arXiv subjects

Marc Arnaudon

Publications and source records attributed to Marc Arnaudon.

At least 19 recordsLinked to original sources

arXiv2026

Large time behaviour for the semigroup of the kinetic Brownian motion in the plane

We establish an integration by parts formula for the semi-group in time $T > 0$ of the kinetic Brownian motion in the Euclidean plane together with its speed in the circle. The stochastic differential equation of our kinetic Brownian motion is driven here by one real-valued Brownian motion constructed with an orthonormal basis of $L^2([0,T],\mathbb R)$ and an independent sequence of $\mathscr N(0,1)$ random variables. Our method is based on an explicit computation of a Malliavin dual in the Gaussian space. We are mainly interested in large time $T$. From our integration by parts, we obtain gradient estimates including a reverse Poincar{\'e} inequality for the semi-group. As a direct consequence, we also obtain a Liouville property for the generator of the kinetic Brownian motion and its speed: all bounded harmonic functions are constant.

math.PR↗

arXiv2025

Martingales On A Euclidean Manifold With A Boundary And Reflected BSDES In Non-Convex Domains

The purpose of this paper is twofold. First, we introduce the notion of a $\Gamma$-martingale on a Euclidean manifold with a boundary (i.e., the closure of an open connected domain in R d ), we provide its equivalent characterization through the $\Gamma$-convex functions, and we establish its connection with the reflected backward stochastic differential equations (BSDEs) in the associated domain. Second, we show how the tools of stochastic geometry can be used to develop a new method for proving existence and uniqueness of solutions to reflected BSDEs. We implement this method and obtain a well-posedness result for reflected BSDEs in any bounded, two-dimensional, simply-connected domain that is locally C2 -diffeomorphic to a convex set. This work extends the results of [6] and [16].

math.PR↗

arXiv2025

Continuous-time filtering in Lie groups: estimation via the Fr{\'e}chet mean of solutions to stochastic differential equations

We compute the Fr\'echet mean $\mathscr{E}_t$ of the solution $X_{t}$ to a continuous-time stochastic differential equation in a Lie group. It provides an estimator with minimal variance of $X_{t}$. We use it in the context of Kalman filtering and more precisely to infer rotation matrices. In this paper, we focus on the prediction step between two consecutive observations. Compared to state-of-the-art approaches, our assumptions on the model are minimal.

math.PR↗

arXiv2024

On the separation cut-off phenomenon for Brownian motions on high dimensional rotationally symmetric compact manifolds

Given a family of rotationally symmetric compact manifolds indexed by the dimension and a weight function, the goal of this paper is to investigate the cut-off phenomenon for the Brownian motions on this family. We provide a class of compact manifolds with non-negative Ricci curvatures for which the cut-off in separation with windows occurs, in high dimension, with different explicit mixing times. We also produce counter-examples, still with non-negative Ricci curvatures, where there are no cut-off in separation. In fact we show a phase transition for the cut-off phenomenon concerning the Brownian motions on a rotationally symmetric compact manifolds. Our proof is based on a previous construction of a sharp strong stationary times by the authors, and some quantitative estimates on the two first moments of the covering time of the dual process. The concentration of measure phenomenon for the above family of manifolds appears to be relevant for the study of the corresponding cut-off.

math.PR↗

arXiv2024

Coarse extrinsic curvature of Riemannian submanifolds

We introduce a novel concept of coarse extrinsic curvature for Riemannian submanifolds, inspired by Ollivier's notion of coarse Ricci curvature. This curvature is derived from the Wasserstein 1-distance between probability measures supported in the tubular neighborhood of a submanifold, providing new insights into the extrinsic curvature of isometrically embedded manifolds in Euclidean spaces. The framework also offers a method to approximate the mean curvature from statistical data, such as point clouds generated by a Poisson point process. This approach has potential applications in manifold learning and the study of metric embeddings, enabling the inference of geometric information from empirical data.

math.DG↗

arXiv2024

A coupling strategy for Brownian motions at fixed time on Carnot groups using Legendre expansion

We propose a new simple construction of a coupling at a fixed time of two sub-Riemannian Brownian motions on the Heisenberg group and on the free step 2 Carnot groups. The construction is based on a Legendre expansion of the standard Brownian motion and of the L{é}vy area. We deduce sharp estimates for the decay in total variation distance between the laws of the Brownian motions. Using a change of probability method, we also obtain the log-Harnack inequality, a Bismut type integration by part formula and reverse Poincaré inequalities for the associated semi-group.

math.PR↗

arXiv2024

Optimal matching between curves in a manifold

This paper is concerned with the computation of an optimal matching between two manifold-valued curves. Curves are seen as elements of an infinite-dimensional manifold and compared using a Riemannian metric that is invariant under the action of the reparameterization group. This group induces a quotient structure classically interpreted as the ''shape space''. We introduce a simple algorithm allowing to compute geodesics of the quotient shape space using a canonical decomposition of a path in the associated principal bundle. We consider the particular case of elastic metrics and show simulations for open curves in the plane, the hyperbolic plane and the sphere.

