粗糙随机偏微分方程的可积分约束及其在不变流形和稳定性方面的应用

IF 1.7 2区数学 Q1 MATHEMATICS

Journal of Functional Analysis Pub Date : 2024-09-16 DOI:10.1016/j.jfa.2024.110676

M. Ghani Varzaneh, S. Riedel

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引用次数: 0

摘要

我们研究的是 Gerasimovičs 和 Hairer (2019) [31] 中引入的半线性粗糙随机偏微分方程。我们为方程由合适的高斯过程驱动时的解及其线性化提供了 Lp(Ω)-integrable 先验边界。利用巴拿赫空间的乘法遍历定理，我们可以推导出线性化方程在静止点附近存在李亚普诺夫谱。我们还提供了静止点周围存在的局部稳定流形、不稳定流形和中心流形。在所有 Lyapunov 指数都为负的情况下，可以推导出局部指数稳定性。我们用几个例子来说明我们的发现。

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An integrable bound for rough stochastic partial differential equations with applications to invariant manifolds and stability

We study semilinear rough stochastic partial differential equations as introduced in Gerasimovičs and Hairer (2019) [31]. We provide $L^{p} (Ω)$ -integrable a priori bounds for the solution and its linearization in case the equation is driven by a suitable Gaussian process. Using the multiplicative ergodic theorem for Banach spaces, we can deduce the existence of a Lyapunov spectrum for the linearized equation around stationary points. The existence of local stable, unstable, and center manifolds around stationary points is provided. In the case where all Lyapunov exponents are negative, local exponential stability can be deduced. We illustrate our findings with several examples.

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来源期刊

Journal of Functional Analysis 数学-数学

CiteScore

3.20

自引率

5.90%

发文量

271

审稿时长

7.5 months

期刊介绍： The Journal of Functional Analysis presents original research papers in all scientific disciplines in which modern functional analysis plays a basic role. Articles by scientists in a variety of interdisciplinary areas are published. Research Areas Include: • Significant applications of functional analysis, including those to other areas of mathematics • New developments in functional analysis • Contributions to important problems in and challenges to functional analysis