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日時: 2024年7月26日(金) 15:30-16:30
Date: July 26, 2024 15:30-16:30
会場:東京大学大学院数理科学研究科 大講義室
Place: auditorium, Graduate School of Math. Sci. Bldg.
Juan Manfredi 氏（University of Pittsburgh）
Juan Manfredi (University of Pittsburgh)
Asymptotic Mean Value Expansions for Solutions to General Elliptic and Parabolic Equations (English)
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, L ́evy, 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 the Monge-Amp`ere and k-Hessian equations and to the p-Laplacian in Euclidean space and the Heisenberg group.