# Probability and Statistics Seminar

### 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 $\left(p=1/2\right)$ we will get to the heat equation while in the Asymmetric case $\left(p\ne 1/2\right)$ 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.