arXiv subjects

Averil Aussedat

Publications and source records attributed to Averil Aussedat.

4 recordsLinked to original sources

arXiv2026

The Monge-Kantorovich tangent bundle to a measure

We study the tangent bundle $T_μ$ to a Radon measure $μ$ in Euclidean space, introduced by Bouchitté, Champion and Jimenez (2005). Its construction, inspired by Monge-Kantorovich optimal transport theory, involves the duality between Lipschitz functions and the so-called Arens-Eells space, a Banach subspace of distributions obtained by completing the set of balanced signed measures. Precisely, a velocity field $σ$ is $μ$-tangent when the divergence of $σμ$ lies in the Arens-Eells space. The tangent bundle $T_μ$ defined $μ$-almost everywhere provides a local projection that allows to construct a $μ$-tangential gradient operator on Lipschitz functions that is weakly continuous and enjoys integration by parts. In this paper, we introduce a new quantitative estimate involving the tangential and normal components of a given velocity vector field $σ\in L^1_μ(\mathbb{R}^d)$. Specifically, we show that the tangential condition $σ\in T_μ$ holding $μ$ a.e. is equivalent to each of the following two conditions: the convergence in the Arens-Eells space of $h^{-1} \left((id+h σ)_{\#} μ- μ\right)$ as $h \to 0$, and the differentiability of Lipschitz functions along $σ$. Moreover, given a field of non-tangent directions, we construct a Lipschitz function that is not differentiable on a large set, confirming that $T_μ$ agrees with the decomposability bundle of Alberti and Marchese. We finally relate $T_μ$ to the tangent measures of Preiss and survey further properties of the tangential differential calculus.

math.AP↗

arXiv2026

Quantified rigidity of $\mathcal{A}$-free Young measure concentrations

We establish quantitative stability of the De Philippis--Rindler rigidity on conical convex regions. By constructing $\mathcal{A}$-quasiconvex functions with a tailored concavity---which additionally yields a new, elementary proof of the original theorem for constant-rank operators---we prove a spreading inequality that strictly limits the angular concentration of $\mathcal{A}$-free measures. This geometric bound establishes that singular concentrations cannot cluster arbitrarily close to a direction outside the Tartar wave cone. As a consequence, we obtain a new $L^1$ compensated compactness result: $\mathcal{A}$-free sequences with uniformly bounded mass are forced to be equi-integrable, thereby preventing the formation of mass concentrations, provided their targets are asymptotically restricted to cones whose aperture is controlled by a power of the distance to the wave cone of $\mathcal A$.

math.AP↗

arXiv2026

Locality of centred tangent cones in the Wasserstein space

The geometric tangent cone to a probability measure $μ$ is a set of measure-valued applications that are almost geodesics. This is a nonlocal condition, typically lost when conditioning the measure on a given set. We show that if one removes the barycenter of any element of the tangent cone, then the resulting set of centred measure fields is characterized by a local condition. Precisely, centred tangent fields must be concentrated on a family of vector subspaces attached to any point, and these subspaces correspond to the normal spaces to some sets of ``dimension $k$'' on which the measure $μ$ is concentrated.

math.MG↗

arXiv2026

Characterization of measures on the real line that are critically unstable under small shifts

We study the perturbation of a measure $μ\in \mathscr{P}(\mathbb{R})$ consisting in superposing two copies of $μ$, each slightly shifted by a small distance $\pm h$. The difference between $μ$ and its perturbation is measured with a Wasserstein distance. For any $μ$, this distance is bounded from above by $h$. We show that measures for which this critical rate is achieved when $h$ goes to 0 are characterized as the ones giving most of their mass to some particular porous sets. This is used to identify which measures $μ$ on the real line have a 2-Wasserstein tangent cone equal to the set of directions inducing curves with maximal initial speed.

math.OC↗