Exponential Integrability of Stochastic Convolutions

Seidler, Jan; Sobukawa, Takuya

doi:10.1112/s0024610702003745

Cited by 17 publications

(18 citation statements)

References 17 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…One of the first results is [CM90]; for recent results, see, for instance, [BP00b] and the references therein. A recent new approach relying on Zygmund's interpolation inequality is [SS03].…”

Section: Stochastic Convolutionmentioning

confidence: 99%

Amplitude Equations for Locally Cubic Nonautonomous Nonlinearities

Blömker¹

2003

SIAM J. Appl. Dyn. Syst.

View full text Add to dashboard Cite

Abstract. For systems of partial differential equations (PDEs) with locally cubic nonlinearities, which are perturbed by additive noise, we describe the essential dynamics for small solutions. If the system is near a change of stability, then a natural separation of time-scales occurs and the amplitudes of the dominant modes are given on a long time-scale by a stochastic ordinary differential equation. We consider applications to dynamic pitchfork bifurcation, pattern formation below the threshold of stability, and transient dynamics of stochastic PDEs near this deterministic bifurcations.Key words. amplitude equation, pattern formation, SPDE, slow modes, separation of time-scales, approximate center-manifold, bifurcation, multiple scale analysis AMS subject classifications. 60H15, 60H10, 37L55, 37L65DOI. 10.1137/S11111111034213551. Introduction. Amplitude equations are well known in the physics literature (see, e.g., [H83] or [W97]). They usually describe some order parameter for the system, which evolves on a much slower time-scale. This separation of time-scales occurs, for example, very naturally in a small neighborhood of bifurcations, where a change of stability occurs.Amplitude equations can be used either for spatially extended systems, where they are stochastic partial differential equations (SPDEs), or for systems on bounded domains, where they are given as stochastic ordinary differential equations (SODEs). This paper will focus on the second case, where an SODE describes the dynamics of the amplitudes of dominant modes evolving on some slow time-scale. On the other hand, all nondominant modes evolve rapidly on a fast time-scale, but they stay much smaller than the dominant ones. The modes in our context are given by the Fourier series expansion with respect to the eigenfunctions of the corresponding linearized operator. For deterministic systems the theory is rigorously understood even for spatially extended systems (see, e.g.,[KSM92,vH91] for the first results). However, there is a lack of results for stochastic systems. The only rigorous example is [BMS01] for a stochastic Swift-Hohenberg equation with periodic boundary conditions on a bounded interval. In this example, a complexvalued SODE was derived describing the amplitude of the dominant mode in a standard complex Fourier series on a very long time-scale.Our main theorems will extend the results of [BMS01] to a large class of SPDEs and systems of SPDEs. Moreover, our applications will demonstrate the power of this approach, when describing transient dynamics of stochastic equations.

show abstract

Section: Stochastic Convolutionmentioning

confidence: 99%

Amplitude Equations for Locally Cubic Nonautonomous Nonlinearities

Blömker¹

2003

SIAM J. Appl. Dyn. Syst.

View full text Add to dashboard Cite

show abstract

“…Many results of this type exist in the literature. For example, results similar to Lemma 3.8 below are well-known and can be found in [4,21,24], as well as the references therein. For our applications in Section 4, however, we need to establish the precise dependence of the constants in these results on the various underlying parameters.…”

Section: Probability Estimatesmentioning

confidence: 74%

“…For all 0 ≤ t ≤ min{T c , T q }, the estimates (23), (24), and (26), in combination with the assumption in (22), can now be used to establish…”

Section: Second Decomposition Stagementioning

confidence: 99%

Second phase spinodal decomposition for the Cahn-Hilliard-Cook equation

Blömker

Maier-Paape

Wanner

2008

Trans. Amer. Math. Soc.

View full text Add to dashboard Cite

Abstract. We consider the dynamics of a nonlinear partial differential equation perturbed by additive noise. Assuming that the underlying deterministic equation has an unstable equilibrium, we show that the nonlinear stochastic partial differential equation exhibits essentially linear dynamics far from equilibrium. More precisely, we show that most trajectories starting at the unstable equilibrium are driven away in two stages. After passing through a cylindrical region, most trajectories diverge from the deterministic equilibrium through a cone-shaped region which is centered around a finite-dimensional subspace corresponding to strongly unstable eigenfunctions of the linearized equation, and on which the influence of the nonlinearity is surprisingly small. This abstract result is then applied to explain spinodal decomposition in the stochastic Cahn-Hilliard-Cook equation on a domain G. This equation depends on a small interaction parameter ε > 0, and one is generally interested in asymptotic results as ε → 0. Specifically, we show that linear behavior dominates the dynamics up to distances from the deterministic equilibrium which can reach ε −2+dim G/2 with respect to the H 2 (G)-norm.

show abstract

“…In [5] Da Prato and Zabczyk investigated the stochastic convolution maximal function and established an inequality of type (2.1) where p − 2 appears instead of p − 1. In [15], the authors investigate behaviour of this maximal function near L 2 and they restrict themselves for p ≥ 2 + δ with some positive δ. Rather surprisingly Yano's theorem does not permit a straightforward "shift" of the situation from p → 1 + to p → 2 + .…”

Section: Statement Of Main Resultsmentioning

confidence: 99%