Abstract:
Let (M, g, o) be a pointed Riemannian manifold and W(M) := {σ ∈C ([0, 1], M ) | σ (0) = o} be the corresponding path space. A canonical measure v on W(M) is the Wiener measure, i.e. the law of Brownian motion M. It is natural to develop differential structures that are compatible with Wiener measure. In general, this is challenging because W(M) is nonlinear and infinite dimensional. In this talk we will first report the development of differential geometry in path spaces in the last 30 years. In particular, we will prove an integration by parts formula for a non-adapted Cameron-Martin vector field on W(M) which is obtained by minimizing an energy that includes the damping effect of Ricci curvature. If time permits, we will also introduce two natural dynamics on path (loop) spaces: “Ornstein-Uhlenbeck” like process and Langevin dynamic and present some related open questions.
Biography:
Zhehua Li is currently a Postdoc Fellow at NYU Shanghai. He obtained his Ph.D. from University of California, San Diego under the supervision of Prof. Bruce Driver. His research interests lie in the intersection of probability and differential geometry.
Seminar by the NYU-ECNU Institute of Mathematical Sciences at NYU Shanghai