arXiv subjects

Fabrice Baudoin

Publications and source records attributed to Fabrice Baudoin.

At least 19 recordsLinked to original sources

arXiv2026

Sub-Riemannian Selberg Trace Formulae for Compact Quotients of SL(2,R) and Determinants of Sub-Laplacians

We prove sub-Riemannian Selberg trace formulae for compact quotients of SL(2,R). Using the Fourier decomposition along the SO(2)-fibers, we reduce the heat trace computation to the Selberg trace formula for Maass Laplacians on the hyperbolic plane. The resulting formula has an identity contribution and a hyperbolic contribution, the latter involving a theta factor that records both the twisting character and the central sign of the chosen lift to SL(2,\mathbb R)). We then use this trace formula to compute the zeta-regularized determinant of the sub-Laplacian. The determinant formula is expressed as a universal factor depending only on the base hyperbolic surface times an explicit lift- and character-dependent Selberg-type product.

math.DG↗

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

Geometric functionals of Brownian motion on Hermitian symmetric spaces of non-compact type

We study Brownian motion on Hermitian symmetric spaces of non-compact type in their bounded-domain realization. Using Jordan triple systems, we identify the spectral values after an appropriate change of variables as a Heckman-Opdam diffusion of type $BC_r$. We then analyze two Brownian functionals: the symplectic area associated with the canonical Kähler form, and, in the tube-type case, the winding defined by the Jordan determinant. For the area process we prove a martingale representation, a central limit theorem, and an exact conditional characteristic function expressed as a ratio of Heckman-Opdam heat kernels. For the determinant winding process we obtain analogous heat kernel formulas and prove convergence to a Cauchy law with scale determined by the initial determinant. These results extend classical formulas of Paul Lévy and Marc Yor from the Euclidean setting to the full class of Hermitian symmetric spaces of non-compact type.

math.PR↗

arXiv2026

Sub-Laplacian generalized curvature dimension inequalities on Riemannian foliations

We develop a Bochner theory and Bakry-Emery calculus for horizontal Laplacians associated with general Riemannian foliations. No bundle-like assumption on the metric, nor any total geodesicity or minimality condition on the leaves is imposed. Using a metric connection adapted to the horizontal-vertical splitting, we derive explicit Bochner formulas for the horizontal Laplacian acting on horizontal and vertical gradients, as well as a unified identity for the full gradient. These formulas involve horizontal Ricci curvature, torsion and vertical mean curvature terms intrinsic to the foliated structure. From these identities, we establish generalized curvature dimension inequalities, extending earlier results in sub-Riemannian geometry. As applications, we obtain horizontal Laplacian comparison theorems, Bonnet-Myers type compactness results with explicit diameter bounds, stochastic completeness, first eigenvalue estimates and gradient and regularization estimates for the horizontal heat semigroup. The framework applies, in particular, to contact manifolds and Carnot groups of arbitrary step.

math.DG↗

arXiv2025

Quaternionic stochastic areas on quaternionic full flag manifolds and applications

We show that a Brownian motion on the quaternionic full flag manifold can be represented as a matrix-valued diffusion obtained in a simple way from a symplectic Brownian motion. By relating its radial dynamics to the Brownian motion on the quaternionic sphere, an explicit formula for the characteristic function of the joint distribution of the quaternionic stochastic areas is obtained. The limit law of these quaternionic stochastic areas is shown to be a multivariate normal distribution with zero mean and non-diagonal covariance matrix. These results are subsequently applied to establish new results about simultaneous quaternionic windings on the quaternionic spheres.

math.PR↗

arXiv2025

Extension method in Dirichlet spaces with sub-Gaussian estimates and applications to regularity of jump processes on fractals

We investigate regularity properties of some non-local equations defined on Dirichlet spaces equipped with sub-gaussian estimates for the heat kernel associated to the generator. We prove that weak solutions for homogeneous equations involving pure powers of the generator are actually Hölder continuous and satisfy an Harnack inequality. Our methods are based on a version of the Caffarelli-Silvestre extension method which is valid in any Dirichlet space and our results complement the existing literature on solutions of PDEs on classes of Dirichlet spaces such as fractals.

math.AP↗

arXiv2025

Sub-Laplacian comparison theorems on Riemannian foliations with minimal leaves and applications

We prove comparison theorems for the horizontal Laplacian of the Riemannian distance in the context of Riemannian foliations with minimal leaves. This general framework generalizes previous works and allow us to consider the sub-Laplacian of Carnot groups of arbitrary steps. The comparison theorems yield a Bonnet-Myers type theorem, stochastic completeness and Lipschitz regularization property for the sub-Riemannian semigroup.

math.DG↗

arXiv2025

Topology and bottom spectrum of transversally negatively curved foliations

We show that for any Riemannian foliation with a simply connected and negatively curved leaf space the normal exponential map of a leaf is a diffeomorphism. As an application, if the leaves are furthermore minimal submanifolds, we give a sharp estimate for the bottom of the spectrum of such a Riemannian manifold. Our proof of the spectral estimate also yields an estimate for the bottom of the spectrum of the horizontal Laplacian.

math.DG↗

arXiv2025

Moment estimates for the stochastic heat equation on Cartan-Hadamard manifolds

We study the effect of curvature on the Parabolic Anderson model by posing it over a Cartan-Hadamard manifold. We first construct a family of noises white in time and colored in space parameterized by a regularity parameter $α$, which we use to explore regularity requirements for well-posedness. Then, we show that conditions on the heat kernel imply an exponential in time upper bound for the moments of the solution, and a lower bound for sectional curvature imply a corresponding lower bound. These results hold if the noise is strong enough, where the needed strength of the noise is affected by sectional curvature.

