LiBERA On-line Seminars

Continuous data assimilation methods are known to be highly effective in deterministic settings for asymptotically synchronizing approximate solutions with observed dynamics. In this work, we extend this framework to a stochastic regime by considering the two-dimensional incompressible Navier-Stokes equations subject to either additive or multiplicative noise. We establish sufficient conditions on the nudging parameter and the spatial observation scale that guarantee convergence of the nudged solution to the true stochastic flow. In the case of multiplicative noise, convergence holds in expectation, with exponential or polynomial rates depending on the growth of the noise covariance. For additive noise, we obtain the exponential convergence both in expectation and pathwise. The talk is based on a joint work with H. Bessaih, B. Ferrario and O. Landoulsi.

We establish the existence of the cut-off phenomenon for the Langevin--Kolmogorov random dynamics with monomial convex potentials, possible singular, and driven by a Brownian motion with small strength. We consider a truncated \chi^2-distance, that is, a distance based on Galerkin projections of the eigensystem, and show that not only a refined knowledge of the eigenvalues is needed but also a refined asymptotics of the growth for the eigenfunctions of the Fokker--Planck equations associated to the Langevin--Kolmogorov dynamics. In addition, asymptotics of the mixing times are established.

We construct a stochastic process which is related to the fundamental solution of the parabolic p-Laplace equation in the same way as Brownian motion is related to the heat kernel of the heat equation. More precisely, for the p-Laplace equation we identify an associated McKean—Vlasov SDE, and our constructed stochastic process consists of solutions to this SDE and, moreover, constitutes a nonlinear Markov process. We call this process p-Brownian, which for p=2 coincides with standard Brownian motion. Joint work with Viorel Barbu (A.I. Cuza University) and Michael Röckner (Bielefeld University).

We establish heat kernel bounds and regularity estimates for the transition density of the canonical process associated with the martingale problem corresponding to the generator of a formal multi-dimensional Brownian SDE with distributional drift. We consider both the non degenerate case and the degenerate kinetic case in the so-called Young regime, namely when the drift coefficient is a distribution in a negative Besov space with regularity parameter greater than -1/2. Importantly, the estimates are obtained using as parametrix the transition density of the SDE (with multiplicative noise) without the singular perturbation in place of the usual Gaussian density.

This talk is based on a joint work with Stéphane Menozzi (University of Évry Val d'Essonne).