arXiv subjects

Jonathan Junné

Publications and source records attributed to Jonathan Junné.

8 recordsLinked to original sources

arXiv2026

A counterexample to McKean's conjecture for the Landau-Coulomb equation

We disprove McKean's conjecture, which asserts that the entropy dissipation is monotone nonincreasing along solutions, or equivalently that the entropy is convex in time, for the spatially homogeneous Landau-Coulomb equation. We provide an explicit counterexample which consists of a Maxwellian equilibrium under radially symmetric, smooth perturbation on an annulus of scale $R\gg 1$. Developing the derivative of the entropy dissipation in powers of $R$ reveals that the leading order terms can be positive for a suitably chosen perturbation. Remarkably, the counterexamples are close to equilibrium in relative entropy and standard Sobolev norms. Since the Landau equation arises from the Boltzmann equation in the grazing collisions limit, our counterexample also shows that McKean's conjecture is not true in general for the Boltzmann equation.

math.AP↗

arXiv2026

Asymptotic mean value Laplacian on equiregular sub-Riemannian manifolds

Let $(M,\mathcal{D},g)$ be a smooth equiregular sub-Riemannian manifold equipped with a smooth positive measure $μ$. We study the small-scale limit of the metric-ball mean-value operator \[ A_hf(x)=\frac{1}{h^{2}μ(B(x,h)) }\int_{B(x,h)}(f(q)-f(x))\, dμ(q). \] Exact homogeneity yields the pointwise limit on Carnot groups. On a general equiregular manifold, convergence for every smooth test function is equivalent to convergence of the rescaled horizontal first moments of metric balls in first-kind privileged coordinates. This criterion is independent of $μ$; when it holds, the principal symbol is determined by the normalized second-moment tensor of the tangent unit ball, and the drift satisfies an explicit change-of-measure formula. In step at most two, we verify the criterion by combining a real-analytic finite-jet reduction with tame integration, which rules out oscillation of the normalized moments. We also compute the limit on Lie groups and prove unconditional distributional convergence of the volume-weighted operators on every equiregular manifold.

math.DG↗

arXiv2026

Hydrodynamic limit of the symmetric exclusion process on complete Riemannian manifolds and principal bundles

We prove that the hydrodynamic limit of the symmetric exclusion process (SEP) is a Fokker-Planck equation in the setting of Poisson random neighborhood graphs approximating a weighted Riemannian manifold with Ricci curvature bounded from below. We also consider the lift of the SEP to a principal bundle, and obtain a Fokker-Planck equation with a weighted horizontal Laplacian as its hydrodynamic limit. Both results significantly extend the geometric settings in which one can prove the hydrodynamic limit from duality combined with convergence of the single particle random walk towards a diffusion process.

math.PR↗

arXiv2025

On the existence of solutions to the multi-species Landau equation

We consider the spatially homogeneous Landau equation for multiple species with different masses. As in the single-species case, the singularity of the collision operator is determined by a parameter $γ\in [-3,1]$, where $γ= -3$ corresponds to Coulomb interactions. We prove that if $γ\geq -\sqrt{8}$ in the cross-interaction operators, then there exists a natural multi-species generalization of the Fisher information which is a Lyapunov functional for the multi-species Landau system. On the other hand, we give a counterexample showing that the Fisher information is in general no longer a Lyapunov functional below the threshold $(γ< - \sqrt{8})$ for the two-species system if one species has infinite mass. However, we are able to provide a new method to show global well-posedness, by constructing a different Lyapunov functional based on the spherical Fisher information.

math.AP↗

arXiv2025

Propagation of Velocity Moments for the Magnetized Vlasov-Poisson System with Space-Time Dependent Magnetic Fields

We prove that polynomial velocity moments of solutions to the 2D magnetized Vlasov-Poisson system and the 3D magnetized screened Vlasov-Poisson equation remain finite for all times, provided they are finite initially, even when the external magnetic field $B=B(t,x)$ is space-time dependent. We deduce propagation of regularity, thereby implying the existence of global classical solutions. Moreover, we prove optimal stability estimates in the kinetic-Wasserstein distance on par with the unmagnetised case.

math.AP↗

arXiv2025

Stability Estimates in Kinetic Wasserstein Distances for the Vlasov-Poisson System with Yudovich Density

We investigate the stability of solutions to the Vlasov-Poisson system using the unifying framework of the kinetic Wasserstein distance, introduced by Iacobelli in (Section 4 in Arch. Ration. Mech. Anal. 244 (2022), no. 1, 27-50). This allows us to treat both macroscopic densities that lie in a Yudovich space, as recently considered by Crippa et al. (Theorem 1.6 in Nonlinearity 37 (2024), no. 9, 095015) for the $1$-Wasserstein distance, and higher order Wasserstein distances, for which only bounded macroscopic densities were treated by Iacobelli and the first author (Theorem 1.11 in Bull. Lond. Math. Soc. 56 (2024), 2250-2267). First, we establish an $L^p$-estimate on the difference between two force fields in terms of a suitable nonlinear quantity that controls the kinetic Wasserstein distance between their macroscopic densities. Second, we use this estimate in order to derive a closable Osgood-type inequality for the kinetic Wasserstein distance between two solutions. This enables us to prove our main theorem; for $1 \le p < +\infty$ we show the $p$-Wasserstein stability of solutions to the Vlasov-Poisson system with macroscopic densities belonging to a Yudovich space.

math.AP↗

arXiv2024

Stability estimates for the Vlasov-Poisson system in $p$-kinetic Wasserstein distances

We extend Loeper's $L^2$-estimate relating the electric fields to the densities for the Vlasov-Poisson system to $L^p$, with $1 < p < +\infty$, based on the Helmholtz-Weyl decomposition. This allows us to generalize both the classical Loeper's $2$-Wasserstein stability estimate and the recent stability estimate by the first author relying on the newly introduced kinetic Wasserstein distance to kinetic Wasserstein distances of order $1 < p < +\infty$.

math.AP↗

arXiv2023

Invariance principle for Lifts of Geodesic Random Walks

We consider a certain class of Riemannian submersions $π: N \to M$ and study lifted geodesic random walks from the base manifold $M$ to the total manifold $N$. Under appropriate conditions on the distribution of the speed of the geodesic random walks, we prove an invariance principle; i.e., convergence to horizontal Brownian motion for the lifted walks. This gives us a natural probabilistic proof of the geometric identity relating the horizontal Laplacian $Δ_\H$ on $N$ and the Laplace-Beltrami operator $Δ_M$ on $M$. In particular, when $N$ is the orthonormal frame bundle $O(M)$, this identity is central in the Malliavin-Eells-Elworthy construction of Riemannian Brownian motion.

math.PR↗