Partial Differential Equations ans Stochastic Differential Equations Arising in Particle Systems

In this talk, I will introduce a classical example of Particle System: the Simple Exclusion Process. I will give the notion of hydrodynamic limit, which is a Law of Large Numbers for the empirical measure and I will explain how to derive from the microscopic dynamics between particles a partial differential equation describing the evolution of the density profile. For the Simple Exclusion Process, in the Symmetric case $(p=1/2)$ we will get to the heat equation while in the Asymmetric case $(p\\neq 1/2)$ to the Burgers equation. Finally, I will introduce the Central Limit theorem for the empirical measure and the limiting process turns out to be a solution of a stochastic differential equation.