Large deviation principles of 2D stochastic Navier–Stokes equations with Lévy noises

Taking the consideration of two-dimensional stochastic Navier–Stokes equations with multiplicative Lévy noises, where the noises intensities are related to the viscosity, a large deviation principle is established by using the weak convergence method skillfully, when the viscosity converges to 0. Due to the appearance of the jumps, it is difficult to close the energy estimates and obtain the desired convergence. Hence, one cannot simply use the weak convergence approach. To overcome the difficulty, one introduces special norms for new arguments and more careful analysis.

Large deviation principles for a 2D stochastic Allen–Cahn–Navier–Stokes driven by jump noise

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 Poincarà ⓒ Probab. Stat. 47(3) (2011) 725–747].

The non-Lipschitz stochastic Cahn–Hilliard–Navier–Stokes equations in two space dimensions

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.

Pullback dynamics of a 3D modified Navier-Stokes equations with double delays

<p style='text-indent:20px;'>This paper is concerned with the tempered pullback dynamics for a 3D modified Navier-Stokes equations with double time-delays, which includes delays on external force and convective terms respectively. Based on the property of monotone operator and some suitable hypotheses on the external forces, the existence and uniqueness of weak solutions can be shown in an appropriate functional Banach space. By using the energy equation technique and weak convergence method to achieve asymptotic compactness for the process, the existence of minimal family of pullback attractors has also been derived.</p>

Large deviations for stochastic models of two-dimensional second grade fluids Driven by Lévy Noise

In this paper, we establish a Freidlin–Wentzell-type large deviation principle for stochastic models of two-dimensional second grade fluids driven by Lévy noise. The weak convergence method introduced by Budhiraja, Dupuis and Maroulas plays a key role.

Discrete-Time Semi-Markov Random Evolutions in Asymptotic Reduced Random Media with Applications

This paper deals with discrete-time semi-Markov random evolutions (DTSMRE) in reduced random media. The reduction can be done for ergodic and non ergodic media. Asymptotic approximations of random evolutions living in reducible random media (random environment) are obtained. Namely, averaging, diffusion approximation and normal deviation or diffusion approximation with equilibrium by martingale weak convergence method are obtained. Applications of the above results to the additive functionals and dynamical systems in discrete-time produce the above tree types of asymptotic results.

Moderate deviations for stochastic models of two-dimensional second grade fluids

In this paper, we prove a central limit theorem and establish a moderate deviation principle for stochastic models of incompressible second grade fluids. The weak convergence method introduced by [6] plays an important role.

Moderate deviations for a fractional stochastic heat equation with spatially correlated noise

In this paper, we study the Moderate Deviation Principle for a perturbed stochastic heat equation in the whole space [Formula: see text]. This equation is driven by a Gaussian noise, white in time and correlated in space, and the differential operator is a fractional derivative operator. The weak convergence method plays an important role.

Large deviations for SPDEs of jump type

In this paper, we establish a large deviation principle for a fully nonlinear stochastic evolution equation driven by both Brownian motions and Poisson random measures on a given Hilbert space H. The weak convergence method plays an important role.