Probabilistic analysis of boundary conditions for parabolic and elliptic partial differential equations
The objective of this part of the course is to establish probabilistic representations of solutions of parabolic or elliptic partial differential equations with domain edge conditions of the Dirichlet or Neumannn type. We will start by showing some properties of the time of reaching the edge of a domain by the solution of a stochastic differential equation. We will then show the Itô-Tanaka formula and study the local time of a diffusion process at the edge of a regular domain. Then, we will study in detail the solutions of stochastic differential equations stopped at the edge of a domain and we will establish existence and uniqueness results for reflected stochastic differential equations. It will be shown that these probabilistic objects allow the construction of regular solutions to parabolic or elliptic partial differential equations with Dirichlet or Neumannn boundary conditions. A particular application will be treated: the localization in bounded domains of elliptic or parabolic problems posed in all space. Finally, we will establish convergence speed results for approximation schemes of stopped or reflected diffusion processes.