Title: Mean Value Expansions for Solutions to General Elliptic and Parabolic Equations
Abstract: Harmonic functions in Euclidean space are characterised by the mean value property and are also obtained as expectations of stopped Brownian motion processes. This equivalence has a long history with fundamental contributions by Doob, Hunt, Ito, Kakutani, Kolmogorov, Lévy, and many others. In this lecture, we will present ways to extend this characterisation to solutions of non-linear elliptic and parabolic equations.
The non-linearity of the equation requires that the rigid mean value property be replaced by asymptotic mean value expansions and the Brownian motion by stochastic games, but the main equivalence remains when formulated with the help of the theory of viscosity solutions. Moreover, this local equivalence also holds on more general ambient spaces like Riemannian manifolds and the Heisenberg group.
I will present examples related the Monge-Ampère and k-Hessian equations and to the p-Laplacian in Euclidean space and the Heisenberg group.
