Lecture notes on the DiPerna-Lions theory in abstract measure spaces

These notes closely correspond to a series of lectures given by the first author in Toulouse, on the recent extension of the theory of ODE well-posedness to abstract spaces, jointly obtained by the two authors. In the last part, we describe some further developments with respect to the theory of (possibly degenerate) diffusion processes, in a similar setting, contained in the second author's PhD thesis

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

Well posedness of Lagrangian flows and continuity equations in metric measure spaces

We establish, in a rather general setting, an analogue of DiPerna-Lions theory on well-posedness of flows of ODE's associated to Sobolev vector fields. Key results are a well-posedness result for the continuity equation associated to suitably defined Sobolev vector fields, via a commutator estimate, and an abstract superposition principle in (possibly extended) metric measure spaces, via an embedding into $\mathbb{R}^\infty$. When specialized to the setting of Euclidean or infinite dimensional (e.g. Gaussian) spaces, large parts of previously known results are recovered at once. Moreover, the class of ${\sf RCD}(K,\infty)$ metric measure spaces object of extensive recent research fits into our framework. Therefore we provide, for the first time, well-posedness results for ODE's under low regularity assumptions on the velocity and in a nonsmooth context.Comment: Slightly expanded some remarks on the technical assumption (7.11); Journal reference inserte

Almost everywhere well-posedness of continuity equations with measure initial data

The aim of this note is to present some new results concerning "almost everywhere" well-posedness and stability of continuity equations with measure initial data. The proofs of all such results can be found in \cite{amfifrgi}, together with some application to the semiclassical limit of the Schr\"odinger equation

Surface measures and convergence of the Ornstein-Uhlenbeck semigroup in Wiener spaces

We study points of density 1/2 of sets of finite perimeter in infinite-dimensional Gaussian spaces and prove that, as in the finite-dimensional theory, the surface measure is concentrated on this class of points. Here density 1/2 is formulated in terms of the pointwise behaviour of the Ornstein-Uhlembeck semigroup