In this paper, we introduce a hybrid metric geometry on the space of absolutely continuous probability densities that combines optimal transport (Wasserstein geometry) and log-ratio composition (Aitchison geometry). The hybrid distance $D_\alpha$ is defined through a Benamou--Brenier-type dynamical formulation that couples spatial transport with a centered reaction term preserving total mass.We prove that $D_\alpha$ is a genuine metric and establish comparison estimates with the Wasserstein and Aitchison distances. In particular, we show that the topology induced by $D_\alpha$ is stronger than the narrow topology and weaker than the supremum topology generated by the Wasserstein and Aitchison metrics. We further prove that the metric space is geodesic. Within this framework, we develop the foundations of a gradient flow theory in the sense of Ambrosio--Gigli--Savar\'e, including the characterization of absolutely continuous curves, metric derivatives, metric slopes, and formal Jordan--Kinderlehrer--Otto schemes. We also investigate hybrid barycenters and their connections with Wasserstein barycenters and Aitchison barycenters. Finally, we discuss several partial differential equations, including logistic diffusion, Allen--Cahn equations with log-ratio constraints, and chemotaxis models with logarithmic growth, as formal gradient flows associated with the hybrid geometry, and compare the proposed framework with the Wasserstein--Fisher--Rao metric.
We develop a global differential calculus on the $L^2$-Wasserstein space over a closed Riemannian manifold, based on derivations of cylinder functions rather than on the standard pointwise approach. Within this framework we define some fundamental geometric tools. In particular, we show that the Levi-Civita connection on the base manifold lifts to the unique torsion-free connection compatible with the'extended'Otto metric. The corresponding Riemann tensor is exactly the lift of the base Riemann tensor, which shows that - in this framework - the correction terms of the classical gradient formalism are not intrinsic curvature terms, but arise from the projection onto the measure-dependent gradient distribution. This allows us to revisit with a global and purely differential approach the smooth computations by J. Lott, Comm. Math. Phys., 277(2):423-437, 2007, reaching partially different conclusions.
We introduce a notion of continuity equation on metric spaces that is capable of describing curves of probability measures which are absolutely continuous, and more generally of bounded variation (BV), with respect to the 1-Wasserstein distance. This continuity equation is based on a notion of measure-valued derivations, whose basic theory is also developed in this paper. On $\mathbb{R}^n$, our formulation is consistent with the continuity equation with singular flux introduced by Almi--Rossi--Savar\'e (arXiv:2506.15333), including the corresponding notion of minimal solutions. In this work, we characterize BV-curves in the space of probability measures equipped with the (extended) 1-Wasserstein distance as those curves satisfying the continuity equation with a measure-valued derivation of finite mass. To this aim, we extend our previous work (Calc.Var.(2024)63:16) on probabilistic representations on BV-curves and construct from them measure-valued derivations (resp. flux measures) on geodesic metric spaces (resp. on $\mathbb{R}^n$).
The goal of this work is to introduce a notion of mean curvature for level sets of functions in non-smooth spaces with Ricci curvature bounded below, and to prove that it satisfies sharp geometric inequalities. More precisely, we define a suitable Willmore functional $\mathcal{W}$ on Sobolev functions, whose domain of finiteness is dense in $L^p$ for any $1\le p<\infty$. For any function with finite Willmore energy, we show that almost all of its level sets admit a mean curvature vector satisfying the natural integration by parts formula with respect to the tangential divergence. As a main application, we show that in ${\rm RCD}(0,N)$ spaces with Euclidean volume growth, almost every level set of the electrostatic potential possesses a mean curvature vector in the above sense. Furthermore, we prove that this vector satisfies the same sharp Willmore inequality as in the smooth setting, alongside rigidity and almost-rigidity statements. Finally, as a technical tool, we generalize the sharp isocapacitary inequality to the non-smooth setting.