math.DG↗

arXiv2023

The stochastic renormalized mean curvature flow for planar convex sets

We investigate renormalized curvature flow (RCF) and stochastic renormalized curvature flow (SRCF) for convex sets in the plane.RCF is the gradient descent flow for logarithm of $σ/λ^2$ where $σ$ is the perimeter and $λ$ is the volume. SRCF is RCF perturbated by a Brownian noise and has the remarkable property that it can be intertwined with the Brownian motion, yielding a generalization of Pitman "2M-X" theorem. We prove that along RCF, entropy $\mathcal{E}_t$ for curvature as well as $h_t:=σ_t/λ_t$ are non-increasing. We deduce infinite lifetime and convergence to a disk after normalization.For SRCF the situation is more complicated. The process $(h_t)_t$ is always a supermartingale. For $(\mathcal{E}_t)_t$ to be a supermartingale, we need that the starting set is invariant by the isometry group $G_n$ generated by the reflection with respect to the vertical line and the rotation of angle $2π/n$ with $n\ge 3$. But for proving infinite lifetime, we need invariance of the starting set by $G_n$ with $n\ge 7$. We provide the first SRCF with infinite lifetime which cannot be reduced to a finite dimensional flow. Gage inequality plays a major role in our study of the regularity of flows, as well as a careful investigation of morphological skeletons. We characterize symmetric convex sets with star shaped skeletons in terms of properties of their Gauss map. Finally, we establish a new isoperimetric estimate for these sets, of order $1/n^4$ where $n$ is the number of branches of the skeleton.

math.PR↗

arXiv2023

A Lyapunov approach to stability of positive semigroups: An overview with illustrations

The stability analysis of possibly time varying positive semigroups on non necessarily compact state spaces, including Neumann and Dirichlet boundary conditions is a notoriously difficult subject. These crucial questions arise in a variety of areas of applied mathematics, including nonlinear filtering, rare event analysis, branching processes, physics and molecular chemistry. This article presents an overview of some recent Lyapunov-based approaches, focusing principally on practical and powerful tools for designing Lyapunov functions. These techniques include semigroup comparisons as well as conjugacy principles on non necessarily bounded manifolds with locally Lipschitz boundaries. All the Lyapunov methodologies discussed in the article are illustrated in a variety of situations, ranging from conventional Markov semigroups on general state spaces to more sophisticated conditional stochastic processes possibly restricted to some non necessarily bounded domains, including locally Lipschitz and smooth hypersurface boundaries, Langevin diffusions as well as coupled harmonic oscillators.

math.PR↗

arXiv2023

Coarse Ricci curvature of weighted Riemannian manifolds

We show that the generalized Ricci tensor of a weighted complete Riemannian manifold can be retrieved asymptotically from a scaled metric derivative of Wasserstein 1-distances between normalized weighted local volume measures. As an application, we demonstrate that the limiting coarse curvature of random geometric graphs sampled from Poisson point process with non-uniform intensity converges to the generalized Ricci tensor.

math.DG↗

arXiv2022

Riemannian data-dependent randomized smoothing for neural networks certification

Certification of neural networks is an important and challenging problem that has been attracting the attention of the machine learning community since few years. In this paper, we focus on randomized smoothing (RS) which is considered as the state-of-the-art method to obtain certifiably robust neural networks. In particular, a new data-dependent RS technique called ANCER introduced recently can be used to certify ellipses with orthogonal axis near each input data of the neural network. In this work, we remark that ANCER is not invariant under rotation of input data and propose a new rotationally-invariant formulation of it which can certify ellipses without constraints on their axis. Our approach called Riemannian Data Dependant Randomized Smoothing (RDDRS) relies on information geometry techniques on the manifold of covariance matrices and can certify bigger regions than ANCER based on our experiments on the MNIST dataset.

cs.LG↗

arXiv2022

On the separation cut-off phenomenon for Brownian motions on high dimensional spheres

This note proves that the separation convergence towards the uniform distribution abruptly occurs at times around ln(n)/n for the (time-accelerated by 2) Brownian motion on the sphere with a high dimension n. The arguments are based on a new and elementary perturbative approach for estimating hitting times in a small noise context. The quantitative estimates thus obtained are applied to the strong stationary times constructed in a privious article by the authors to deduce the wanted cut-off phenomenon.

