Relationship between probabilities and partial differential equations

The aim of this article is to show the links that can exist between the theory of stochastic processes and partial differential equations (PDE ). The stochastic processes used are Markov processes, for which we distinguish between discrete-time processes (Markov chains) and continuous-time processes (Markov processes). The main idea is to show that the mathematical expectation of functionals of these processes provides a probabilistic representation of solutions to certain equations. Markov chains will thus be associated with discrete, stationary or evolutionary equations, with various types of edge conditions 1 . Markov processes (continuous-time) will be divided into two classes: continuous processes (diffusions) and jump processes. In paragraph 2 , we introduce the prototype of diffusions (the Wiener process) and show the link with the Laplacian. In paragraph 3 , we give the tools needed to handle diffusions...