Rectifiability of sets of finite perimeter in Carnot groups: existence of a tangent hyperplane

We consider sets of locally finite perimeter in Carnot groups. We show that if E is a set of locally finite perimeter in a Carnot group G, then for almost every x in G with respect to the perimeter measure of E, some tangent of E at x is a vertical halfspace. This is a partial extension of a theorem of Franchi-Serapioni-Serra Cassano in step 2 Carnot groups: they have shown that, for almost every x, E has a unique tangent at x, and this tangent is a vertical halfspace.

Leaf superposition property for integer rectifiable currents

We consider the class of integer rectifiable currents without boundary satisfying a positivity condition. We establish that these currents can be written as a linear superposition of graphs of finitely many functions with bounded variation.

Uniqueness of signed measures solving the continuity equation for Osgood vector fields

Nonnegative measure-valued solutions of the continuity equation are uniquely determined by their initial condition, if the characteristic ODE associated to the velocity field has a unique solution. In this paper we give a partial extension of this result to signed measure-valued solutions, under a quantitative two-sided Osgood condition on the velocity field. Our results extend those obtained for log-Lipschitz vector fields by Bahouri and Chemin.

On flows associated to Sobolev vector fields in Wiener spaces: an approach `a la DiPerna-Lions

In this paper we extend the DiPerna-Lions theory on ODEs with Sobolev vector fields to the setting of abstract Wiener spaces.

Existence and Stability for Fokker-Planck equations with log-concave reference measure

We study Markov processes associated with stochastic differential equations, whose non-linearities are gradients of convex functionals. We prove a general result of existence of such Markov processes and a priori estimates on the transition probabilities. The main result is the following stability property: if the associated invariant measures converge weakly, then the Markov processes converge in law. The proofs are based on the interpretation of a Fokker-Planck equation as the steepest descent flow of the relative Entropy functional in the space of probability measures, endowed with the Wasserstein distance. Applications include stochastic partial differential equations and convergence of equilibrium fluctuations for a class of random interfaces.