Stochastics and Dynamics

In this paper, we derive a large deviation principle for a stochastic 2D Allen–Cahn–Navier–Stokes system with a multiplicative noise of Lévy type. The model consists of the Navier–Stokes equations for the velocity, coupled with a Allen–Cahn system for the order (phase) parameter. The proof is based on the weak convergence method introduced in [A. Budhiraja, P. Dupuis and V. Maroulas, Variational representations for continuous time processes, Ann. Inst. Henri Poincar Probab. Stat. 47(3) (2011) 725747].

We study the long-time behavior of Kardar–Parisi–Zhang (KPZ)-like equations:

This paper deals with the Wong–Zakai approximations and random attractors for stochastic Ginzburg–Landau equations with a white noise. We first prove the existence of a pullback random attractor for the approximate equation under much weaker conditions than the original stochastic equation. In addition, when the stochastic Ginzburg–Landau equation is driven by an additive white noise, we establish the convergence of solutions of Wong–Zakai approximations and the upper semicontinuity of random attractors of the approximate random system as the size of approximation tends to zero.

The stochastic two-dimensional Cahn–Hilliard–Navier–Stokes equations under non-Lipschitz conditions are considered. This model consists of the Navier–Stokes equations controlling the velocity and the Cahn–Hilliard model controlling the phase parameters. By iterative techniques, a priori estimates and weak convergence method, the existence and uniqueness of an energy weak solution to the equations under non-Lipschitz conditions have been obtained.

In this paper, we establish an averaging principle for neutral stochastic fractional differential equations with non-Lipschitz coefficients and with variable delays, driven by Lévy noise. Our result shows that the solutions of the equations concerned can be approximated by the solutions of averaged neutral stochastic fractional differential equations in the sense of convergence in mean square. As an application, we present an example with numerical simulations to explore the established averaging principle.

Using the method of Girsanov’s transformation, we investigate Talagrand’s quadratic transportation cost inequalities for the laws of the solutions of stochastic partial differential equations (SPDEs) with two reflection walls under the uniform norm on the continuous functions space. These equations are driven by fractional noises.

The Mean Field Games PDEs system is at the heart of the Mean Field Games theory initiated by [J.-M. Lasry and P.-L. Lions, Jeux à champ moyen. I–le cas stationnaire, C. R. Math. 343 (2006) 619–625; J.-M. Lasry and P.-L. Lions, Jeux à champ moyen. II–horizon fini et contrôle optimal, C. R. Math. 343 (2006) 679–684; J.-M. Lasry and P.-L. Lions, Mean field games, Jpn. J. Math. 2 (2007) 229–260] which constitutes a seminal contribution to the modeling and analysis of games with a large number of players. We propose here a numerical method of resolution of such systems based on the construction of a discrete mean field game where the controlled state-variable is a Markov chain approximating the controlled stochastic differential equation [H. Kushner and P. G. Dupuis, Numerical Methods for Stochastic Control Problems in Continuous Time, Stochastic Modeling and Applied Probability, Vol. 24 (Springer Science & Business Media, 2013)]. In particular, existence and uniqueness properties of the discrete MFG are investigated with convergence results under adequate assumptions.