Aula 1B1, pal. RM002
Maria Gordina (University of Connecticut)
Stochastic analysis and geometric functional inequalities
Abstract: We start by recalling that on a Euclidean space there is a connection between the spectrum of the Laplacian and the speed of heat diffusion, which leads to several functional inequalities, such as Poincare, Nash etc. Moving to a curved space, we see that the geometry of the underlying space plays an important role in such an analysis. If, in addition, the state space is infinite-dimensional, the log-Sobolev inequality becomes a useful fact which can be applied to describe entropic convergence of the heat flow to an equilibrium. A probabilistic point of view comes from a path integral representation of the heat flow for stochastic differential equations driven by a Brownian motion. In particular, we will discuss how the Cameron-Martin-Girsanov type theorem is related to certain functional inequalities. The talk will review recent advances in the field, including elliptic and hypo-elliptic settings over both finite- and infinite-dimensional spaces.