A dynamical approximation for stochastic partial differential equations

Abstract

Random invariant manifolds provide geometric structures for understanding stochastic dynamics. In this paper, a dynamical approximation estimate is derived for a class of stochastic partial differential equations, by showing that the random invariant manifold is almost surely asymptotically complete. The asymptotic dynamical behavior is thus described by a stochastic ordinary differential system on the random invariant manifold, under suitable conditions. As an application, stationary states ͑invariant measures͒ are considered for a class of stochastic hyperbolic partial differential equations.

Approximation of Stochastic Invariant Manifolds

“…Even so, the stochastic theory is still much less complete than its deterministic counterpart. For instance the reduction problem of a stochastic partial differential equation (SPDE) to its corresponding stochastic invariant manifolds has been much less studied and only a few works in that direction are available [18,29,40,67,103,145,155]. The practical aspects of the reduction problem of a deterministic dynamical system to its corresponding (local) center, center-unstable or unstable manifolds have been well investigated in various finite-and infinite-dimensional settings; see, e.g., [13,23,30,59,70,74,75,90,91,92,98,100,108,111,117,121,133,134].…”

Averaging, Homogenization and Slow Manifolds for Stochastic Partial Differential Equations

“…For the random dynamical system ϕ(t, ω) generated by the random partial differential equations (3.10), we have the following result [29,Lemma 4.5] . Before we prove Theorem 3.1, we need the following lemma, which implies the backward solvability of the system (3.10) restricted on the invariant manifold M(ω).…”

“…So we further need a detail description of the system on the random invariant manifold. 29,6 However the inclusion of stochastic force leads to different model from the deterministic case, especially when we are concerned with the behavior of the system on a long time scale. For example, due to coupling between fast and slow parts, the stochastic force on microscopic scale may be fed into the macroscopic scale during the approach to derive the lower dimensional simplified system and on a long time scale, such stochastic force appears as Gaussian white noise.…”

“…The reduced model is derived by using the notion of random asymptotic completeness of random invariant manifold. 29 The result is further applied to derive a local reduced model for SPDEs with non-Lipschitz nonlinearity, quadratic nonlinearity, and the shape of the local random invariant manifold is also described in section 3.2.2. 6 The lower dimensional reduced model obtained by random slow manifold is in fact a non-autonomous stochastic differential equation in a lower dimensional space, 29 which is usually not easy to be treated to study dynamics of the system on a long time scale.…”

Stochastic D-bifurcation for a damped sine-Gordon equation with noise

“…3 Blömker 4 studied the amplitude equations approximating the dynamics of the original stochastic partial differential equations with boundary conditions by the multiscale method. The stochastic center manifold theory has been used to reduce dimensions of the infinite dimensional dynamical system perturbed by noises in Wang et al, 5 and the bifurcation behaviors can be analyzed locally without loss of information.…”