math.PR↗

arXiv2022

Construction of set-valued dual processes on manifolds

The purpose of this paper is to construct a Brownian motion $X := (X_t)_{t\geq 0}$ taking values in a Riemannian manifold $M$, together with a compact valued process $D:= (D_t)_{t\geq 0}$ such that, at least for small enough ${\mathscr F}^D$-stopping time $τ> 0$ and conditioned by ${\mathscr F}_τ^D$, the law of $X_τ$ is the normalized Lebesgue measure on $D_τ$. This intertwining result is a generalization of Pitman theorem. We first construct regular intertwined processes related to Stokes' theorem. Then using several limiting procedures we construct synchronous intertwined, free intertwined, mirror intertwined processes. The local times of the Brownian motion on the (morphological) skeleton or the boundary of $D$ plays an important role. Several examples with moving intervals, discs, annulus, symmetric convex sets are investigated. KEYWORDS: Brownian motions on Riemannian manifolds, intertwining relations, set-valued dual processes, couplings of primal and dual processes, stochastic mean curvature evolutions, boundary and skeleton local times, generalized Pitman theorem.

math.PR↗

arXiv2020

A duality formula and a particle Gibbs sampler for continuous time Feynman-Kac measures on path spaces

Continuous time Feynman-Kac measures on path spaces are central in applied probability, partial differential equation theory, as well as in quantum physics. This article presents a new duality formula between normalized Feynman-Kac distribution and their mean field particle interpretations. Among others, this formula allows us to design a reversible particle Gibbs-Glauber sampler for continuous time Feynman-Kac integration on path spaces. This result extends the particle Gibbs samplers introduced by Andrieu-Doucet-Holenstein [2] in the context of discrete generation models to continuous time Feynman-Kac models and their interacting jump particle interpretations. We also provide new propagation of chaos estimates for continuous time genealogical tree based particle models with respect to the time horizon and the size of the systems. These results allow to obtain sharp quantitative estimates of the convergence rate to equilibrium of particle Gibbs-Glauber samplers. To the best of our knowledge these results are the first of this kind for continuous time Feynman-Kac measures.

math.PR↗

arXiv2019

A variational approach to nonlinear and interacting diffusions

The article presents a novel variational calculus to analyze the stability and the propagation of chaos properties of nonlinear and interacting diffusions. This differential methodology combines gradient flow estimates with backward stochastic interpolations, Lyapunov linearization techniques as well as spectral theory. This framework applies to a large class of stochastic models including non homogeneous diffusions, as well as stochastic processes evolving on differentiable manifolds, such as constraint-type embedded manifolds on Euclidian spaces and manifolds equipped with some Riemannian metric. We derive uniform as well as almost sure exponential contraction inequalities at the level of the nonlinear diffusion flow, yielding what seems to be the first result of this type for this class of models. Uniform propagation of chaos properties w.r.t. the time parameter are also provided. Illustrations are provided in the context of a class of gradient flow diffusions arising in fluid mechanics and granular media literature. The extended versions of these nonlinear Langevin-type diffusions on Riemannian manifolds are also discussed.

math.PR↗

arXiv2018

Gradient Estimates on Dirichlet Eigenfunctions

By methods of stochastic analysis on Riemannian manifolds, we derive explicit constants $c\_1(D)$ and $c\_2(D)$ for a $d$-dimensional compact Riemannian manifold $D$ with boundary such that $c\_1(D)\sqrtλ\|ϕ\|\_\infty \le \|\nabla ϕ\|\_\infty\le c\_2(D)\sqrtλ \|ϕ\|\_\infty$ holds for any Dirichlet eigenfunction $ϕ$ of $-Δ$ with eigenvalue $λ$. In particular, when $D$ is convex with nonnegative Ricci curvature, this estimate holds for $c\_1(D)=\frac{1}{de}$ and $c\_2(D)=\sqrt{e}\left(\frac{\sqrt{2}}{\sqrtπ}+\frac{\sqrtπ}{4\sqrt{2}}\right)$. Corresponding two-sided gradient estimates for Neumann eigenfunctions are derived in the second part of the paper.

math.PR↗

arXiv2017

On a coupling of solutions to the interface SDE on a star graph

Inspired by Tsirelson proof of the non Brownian character of WalshBrownian motion ltration on three or more rays, we prove some results on aparticular coupling of solutions to the interface SDE on a star graph, recentlyintroduced. This coupling consists in two solutions which are independentgiven the driving Brownian motion. As a consequence, we deduce that if the stargraph contains 3 or more rays, the argument of the solution at a xed time isindependent of the driving Brownian motion.

math.PR↗

arXiv2017

An entropic interpolation problem for incompressible viscid fluids

In view of studying incompressible inviscid fluids, Brenier introduced in the late 80's a relaxation of a geodesic problem addressed by Arnold in 1966. Instead of inviscid fluids, the present paper is devoted to incompressible viscid fluids. A natural analogue of Brenier's problem is introduced, where generalized flows are no more supported by absolutely continuous paths, but by Brownian sample paths. It turns out that this new variational problem is an entropy minimization problem with marginal constraints entering the class of convex minimization problems. This paper explores the connection between this variational problem and Brenier's original problem. Its dual problem is derived and the general shape of its solution is described. Under the restrictive assumption that the pressure is a nice function, the kinematics of its solution is made explicit and its connection with the Navier-Stokes equation is established.

math.PR↗