Given a complete doubling metric measure space $(X,\rho,\mu)$ supporting a Poincar\'e inequality, we prove weak-type characterizations of the Sobolev space $\dot{W}^{1,p}(\mu)$ and the space of functions of bounded variation, achieving a full analogy in general Poincar\'e spaces with the Euclidean results of Brezis et al. [Anal. PDE 17 (2024), 943-979]. The main novelty is that the finiteness of a weak-type norm, which only refers to differences or mean oscillations of $f$ without assuming any smoothness a priori, already guarantees the membership of $f$ in the relevant Sobolev or BV space. This distinguishes our contribution from the recent work of F. Dai et al. [Adv. Math. 502 (2026), Paper No. 111153], where the related norm-equivalence was obtained under the a priori Lipschitz assumption on $f$. A key intermediate step in our approach is a new localized Bourgain-Brezis-Mironescu type characterization. More precisely, we prove that, if $p\in(1,\infty)$ and $\gamma\in\mathbb R\setminus\{0\}$, then, for any $f\in L^1_{\mathrm{loc}}(\mu)$, \begin{equation*}\tag{$*$} \|f\|_{\dot W^{1,p}(\mu)} \sim \|\rho^{-1}\phi^{-\gamma}F\|_{L^{p,\infty}(\phi^{\gamma p}V^{-1})}, \qquad F\in\{\Delta f,m_f\},\quad \phi\in\{\rho,V\}, \end{equation*} where the homogeneous Sobolev space $\dot{W}^{1,p}(\mu)$ is defined by the minimal $p$-weak upper gradient and, for any $x,y\in X$, we denote $V(x,y):=\mu(B(x,\rho(x,y)))$ and $\Delta f(x,y):=|f(x) - f(y)|$, and $m_f(x,y)$ is the mean oscillation of $f$ on the ball $B(x,\rho(x,y))$. For $p=1$, the equivalence $(*)$ holds after replacing $\|f\|_{\dot W^{1,1}(\mu)}$ by a bounded variation norm and restricting the parameters to the optimal ranges $\gamma\in(-\infty,-1)\cup(0,\infty)$ for $\phi=\rho$ or $\gamma\in (-\infty,-\frac1d)\cup(0,\infty)$ for $\phi=V$, where $d\in(0,\infty)$ is the lower dimension of $X$.
T. Hytönen, Dachun Yang, Wen Yuan et al.· 0 citations
Optimal transport asks for the cheapest way of rearranging one distribution of mass into another, and the answer supplies a metric on probability measures that respects the geometry of the underlying space. This review traces the subject from the formulation posed by Monge in 1781, through the linear relaxation introduced by Kantorovich, to the characterisation of optimal maps as gradients of convex functions. The resulting Wasserstein distances behave differently from divergences based on pointwise comparison of densities: they remain finite and informative for measures with disjoint support, and the shortest paths they induce displace mass rather than fade one profile into another. The dynamic formulation connects these paths to a fluid mechanical problem and identifies several diffusion equations as gradient flows of familiar energies. Numerical work has changed the field. Entropic regularisation reduces the linear program to matrix scaling iterations that are simple, parallel and differentiable, at the cost of blurring the transport plan by an amount controlled by the regularisation parameter. Semi-discrete and sliced methods offer alternatives when accuracy matters more than speed. Applications now include image retrieval, generative modelling, domain adaptation and the reconstruction of developmental trajectories from single-cell measurements.
Assanu Augustine, Assanu Augustine· International Journal of Pur...· 0 citations
We propose a meshless Lagrangian dynamical method for approximating Wasserstein gradient flows (WGFs). The evolving measure is represented as the pushforward of the initial measure $\mu_0$ through a transport map in the weighted Hilbert space $L^2_{\mu_0}$. We approximate this map in time-dependent linear subspaces of $L^2_{\mu_0}$, whose orthonormal frames are evolved by a Dirac--Frenkel dynamical principle on a Stiefel manifold constrained to a finite-dimensional background space, adaptively constructed via local approximations of the WGF velocity field. We prove that the resulting transport map induces an absolutely continuous curve of probability measures in Wasserstein space, whose velocity is obtained by projecting the exact WGF velocity onto the background space, and we show that the approximation preserves the energy dissipation structure up to the projection error of the velocity. Moreover, for geodesically convex energies, we derive an a posteriori estimate controlling such projection error through the adaptive construction of the background space, yielding as well a bound on the approximation error of the pushforward measure in the Wasserstein metric. Numerical experiments on linear and nonlinear Fokker--Planck equations, porous-medium diffusion, and interaction energies demonstrate the accuracy of the method, its energy-dissipation properties, and the advantages of the adaptive construction.
Isabella Carla Gonnella, O. Mula, F. Pichi et al.· arXiv.org· 0 citations
We use cookies to run the site and, with your consent, for analytics and to show ads.
See our Cookie Policy.
