Searcharxiv⌕ Search

arXiv subjects

Yann Chaubet

Publications and source records attributed to Yann Chaubet.

11 recordsLinked to original sources
arXiv2026

Geodesic L\'evy flights on Zoll surfaces

We study the mean first capture time of isotropic L\'evy flights on Zoll surfaces, namely the expected time for a geodesic L\'evy process to reach a shrinking geodesic ball. While the leading-order asymptotics are universal, we prove that the first correction term encodes subtle geometric information. More precisely, it is completely determined by the local singularity type of the conjugate locus, quantified by the degree of the conjugate point. This yields a hierarchy of asymptotic regimes governed by the L\'evy exponent.

math.DG↗
arXiv2025

Minimax spectral estimation of weighted Laplace operators

Given $n$ i.i.d. observations, we study the problem of estimating the spectrum of weighted Laplace operators of the form $\Delta_f=\Delta + \alpha \nabla \log f\cdot \nabla$, where $f$ is a positive probability density on a known compact $d$-dimensional manifold without boundary and $\alpha\in \mathbb{R}$ is a hyperparameter. These operators arise as continuum limits of graph Laplacian matrices and provide valuable geometric information on the underlying data distribution. We establish the exact minimax rates of estimation for this problem, by exhibiting two different rates of convergence for eigenfunctions and eigenvalues. When $f$ belongs to a H\"older-Zygmund class $\mathscr{C}^s$ of regularity $s\geqslant 2$, the eigenfunctions can be estimated with respect to the $\mathrm{L}^q$-norm ($q\geqslant 1$) via plug-in methods at the minimax rate $n^{-\frac{s+1}{2s+d}}$ for $d\geqslant 3$ (with different rates for $d\leqslant 2$). Moreover, eigenvalues can be estimated at the minimax rate $n^{-\frac{4s}{4s+d}}+n^{-\frac 12}$. In the regime $s>\frac d4$, we further show that asymptotically efficient estimators exist. We also present a general framework for estimating nonlinear functionals over H\"older-Zygmund spaces, with potential applications to a broad class of statistical problems.

math.ST↗
arXiv2024

Nonlinear chaotic Vlasov equations

In this article, we study nonlinear Vlasov equations with a smooth interaction kernel on a compact manifold without boundary where the geodesic flow exhibits strong chaotic behavior, known as the Anosov property. We show that, for small initial data with finite regularity and supported away from the null section, there exist global solutions to the nonlinear Vlasov equations which weakly converge to an equilibrium of the free transport equation, and whose potential strongly converges to zero, both with exponential speed. Central to our approach are microlocal anisotropic Sobolev spaces, originally developed for studying Pollicott-Ruelle resonances, that we further refine to deal with the geometry of the full cotangent bundle, which paves the way to the analysis of nonlinear Vlasov equations.

math.AP↗
arXiv2024

Poincaré series for surfaces with boundary

We show that the Poincaré series counting orthogeodesics of a negatively curved surface with totally geodesic boundary extends meromorphically to the whole complex plane, as well as the series counting geodesic arcs linking two points; we also give their value at zero.

math.DG↗
arXiv2024

Dynamical zeta functions for billiards

Let $D \subset {\mathbb R}^d,\: d \geqslant 2,$ be the union of a finite collection of pairwise disjoint strictly convex compact obstacles. Let $μ_j \in {\mathbb C},\: {\rm Im}\: μ_j > 0,$ be the resonances of the Laplacian in the exterior of $D$ with Neumann or Dirichlet boundary condition on $\partial D$. For $d$ odd, $u(t) = \sum_j e^{i |t| μ_j}$ is a distribution in $ \mathcal{D}'({\mathbb R} \setminus \{0\})$ and the Laplace transforms of the leading singularities of $u(t)$ yield the dynamical zeta functions $η_{\mathrm N},\: η_{\mathrm D}$ for Neumann and Dirichlet boundary conditions, respectively. These zeta functions play a crucial role in the analysis of the distribution of the resonances. Under the non-eclipse condition (1.1), for $d \geqslant 2$ we show that $η_{\mathrm N}$ and $η_\mathrm D$ admit a meromorphic continuation to the whole complex plane. In the particular case when the boundary $\partial D$ is real analytic, by using a result of Fried (1995), we prove that the function $η_\mathrm{D}$ cannot be entire. Following the result of Ikawa (1988), this implies the existence of a strip $\{z \in {\mathbb C}: \: 0 < {\rm Im}\: z \leqα\}$ containing an infinite number of resonances $μ_j$ for the Dirichlet problem. Moreover, for $α\gg 1$ we obtain a lower bound for the resonances lying in this strip.

