A gradient flow approach to the Boltzmann equation

We show that the spatially homogeneous Boltzmann equation evolves as the gradient flow of the entropy with respect to a suitable geometry on the space of probability measures which takes the collision process into account. This gradient flow structure allows to give a new proof for the convergence of Kac's random walk to the homogeneous Boltzmann equation, exploiting the stability of gradient flows.Comment: Presentation reworked and streamlined. Variational characterization of the Boltzmann equation simplified using the action of curve without referring to the associated distance function. Discussion of the distance moved to appendix. Additional assumption missing in previous version on moment bounds of order higher than 2 for Kac walk added in Thm 1.

Curvature bounds for configuration spaces

We show that the configuration space over a manifold M inherits many curvature properties of the manifold. For instance, we show that a lower Ricci curvature bound on M implies for the configuration space a lower Ricci curvature bound in the sense of Lott-Sturm-Villani, the Bochner inequality, gradient estimates and Wasserstein contraction. Moreover, we show that the heat flow on the configuration space, or the infinite independent particle process, can be identified as the gradient flow of the entropy.Comment: 34 page

The heat equation on manifolds as a gradient flow in the Wasserstein space

Erbar M. The heat equation on manifolds as a gradient flow in the Wasserstein space. Ann. Inst. Henri Poincaré Probab. Stat. 2010;46(1):1-23

Discrete Ricci curvature bounds for Bernoulli-Laplace and random transposition models

We calculate a Ricci curvature lower bound for some classical examples of random walks, namely, a chain on a slice of the n-dimensional discrete cube (the so-called Bernoulli-Laplace model) and the random transposition shuffle of the symmetric group of permutations on n letters

Gradient flows of the entropy for jump processes

Erbar M. Gradient flows of the entropy for jump processes. Ann. Inst. Henri Poincaré Probab. Stat. 2014;50(3):920-945