arXiv subjects

Ambroise Soglo

Publications and source records attributed to Ambroise Soglo.

3 recordsLinked to original sources

arXiv2026

A new Geometric Setting for the Analysis of Partial Differential Equations

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 $D_α$ is defined through a Benamou--Brenier-type dynamical formulation that couples spatial transport with a centered reaction term preserving total mass.We prove that $D_α$ is a genuine metric and establish comparison estimates with the Wasserstein and Aitchison distances. In particular, we show that the topology induced by $D_α$ 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--Savaré, 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.

math.AP↗

arXiv2026

Variable Exponent Wasserstein Spaces: Stability of Entropy Convexity and Modified Rényi Entropy

We study the Wasserstein space $\mathcal{P}(M)$ equipped with a distance $\Wp$ constructed from the Lagrangian $L(x,v)=|v|^{p(x)}$ where $p(x)=2+\varepsilon(x)$ with $\varepsilon$ small. Building on the fundamental work of Lott and Villani on the $K$-geodesic convexity of the Boltzmann entropy in $(\mathcal{P}(M),\Wb)$, we establish a generalized inequality showing that the entropy remains $\left(K - C\|\varepsilon\|_\infty\right)$-convex along $\Wp$-geodesics. We then introduce a modified Rényi entropy that exactly compensates the logarithmic divergence that appears in the expansions of $\Wp^2$, obtaining thus a sharp equivalence that reaveals the Bakry-Émery tensor as the effective curvature in the variable exponent setting. As applications, we derive perturbed versions of the Log-Sobolev and Talagrand inequalities in variable exponent Wasserstein spaces, showing that these fundamental functional inequalities are robust under small perturbations of the transport exponent. This work generalizes the Lott-Villani theorem and its consequences (\emph{J. Lott and C. Villani, Ann. of Math. \textbf{169} (2009), 903-991}) to situations where the transport metric varies spatially.

math.NA↗

arXiv2019

Finsler structure for variable exponent Wasserstein space and gradient flows

In this paper, we propose a variational approach based on optimal transportation to study the existence and unicity of solution for a class of parabolic equations involving $q(x)$-Laplacian operator \begin{equation*}\label{equation variable q(x)} \frac{\partial ρ(t,x)}{\partial t}=div_x\left(ρ(t,x)|\nabla_x G^{'}(ρ(t,x))|^{q(x)-2}\nabla_x G^{'}(ρ(t,x)) \right) .\end{equation*} The variational approach requires the setting of new tools such as appropiate distance on the probability space and an introduction of a Finsler metric in this space. The class of parabolic equations is derived as the flow of a gradient with respect the Finsler structure. For $q(x)\equiv q$ constant, we recover some known results existing in the literature for the $q$-Laplacian operator.

math.AP↗