arXiv subjects

Magalie Bénéfice

Publications and source records attributed to Magalie Bénéfice.

7 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 semigroup in time $T>0$ of the kinetic Brownian motion in the Euclidean plane together with its velocity in the circle. The stochastic differential equation of our kinetic Brownian motion is driven here by one real-valued Brownian motion constructed from an orthonormal basis of $L^2([0,T],\R)$ and an independent sequence of $\SN(0,1)$ 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{é} inequality for the semigroup. 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↗

arXiv2026

Cutoff in separation for compact harmonic manifold

We show that cutoff in separation occurs for Brownian motion in some families of compact harmonic manifolds. We compute the cutoff time and windows in four families of compact harmonic manifold namely S n , CP n , HP n and RP n (the first three families are the only families of compact simply connected harmonic manifolds, see [19]). The proof is based on sharp strong stationary times and sufficiently accurate asymptotic expansions of their means and variances.

math.PR↗

arXiv2025

Non co-adapted couplings of Brownian motions on free, step 2 Carnot groups

On the free, step $2$ Carnot groups of rank $n$ $\mathbb{G}_n$, the subRiemannian Brownian motion consists in a $\mathbb{R}^n$-Brownian motion together with its $\frac{n(n-1)}{2}$ L{é}vy areas. In this article we construct an explicit successful non co-adapted coupling of Brownian motions on $\mathbb{G}_n$. We use this construction to obtain gradient inequalities for the heat semi-group on all the homogeneous step $2$ Carnot groups. Comparing the first coupling time and the first exit time from a domain, we also obtain gradient inequalities for harmonic functions on $\mathbb{G}_n$. These results generalize the coupling strategy by Banerjee, Gordina and Mariano on the Heisenberg group.

math.PR↗

arXiv2025

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

We compute the Fréchet 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

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

Couplings of Brownian motions on $SU(2,\mathbb{C})$ and $SL(2,\mathbb{R})$

The Lie groups $SU(2)$ and $SL(2,\mathbb{R})$ can be viewed as model spaces in subRiemannian geometry. Coupling two subelliptic Brownian motions on $SU(2)$ (resp. $SL(2,\mathbb{R})$) consists in coupling two Brownian motions on the sphere (resp. the hyperbolic plane) and simultaneously their swept areas. Using this approach we propose an explicit construction of a co-adapted successful coupling on $SU(2)$. The strategy is to alternate between reflection and synchronous (with noise) coupling on the sphere. We also describe some more general constructions of co-adapted couplings on $SU(2)$ and also on $SL(2,\mathbb{R})$.

math.PR↗

arXiv2023

Non co-adapted couplings of Brownian motions on subRiemannian manifolds

In this article we continue the study of couplings of subelliptic Brownian motions on the subRiemannian manifolds SU (2) and SL(2, R). Similar to the case of the Heisenberg group, this subelliptic Brownian motion can be considered as a Brownian motion on the sphere (resp. the hyperbolic plane) together with its swept area modulo 4$π$. Using this structure, we construct an explicit non co-adapted successful coupling on SU (2) and, under strong conditions on the starting points, on SL(2, R) too. This strategy uses couplings of Brownian bridges, taking inspiration into the work from Banerjee, Gordina and Mariano [3] on the Heisenberg group. We prove that the coupling rate associated to these constructions is exponentially decreasing in time and proportionally to the subRiemannian distance between the starting points. We also give some gradient inequalities that can be deduced from the estimation of this coupling rate.

math.PR↗