Superposition principle and schemes for measure differential equations

Abstract

Measure Differential Equations (MDE) describe the evolution of probability measures driven by probability velocity fields, i.e. probability measures on the tangent bundle. They are, on one side, a measure-theoretic generalization of ordinary differential equations; on the other side, they allow to describe concentration and diffusion phenomena typical of kinetic equations. In this paper, we analyze some properties of this class of differential equations, especially highlighting their link with nonlocal continuity equations. We prove a representation result in the spirit of the Superposition Principle by Ambrosio-Gigli-Savaré, and we provide alternative schemes converging to a solution of the MDE, with a particular view to uniqueness/non-uniqueness phenomena.

Dissipative probability vector fields and generation of evolution semigroups in Wasserstein spaces

“…We remind that the relation between our solution and the weaker notion studied in [Pic19] was exploited in Section 6.5. Here, we conclude with a further remark coming from the connections between our approximating scheme proposed in (EE) and the schemes proposed in [Cam+21] and [Pic19].…”

“…If we assume that the MPVF F is a sequentially closed subset of P sw 2 (TX) with convex sections, we are able to provide a stronger result showing a particular property satisfied by the solutions of (6.1) (see Theorem 6.27). This is called barycentric property and it is strictly connected with the weaker definition of solution discussed in [Pic19;Pic18;Cam+21]. We first introduce a directional closure of F along smooth cylindrical deformations.…”

“…In these papers, the author aims to describe the evolution of curves in the space of probability measures under the action of a so called probability vector field F (see Definition 4.1 and Remark 4.2). However, as exploited also in [Cam+21], the definition of solution to (6.1) given in [Pic19; Pic18; Cam+21] is too weak and it does not enjoy uniqueness property which is recovered only at the level of the semigroup through an approximation procedure.…”

“…where cl(F) is the sequential closure of the graph of F in the strong-weak topology of P sw 2 (TX) (see [NS21] and Section 2.2 for more details; in fact, a more restrictive "directional" closure could be considered, see (6.34)) and co(cl(F) [µ]) denotes the closed convex hull of the given section cl(F) [µ]. However, even in the case of a single valued map, (1.12) is not enough to characterize the limit solution, as it has been shown by an interesting example in [Pic19;Cam+21] (see also the gradient flow of Example 6.34).…”

Measure Differential Equation with a Nonlinear Growth/Decay Term

“…Roughly speaking, if (x, v) belongs to the support of V [µ], µ at position x evolves with velocity v. We refer to Section 5 and [25,Section 7.1] for examples of Measure Vector Fields and to [5] for recent work on numerical schemes for MDE.…”