Abstract:
(joint work with Martin Huesmann)
This talk is concerned with the geometry of configuration spaces, the natural state
spaces of infinite particle systems. We consider a system of independent Brownian
particles on a manifold M. The configuration space Conf(M) of locally finite
counting measures on M inherits a differentiable structure from the base space in
such a way that the natural diffusion on this infinite-dim. Riemannian manifold is
the particle system. We are interested in curvature properties of the manifold
Conf(M). More precisely, we will show that various manifestations of lower bounded
(Ricci-) curvature extend from the base space to the configuration space. This
includes gradient estimates for heat semigroup and convexity of the entropy along
L^2 transport geodesics. Our motivation comes from interacting particle systems and
the paradigm that lower curvature bounds are useful in the study of the trend to
equilibrium. Understanding the geometry of non-interacting systems is a first step
in this direction.