arXiv subjects
Publications and source records attributed to Fabrice Baudoin.
In this short note, using the Kendall-Cranston coupling, we study on Kähler (resp. quaternion Kähler) manifolds first eigenvalue estimates in terms of dimension, diameter, and lower bounds on the holomorphic (resp. quaternionic) sectional curvature.
We prove Cameron-Martin type quasi-invariance results for the heat kernel measure of infinite-dimensional Kolmogorov and related diffusions. We first study quantitative functional inequalities for appropriate finite-dimensional approximations of these diffusions, and we prove these inequalities hold with dimension-independent coefficients. Applying an approach developed by Baudoin, Driver, Gordina, and Melcher previously, these uniform bounds may then be used to prove that the heat kernel measure for certain of these infinite-dimensional diffusions is quasi-invariant under changes of the initial state.
We develop a new approach to prove multiplier theorems in various geometric settings. The main idea is to use martingale transforms and a Gundy-Varopoulos representation for multipliers defined via a suitable extension procedure. Along the way, we provide a probabilistic proof of a generalization of a result by Stinga and Torrea, which is of independent interest. Our methods here also recover the sharp $L^p$ bounds for second order Riesz transforms by a liming argument.
We study the Brownian motion on the non-compact Grassmann manifold $\frac{\mathbf{U}(n-k,k)} {\mathbf{U}(n-k)\mathbf{U}(k)}$ and some of its functionals. The key point is to realize this Brownian motion as a matrix diffusion process, use matrix stochastic calculus and take advantage of the hyperbolic Stiefel fibration to study a functional that can be understood in that setting as a generalized stochastic area process. In particular, a connection to the generalized Maass Laplacian of the complex hyperbolic space is presented and applications to the study of Brownian windings in the Lie group $\mathbf{U}(n-k,k)$ are then given.
Under suitable conditions, we show that the Euler characteristic of a foliated Riemannian manifold can be computed only from curvature invariants which are transverse to the leaves. Our proof uses the hypoelliptic sub-Laplacian on forms recently introduced by two of the authors.
This paper studies convergence to equilibrium for second-order Langevin dynamics under general growth conditions on the potential. Although we are principally motivated by the case when the potential is singular, e.g. when the dynamics has repulsive forces and/or interactions, the results presented in this paper hold more generally. In particular, our main result is that, given (very) basic structural and growth conditions on the potential, the dynamics relaxes to equilibrium exponentially fast in an explicitly measurable way. The ``explicitness" of this result comes directly from the constants appearing in the growth conditions, which can all be readily estimated, and a local Poincaré constant for the invariant measure $μ$. This result is applied to the specific situation of a singular interaction and polynomial confining well to provide explicit estimates on the exponential convergence rate $e^{-σ}$ in terms of the number $N \geqslant 1$ of particles in the system. We will see that $σ\geqslant c/(ρ\vee N^p)$, where $ρ>0$ is the local Poincaré constant for $μ$ and $c>0, p\geqslant 1$ are constants that are independent of $N$.
We study the relationship between functional inequalities for a Markov kernel on a metric space $X$ and inequalities of transportation distances on the space of probability measures $\mathcal{P}(X)$. Extending results of Luise and Savaré on Hellinger--Kantorovich contraction inequalities for the particular case of the heat semigroup on an $RCD(K,\infty)$ metric space, we show that more generally, such contraction inequalities are equivalent to reverse Poincaré inequalities. We also adapt the "dynamic dual" formulation of the Hellinger--Kantorovich distance to define a new family of divergences on $\mathcal{P}(X)$ which generalize the Rényi divergence, and we show that contraction inequalities for these divergences are equivalent to the reverse logarithmic Sobolev and Wang Harnack inequalities. We discuss applications including results on the convergence of Markov processes to equilibrium, and on quasi-invariance of heat kernel measures in finite and infinite-dimensional groups.
In this note, we study the sub-Laplacian of the 15-dimensional octonionic anti-de Sitter space which is obtained by lifting with respect to the anti-de Sitter fibration the Laplacian of the octonionic hyperbolic space $\mathbb{O}H^1$. We also obtain two integral representations for the corresponding subelliptic heat kernel.
This paper provides a characterization of functions of bounded variation (BV) in a compact Riemannian manifold in terms of the short time behavior of the heat semigroup. In particular, the main result proves that the total variation of a function equals the limit characterizing the space BV. The proof is carried out following two fully independent approaches, a probabilistic and an analytic one. Each method presents different advantages.
