arXiv subjects

Koléhè Coulibaly-Pasquier

Publications and source records attributed to Koléhè Coulibaly-Pasquier.

6 recordsLinked to original sources

arXiv2026

Cutoff in separation profile for the flat torus, sphere, and projective spaces

In this paper we show that the cutoff in separation profile for Brownian motion on flat torus T n\,; on spheres S n\,; on real, complex and quaternionic projective space resp. P n pRq, P n pCq and P n pHq, is the tail distribution of some explicit Gumbel distribution. The proof is based on intertwining, dual process together with a representation formula of large moments of the covering time of the dual process.

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↗

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↗

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↗

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↗

arXiv2018

Criteria for exponential convergence to quasi-stationary distributions and applications to multi-dimensional diffusions

We consider general Markov processes with absorption and provide criteria ensuring the exponential convergence in total variation of the distribution of the process conditioned not to be absorbed. The first one is based on two-sided estimates on the transition kernel of the process and the second one on gradient estimates on its semigroup. We apply these criteria to multi-dimensional diffusion processes in bounded domains of $R^d$ or in compact Riemannian manifolds with boundary, with absorption at the boundary.

math.PR↗