Related papers: The superposition principle for the continuity equ…

We establish a general superposition principle for curves of measures solving a continuity equation on metric spaces without any smooth structure nor underlying measure, representing them as marginals of measures concentrated on the…

We extend the result of Lisini (Calc Var Partial Differ Equ 28:85-120, 2007) on the superposition principle for absolutely continuous curves in $p$-Wasserstein spaces to the special case of $p=1$. In contrast to the case of $p>1$, it is not…

We consider the continuity equation $\partial_t \mu_t + \mathop{\mathrm{div}}(b \mu_t) = 0$, where $\{\mu_t\}_{t \in \mathbb R}$ is a measurable family of (possibily signed) Borel measures on $\mathbb R^d$ and $b \colon \mathbb R \times…

We study the geometric structure of the space of random measures $\mathcal{P}_p(\mathcal{P}_p(X))$, endowed with the Wasserstein on Wasserstein metric, where $(X, d)$ is a complete separable metric space. In this setting, we prove a metric…

We study the continuity equation on the metric measure space $(\mathcal{P}_p(X),W_p,Q)$, when $X$ is either the Euclidean space or a compact, oriented, and boundaryless Riemannian manifold, for some suitable reference measure $Q \in…

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…

A metric measure space $(X,d,\mu)$ is said to be $A_{\infty}$ on curves if there exist constants $\tau < 1$ and $\theta > 0$ with the following property. For every $x \in X$, $0 < r \leq \mathrm{diam}(X)$, and a Borel set $S \subset B(x,r)$…

The goal of this paper is to define an evolution equation for a curve of random probability measures $(M_t)_{t\in[0,T]}\subset \mathcal{P}(\mathcal{P}(\mathbb{R}^d))$ associated to a non-local drift $b:[0,T]\times\mathbb{R}^d \times…

We introduce principal curves in Wasserstein space, and in general compact metric spaces. Our motivation for the Wasserstein case comes from optimal-transport-based trajectory inference, where a developing population of cells traces out a…

Let $\mu$ = ($\mu$t)t$\in$R be a 1-parameter family of probability measures on R. In [11] we introduced its ``Markov-quantile''process: a process X= (Xt)t$\in$R that resembles as much as possible the quantile process attached to $\mu$,…

We study measure-valued solutions of the inhomogeneous continuity equation $\partial_t \rho_t + {\rm div}(v\rho_t) = g \rho_t$ where the coefficients $v$ and $g$ are of low regularity. A new superposition principle is proven for positive…

We prove a superposition principle for nonlinear Fokker-Planck-Kolmogorov equations on Euclidean spaces and their corresponding linearized first-order continuity equation over the space of Borel (sub-)probability measures. As a consequence,…

We prove that every one-dimensional locally normal metric current, intended in the sense of U. Lang and S. Wenger, admits a nice integral representation through currents associated to (possibly unbounded) curves with locally finite length,…

We consider a class of nonlocal conservation laws with an interaction kernel supported on the negative real half-line and featuring a decreasing jump at the origin. We provide, for the first time, an existence and uniqueness theory for said…

We study absolutely continuous curves in the adapted Wasserstein space of filtered processes. We provide a probabilistic representation of such curves as flows of adapted processes on a common filtered probability space, extending classical…

We consider a macroscopic quantum system with unitarily evolving pure state $\psi_t\in \mathcal{H}$ and take it for granted that different macro states correspond to mutually orthogonal, high-dimensional subspaces $\mathcal{H}_\nu$ (macro…

Nonnegative measures that are solutions to a transport equation with continuous coefficients have been widely studied. Because of the low regularity of the associated vector field, there is no natural flow since nonuniqueness of integral…

We propose a new method for smoothly interpolating probability measures using the geometry of optimal transport. To that end, we reduce this problem to the classical Euclidean setting, allowing us to directly leverage the extensive toolbox…

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…

We seek to define statistical solutions of hyperbolic systems of conservation laws as time-parametrized probability measures on $p$-integrable functions. To do so, we prove the equivalence between probability measures on $L^p$ spaces and…