COULIBALY-PASQUIER Koléhè

1. Part of my work concerns the study of geometric flows and the heat equation coupled with a variation of metrics. The main object of study is g(t)-Brownian motion.

• M. Arnaudon, K. A. Coulibaly and A. Thalmaier, Brownian motion with respect to a metric depending on time ; definition, existence and applications to Ricci flow, C. R. Math. Acad. Sci. Paris 346 (2008), 773–778.
• K. A. Coulibaly, Brownian motion with respect to time-changing Riemannian metrics, applications to Ricci flow (2011), A.I.H.P., 515-538
• K. A. Coulibaly-Pasquier, Some stochastic process without birth, linked to the mean curvature flow, (2011) Annals of proba., 1305–1331

The behavior of the Wasserstein distance between two measurements evolves according to the conjugate heat equation along the Ricci flow:

• M. Arnaudon, K. A. Coulibaly and A. Thalmaier, Horizontal diffusion in C^1 path space, Séminaire de Probabilité (2011), 73–94.

The local behavior of g(t) Brownian motion and its deviations, through Onsager-Machlup functionals:

• Coulibaly-Pasquier, Koléhè A., Onsager-Machlup functional for uniformly elliptic timeinhomogeneous diffusion. Séminaire de Probabilité (2014), 105–123.

The heat kernel coupled to the Ricci flow

• Coulibaly-Pasquier, Koléhè A., Heat Kernel coupled with geometric flow, and application to Ricci flow. Séminaire de Probabilité (2019), 221–256.

2 Another part of my work deals with intertwined processes with diffusion. It turns out that the boundary of the dual process of a Brownian motion follows a mean-curvature  equation with a stochastic perturbation and a Cheeger drift. A notable fact is that whatever the geometry and dimension of the underlying variety, the volume of the dual is a time-changed Bessel 3.

• Coulibaly-Pasquier, Koléhè A. ; Laurent Miclo, On the evolution by duality of domains on manifolds, Mémoires de la Société Mathématique de France, Volume 171, (2021), 1–110.
• Marc Arnaudon ; Coulibaly-Pasquier, Koléhè A. ; Laurent Miclo, On Markov intertwining relations and primal conditioning,  Journal of Theoretical Probability,  (2023).
• Marc Arnaudon ; Coulibaly-Pasquier, Koléhè A. ; Laurent Miclo, COUPLINGS OF BROWNIAN MOTIONS WITH SET-VALUED DUAL PROCESSES ON RIEMANNIAN MANIFOLDS , Journal de l’école Polytechnique (Mathématiques) (2024).

3. Cut-off phenomenon

• Marc Arnaudon ; Coulibaly-Pasquier, Koléhè A. ; Laurent Miclo, On the separation cut-off phenomenon for Brownian motions on high dimensional rotationally symmetric compact manifolds (Prepint 2024)
• Marc Arnaudon ; Coulibaly-Pasquier, Koléhè A. ; Laurent Miclo, On the separation cut-off phenomenon for Brownian motions on high dimensional spheres, Bernoulli, (2024).
• Marc Arnaudon ; Coulibaly-Pasquier, Koléhè A. ; Laurent Miclo, THE STOCHASTIC RENORMALIZED MEAN CURVATURE FLOW FOR PLANAR CONVEX SETS.(Prepint 2024)

4.Convergence criteria to quasi-stationary measure for multi-dimensional diffusion processes.

• Champagnat, Nicolas ; Coulibaly-Pasquier, Koléhè Abdoulaye ; Villemonais, Denis, Criteria for exponential convergence to quasistationary distributions and applications to multi-dimensional diffusions. Séminaire de Probabilités (2018), 165–182.

The following simulations illustrate an asymptotic convergence theorem obtained with Laurent Miclo in On the evolution by duality of domains on manifolds,

The long-time existence of dual processes, with assumptions on symmetries, is a joint work with Marc Arnaudon and Laurent Miclo.

• This video shows the evolution of the dual of planar Brownian motion, starting from the “star” domain (in blue), and illustrates an asymptotic shape theorem i.e. convergence towards the red disk of the same volume as that of the blue domain, this volume being a time-changed Bessel 3 (the numerous symmetries give very rapid convergence).

• This video shows the evolution of the dual of planar Brownian motion, starting from the “peanut” domain (in blue), and illustrates an asymptotic shape theorem i.e. convergence towards the red disk of the same volume as that of the blue domain, this volume is a time-changed Bessel 3 (convergence is less rapid than previously!).

• This video shows the evolution of the dual of Brownian motion on the 2-dimensional sphere, starting from the cap-shaped domain (in red), the volume of the dual in red is a time-changed and stopped Bessel 3. As expected, this dual fills the sphere, in a time that is a strong stationary time.