math.PR↗

arXiv2025

Korevaar-Schoen and heat kernel characterizations of Sobolev and BV spaces on local trees

We study Sobolev and BV spaces on local trees which are metric spaces locally isometric to real trees. Such spaces are equipped with a Radon measure satisfying a locally uniform volume growth condition. Using the intrinsic geodesic structure, we define weak gradients and develop from it a coherent theory of Sobolev and BV spaces. We provide two main characterizations: one via Korevaar-Schoen-type energy functionals and another via the heat kernel associated with the natural Dirichlet form. Applications include interpolation results for Besov-Lipschitz spaces, critical exponents computations, and a Nash inequality. In globally tree-like settings we also establish $L^p$ gradient bounds for the heat semigroup.

math.AP↗

arXiv2025

Orlicz-Sobolev embeddings and heat kernel based Besov classes

This paper investigates functional inequalities involving Besov spaces and functions of bounded variation, when the underlying metric measure space displays different local and global structures. Particular focus is put on the $L^1$ theory and its applications to sets of finite perimeter and isoperimetric inequalities, which can now capture such structural differences.

math.FA↗

arXiv2025

Brownian motion and stochastic areas on complex full flag manifolds

We show that the Brownian motion on the complex full flag manifold can be represented by a matrix-valued diffusion obtained from the unitary Brownian motion. This representation actually leads to an explicit formula for the characteristic function of the joint distribution of the stochastic areas on the full flag manifold. The limit law for those stochastic areas is shown to be a multivariate Cauchy distribution with independent and identically distributed entries. Using a deep connection between area functionals on the flag manifold and winding functionals on complex spheres, we establish new results about simultaneous Brownian windings on the complex sphere and their asymptotics. As a byproduct, our work also unveils a new probabilistic interpretation of the Jacobi operators and polynomials on simplices.

math.PR↗

arXiv2025

Comparison theorems on H-type sub-Riemannian manifolds

On H-type sub-Riemannian manifolds we establish sub-Hessian and sub-Laplacian comparison theorems which are uniform for a family of approximating Riemannian metrics converging to the sub-Riemannian one. We also prove a sharp sub-Riemannian Bonnet-Myers theorem that extends to this general setting results previously proved on contact and quaternionic contact manifolds.

math.DG↗

arXiv2025

Weighted Besov spaces on Heisenberg groups and applications to the Parabolic Anderson model

This article aims at a proper definition and resolution of the parabolic Anderson model on Heisenberg groups $\mathbf{H}_{n}$. This stochastic PDE is understood in a pathwise (Stratonovich) sense. We consider a noise which is smoother than white noise in time, with a spatial covariance function generated by negative powers $(-Δ)^{-α}$ of the sub-Laplacian on $\mathbf{H}_{n}$. We give optimal conditions on the covariance function so that the stochastic PDE is solvable. A large portion of the article is dedicated to a detailed definition of weighted Besov spaces on $\mathbf{H}_{n}$. This definition, related paraproducts and heat flow smoothing properties, forms a necessary step in the resolution of our main equation. It also appears to be new and of independent interest. It relies on a recent approach, called projective, to Fourier transforms on $\mathbf{H}_{n}$.

math.PR↗

arXiv2024

Dimension-independent functional inequalities by tensorization and projection arguments

We study stability under tensorization and projection-type operations of gradient-type estimates and other functional inequalities for Markov semigroups on metric spaces. Using transportation-type inequalities obtained by F. Baudoin and N. Eldredge in 2021, we prove that constants in the gradient estimates can be chosen to be independent of the dimension. Our results are applicable to hypoelliptic diffusions on sub-Riemannian manifolds and some hypocoercive diffusions. As a byproduct, we obtain dimension-independent reverse Poincaré, reverse logarithmic Sobolev, and gradient bounds for Lie groups with a transverse symmetry and for non-isotropic Heisenberg groups.

math.PR↗

arXiv2024

On the law of the index of Brownian loops related to the Hopf and anti-de Sitter fibrations

We give explicit formulas and asymptotics for the distribution of the index of the Brownian loop in the following geometrical settings: the complex projective line from which two points have been removed; the complex hyperbolic line from which one point has been removed; the odd dimensional spheres from which a great hypersphere has been removed; and the complex anti-de Sitter spaces. Our analysis is based on the geometry of the Hopf and anti-de Sitter fibrations, and on the relationship between winding and area forms.

math.PR↗

arXiv2024

Korevaar-Schoen-Sobolev spaces and critical exponents in metric measure spaces

We present developments in the theory of Korevaar-Schoen-Sobolev spaces on metric measure spaces. While this theory coincides with those of Cheeger and Shanmugalingam if the space is doubling and satisfies a Poincaré inequality, it offers new perspectives in the context of fractals for which the approach by weak upper gradients is inadequate.

math.MG↗

arXiv2024

Heat kernel gradient estimates for the Vicsek set

We prove pointwise and $L^p$ gradient estimates for the heat kernel on the bounded and unbounded Vicsek set and applications to Sobolev inequalities are given. We also define a Hodge semigroup in that setting and prove estimates for its kernel.

math.AP↗