Nonlinear Stochastic Partial Differential Equations
Nonlinear stochastic partial differential equations of first- and second-order are used to describe models in phase transitions, patwhise stochastic control theory, and mean-field games. Examples are Hamilton-Jacobi, Hamilton-Jacobi-Bellman/Isaacs, level-set equations and scalar conservation laws.The study of such partial differential equations requires a novel approach to define solutions as well as to develop a well-posedness theory. In this lecture I describe the context in which these equations arise, explain the major difficulties and discuss the new notions. If time permits, I will also present results about the qualitative (stochastic) properties of the solutions.