Abstract
Harmonic functions in Euclidean space are characterized 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, I will present ways to extend this characterization 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 to the Monge-Ampère and k-Hessian equations and to the p-Laplacian in Euclidean space and the Heisenberg group.
Brief Biography
Juan J. Manfredi is a Professor of Mathematics at the University of Pittsburgh, working on Nonlinear PDEs, Subelliptic Analysis and Stochastic Games. He is the author of around 100 papers, with several important contributions to the theory of degenerate elliptic equations and systems, namely of p-Laplacian and infinity-Laplacian type. He received his bachelor's degree from Universidad Complutense de Madrid in 1979 and his PhD from Washington University in St. Louis in 1986. He has served as an Associate Dean and Vice-Provost for undergraduate studies at the University of Pittsburgh. In 2017, he was elected a foreign member of the Royal Norwegian Society of Sciences and Letters.