General Prerequisites: Integration and measure theory, martingales in discrete and continuous time, stochastic calculus. Functional analysis is useful but not essential.

Course Overview: Stochastic analysis and partial differential equations are intricately connected. This is exemplified by the celebrated deep connections between Brownian motion and the classical heat equation, but this is only a very special case of a general phenomenon. We explore some of these connections, illustrating the benefits to both analysis and probability.

Learning Outcomes: The student will have developed an understanding of the deep connections between concepts from probability theory, especially diffusion processes and their transition semigroups, and partial differential equations.

Course Synopsis: Feller processes and semigroups. Resolvents and generators. Hille-Yosida Theorem (without proof). Diffusions and elliptic operators, convergence and approximation. Stochastic differential equations and martingale problems. Duality. Speed and scale for one dimensional diffusions.

Green's functions as occupation densities. The Dirichlet and Poisson problems. Feynman-Kac formula.