math.DS↗
arXiv2023

Combinatorial zeta functions counting triangles

In this paper, we compute special values of certain combinatorial zeta functions counting geodesic paths in the (n-1)-skeleton of a triangulation of a n-dimensional manifold. We show that they carry a topological meaning. As such, we recover the first Betti number and L2-Betti number of compact manifolds, and the linking number of pairs of null-homologous knots in a 3-manifold. The tool to relate the two sides (counting geodesics/topological invariants) are random walks on higher dimensional skeleta of the triangulation.

math.GT↗
arXiv2022

Geodesic Lévy flights and expected stopping time for random searches

We give an analytic description for the infinitesimal generator constructed by Applebaum-Estrade for Lévy flights on a broad class of closed Riemannian manifolds including all negatively-curved manifolds, the flat torus and the sphere. Various properties of the associated semigroup and the asymptotics of the expected stopping time for Lévy flight based random searches for small targets, also known as the narrow capture problem, are then obtained using our newfound understanding of the infinitesimal generator. Our study also relates to the Lévy flight foraging hypothesis in the field of biology as we compute the expected time for finding a small target by using the Lévy flight random search. A similar calculation for Brownian motion on surfaces was done in [arXiv:2209.12425].

math.PR↗
arXiv2022

Resolvent of vector fields and Lefschetz numbers

Dynamical series such as the Ruelle zeta function have become a staple in the study of hyperbolic flows. They are usually analyzed by relating them to the resolvent of the vector field. In this paper we give the general form of such relations, which involves the intersection of the kernel of said resolvent with integration currents. Our formula is actually valid for any smooth flow, not necessarily hyperbolic. As an application, we introduce certain dynamical series that had not appeared before. Finally, we compute their value at zero, and their relation with topological invariants.

math.DS↗
arXiv2021

Closed geodesics with prescribed intersection numbers

Let $(Σ, g)$ be a closed, oriented, negatively curved surface, and fix pairwise disjoint simple closed geodesics $γ_{\star,1}, \dots γ_{\star, r}$. We give an asymptotic growth as $L \to +\infty$ of the number of primitive closed geodesic of length less than $L$ intersecting $γ_{\star,j}$ exactly $n_j$ times, where $n_1, \dots, n_r$ are fixed nonnegative integers. This is done by introducing a dynamical scattering operator associated to the surface with boundary obtained by cutting $Σ$ along $γ_{\star,1}, \dots, γ_{\star, r}$ and by using the theory of Pollicott-Ruelle resonances for open systems.

math.DS↗
arXiv2019

Dynamical torsion for contact Anosov flows

We introduce a new object, the dynamical torsion, which extends the potentially ill-defined value at $0$ of the Ruelle zeta function of a contact Anosov flow twisted by an acyclic representation of the fundamental group. We show important properties of the dynamical torsion: it is invariant under deformations among contact Anosov flows, it is holomorphic in the representation and it has the same logarithmic derivative as some refined combinatorial torsion of Turaev. This shows that the ratio between this torsion and the Turaev torsion is locally constant on the space of acyclic representations. In particular, for contact Anosov flows path connected to the geodesic flow of some hyperbolic manifold among contact Anosov flows, we relate the leading term of the Laurent expansion of $\zeta$ at the origin, the Reidemeister torsion and the torsions of the finite dimensional complexes of the generalized resonant states of both flows for the resonance $0$. This extends previous work of~\cite{dang2018fried} on the Fried conjecture near geodesic flows of hyperbolic $3$--manifolds, to hyperbolic manifolds of any odd dimension.

math.DS↗