We provide a general framework for the realization of powers or functions of suitable operators on Dirichlet spaces. The first contribution is to unify the available results dealing with specific geometries; a second one is to provide new results on rather general metric measured spaces that were not considered before and fall naturally in the theory of Dirichlet spaces. The main tool is using the approach based on subordination and semi-groups by Stinga and Torrea. Assuming more on the Dirichlet space, we derive several applications to PDEs such as Harnack and Boundary Harnack principles.
We study several matrix diffusion processes constructed from a unitary Brownian motion. In particular, we use the Stiefel fibration to lift the Brownian motion of the complex Grassmannian to the complex Stiefel manifold and deduce a skew-product decomposition of the Stiefel Brownian motion. As an application, we prove asymptotic laws for the determinants of the block entries of the unitary Brownian motion.
We introduce the class of bounded variation (BV) functions in a general framework of strictly local Dirichlet spaces with doubling measure. Under the 2-Poincaré inequality and a weak Bakry-Émery curvature type condition, this BV class is identified with the heat semigroup based Besov class $\mathbf{B}^{1,1/2}(X)$ that was introduced in our previous paper. Assuming furthermore a quasi Bakry-Émery curvature type condition, we identify the Sobolev class $W^{1,p}(X)$ with $\mathbf{B}^{p,1/2}(X)$ for $p>1$. Consequences of those identifications in terms of isoperimetric and Sobolev inequalities with sharp exponents are given.
We introduce heat semigroup-based Besov classes in the general framework of Dirichlet spaces. General properties of those classes are studied and quantitative regularization estimates for the heat semigroup in this scale of spaces are obtained. As a highlight of the paper, we obtain a far reaching $L^p$-analogue, $p \ge 1$, of the Sobolev inequality that was proved for $p=2$ by N. Varopoulos under the assumption of ultracontractivity for the heat semigroup. The case $p=1$ is of special interest since it yields isoperimetric type inequalities.
We prove explicit $L^p$ bounds for second order Riesz transforms of the sub-Laplacian in the Lie groups $\mathbb H$, $\mathbb{SU}(2)$ and $\mathbb{SL}(2)$
We study the radial parts of the Brownian motions on Kähler and quaternion Kähler manifolds. Thanks to sharp Laplacian comparison theorems, we deduce as a consequence a sharp Cheeger-Yau type lower bound for the heat kernels of such manifolds and also sharp Cheng's type estimates for the Dirichlet eigenvalues of metric balls.
We define and study a fractional Gaussian field $X$ with Hurst parameter $H$ on the Sierpiński gasket $K$ equipped with its Hausdorff measure $μ$. It appears as a solution, in a weak sense, of the equation $(-Δ)^s X =W$ where $W$ is a Gaussian white noise on $L_0^2(K,μ)$, $Δ$ the Laplacian on $K$ and $s= \frac{d_h+2H}{2d_w}$, where $d_h$ is the Hausdorff dimension of $K$ and $d_w$ its walk dimension. The construction of those fields is then extended to other fractals including the Sierpiński carpet.
We study the radial part of sub-Riemannian Brownian motion in the context of totally geodesic foliations. Itô's formula is proved for the radial processes associated to Riemannian distances approximating the Riemannian one. We deduce very general stochastic completeness criteria for the sub-Riemannian Brownian motion. In the context of Sasakian foliations and H-type groups, one can push the analysis further, and taking advantage of the recently proved sub-Laplacian comparison theorems one can compare the radial processes for the sub-Riemannian distance to one-dimensional model diffusions. As a geometric application, we prove Cheng's type estimates for the Dirichlet eigenvalues of the sub-Riemannian metric balls, a result which seems to be new even in the Heisenberg group.
With a view towards Riemannian or sub-Riemannian manifolds, RCD metric spaces and specially fractals, this paper makes a step further in the development of a theory of heat semigroup based $(1,p)$ Sobolev spaces in the general framework of Dirichlet spaces. Under suitable assumptions that are verified in a variety of settings, the tools developed by D. Bakry, T. Coulhon, M. Ledoux and L. Saloff-Coste in the paper "Sobolev inequalities in disguise" allow us to obtain the whole family of Gagliardo-Nirenberg and Trudinger-Moser inequalities with optimal exponents. The latter depend not only on the Hausdorff and walk dimensions of the space but also on other invariants. In addition, we prove Morrey type inequalities and apply them to study the infimum of the exponents that ensure continuity of Sobolev functions. The results are illustrated for fractals using the Vicsek set, whereas several conjectures are made for nested fractals and the Sierpinski carpet.