Large deviations and approximations for slow–fast stochastic reaction–diffusion equations

Abstract

A large deviation principle is derived for a class of stochastic reaction-diffusion partial differential equations with slow-fast components. The result shows that the rate function is exactly that of the averaged equation plus the fluctuating deviation which is a stochastic partial differential equation with small Gaussian perturbation. This result also confirms the effectiveness of the approximation of the averaged equation plus the fluctuating deviation to the slow-fast stochastic partial differential equations.

Averaging principle for slow-fast stochastic differential equations with time dependent locally Lipschitz coefficients

“…Cerrai and Lunardi studied the averaging principle for nonautonomous slow-fast systems of stochastic reaction-diffusion equations in [7]. For some further results on this topic, we refer to [4,12,22,23] and the references therein.However, the references we mentioned above always assume that the coefficients satisfy Lipschitz conditions, and there are few results on the average principle for SPDEs with nonlinear terms. For example, stochastic reaction-diffusion equations with polynomial coefficients [5], stochastic Burgers equation [9], stochastic two dimensional Navier-Stokes equations [19], stochastic Kuramoto-Sivashinsky equation [14], stochastic Schrödinger equation [15] and stochastic Klein-Gordon equation [16].…”

“…Cerrai and Lunardi studied the averaging principle for nonautonomous slow-fast systems of stochastic reaction-diffusion equations in [7]. For some further results on this topic, we refer to [4,12,22,23] and the references therein.…”

Averaging principle for stochastic real Ginzburg-Landau equation driven by $ \alpha $-stable process

“…Bogoliubov and Mitropolsky [2] first studied the averaging principle for the deterministic systems. The averaging principle for stochastic differential equations was proved by Khasminskii [25], see, e.g., [3,4,5,6,10,11,12,17,18,19,22,23,27,31,32,35] for further generalization. In most of the existing literature, Wiener noises are considered.…”

Averaging principle for slow–fast stochastic 2D Navier–Stokes equation driven by Lévy noise

“…For instance, stochastic Burgers equations, 21 stochastic wave equation, 22,23 stochastic Kuramoto–Sivashinsky equation, 24 stochastic Schrödinger equation, 25 stochastic Klein–Gordon equation, 26 stochastic Korteweg–de Vries equation, 27 stochastic 2D Navier–Stokes equation, 28 stochastic 3D fractional Leray‐ α model, 29 and SPDE with locally monotone coefficients 30 . The reader might refer to other studies 31‐34 and references therein for further results on this subject.…”