirs. In this process, particles jump in the bulk to one of its nearest neighbor sites with rate 1/2. At the boundaries particles are injected or removed at a rate which depends on a parameter that can be chosen in such a way that the boundary dynamics is either slow or fast. We will explore the hydrodynamic limit scenario and we will see a phase transition for the heat equation with different types of boundary conditions. This lecture is based on the article at arxiv. 1407. 7918. In the second lecture we will consider a model similar to the one of the first lecture but in which particles can jump in the bulk according to a symmetric transition probability rate which allows long jumps, but has finite variance. Moreover, infinitely many reservoirs are added at each end point of the bulk and now particles can be injected or removed from any site of the bulk. As in the previous lecture, the rates of injection and removal depend on a parameter which turn the boundary dynamics slowed of fast depending on the range of this parameter. We will analyze the hydrodynamic limit and we will see a phase transition of a reaction diffusion equation with different types of boundary conditions. This lecture is based on the article arxiv. 1702. 07216. In the third and last lecture we will analyze the model of the previous lecture, for a certain choice of the transition probability rate which has infinite variance and in the hydrodynamic limit we will get a fractional heat equation with Dirichlet boundary conditions written in terms of the regional fractional Laplacian.