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Register a new userLarge deviations for slow-fast stochastic partial differential equations
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A large deviation principle is derived for stochastic 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 also confirms the effectiveness of the approximation of the averaged equation plus the fluctuating deviation to the slow-fast stochastic partial differential equations.
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This paper is devoted to proving the strong averaging principle for slow-fast stochastic partial differential equations with locally monotone coefficients, where the slow component is a stochastic partial differential equations with locally monotone coefficients and the fast component is a stochastic partial differential equations (SPDEs) with strongly monotone coefficients. The result is applicable to a large class of examples, such as the stochastic porous medium equation, the stochastic $p$-Laplace equation, the stochastic Burgers type equation and the stochastic 2D Navier-Stokes equation, which are the nonlinear stochastic partial differential equations. The main techniques are based on time discretization and the variational approach to SPDEs.
By using the technique of the Zvonkins transformation and the classical Khasminkiis time discretization method, we prove the averaging principle for slow-fast stochastic partial differential equations with bounded and H{o}lder continuous drift coefficients. An example is also provided to explain our result.
Averaging is an important method to extract effective macroscopic dynamics from complex systems with slow modes and fast modes. This article derives an averaged equation for a class of stochastic partial differential equations without any Lipschitz assumption on the slow modes. The rate of convergence in probability is obtained as a byproduct. Importantly, the deviation between the original equation and the averaged equation is also studied. A martingale approach proves that the deviation is described by a Gaussian process. This gives an approximation to errors of $mathcal{O}(e)$ instead of $mathcal{O}(sqrt{e})$ attained in previous averaging.
Models of self-organized criticality, which can be described as singular diffusions with or without (multiplicative) Wiener forcing term (as e.g. the Bak/Tang/Wiesenfeld- and Zhang-models), are analyzed. Existence and uniqueness of nonnegative strong solutions are proved. Previously numerically predicted transition to the critical state in 1-D is confirmed by a rigorous proof that this indeed happens in finite time with high probability.
In this paper, we consider the averaging principle for a class of McKean-Vlasov stochastic differential equations with slow and fast time-scales. Under some proper assumptions on the coefficients, we first prove that the slow component strongly converges to the solution of the corresponding averaged equation with convergence order $1/3$ using the approach of time discretization. Furthermore, under stronger regularity conditions on the coefficients, we use the technique of Poisson equation to improve the order to $1/2$, which is the optimal order of strong convergence in general.
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