论文标题
随机PDE和积分方程的等等相：亚稳定性和其他属性
The Isochronal Phase of Stochastic PDE and Integral Equations: Metastability and Other Properties
论文作者
论文摘要
我们研究了随机演化系统中波浪，振荡和其他时空模式的动力学，包括SPDE和随机积分方程。将给定模式表示为确定性动力学的平滑，稳定的不变歧管，我们使用等等相位将随机动力学减少到该歧管上有限的尺寸SDE。等位相的定义是通过将歧管的邻域映射到歧管本身上来定义，类似于A.T. 〜Winfree和J.〜Guckenheimer定义为有限维振荡器定义的等质相。然后，我们确定一种概率度量，指示图案/波在歧管上徘徊时的随机扰动的平均位置。事实证明，该概率度量在时间尺度上是准确的，大于$ o（σ^{ - 2}）$，但小于$ o（\ exp（cσ^{ - 2}））$，其中$σ\ ll1 $是随机扰动的幅度。此外，使用此度量，我们确定了歧管上确定性和随机运动之间差异的预期速度。
We study the dynamics of waves, oscillations, and other spatio-temporal patterns in stochastic evolution systems, including SPDE and stochastic integral equations. Representing a given pattern as a smooth, stable invariant manifold of the deterministic dynamics, we reduce the stochastic dynamics to a finite dimensional SDE on this manifold using the isochronal phase. The isochronal phase is defined by mapping a neighbourbhood of the manifold onto the manifold itself, analogous to the isochronal phase defined for finite-dimensional oscillators by A.T.~Winfree and J.~Guckenheimer. We then determine a probability measure that indicates the average position of the stochastic perturbation of the pattern/wave as it wanders over the manifold. It is proved that this probability measure is accurate on time-scales greater than $O(σ^{-2})$, but less than $O(\exp(Cσ^{-2}))$, where $σ\ll1$ is the amplitude of the stochastic perturbation. Moreover, using this measure, we determine the expected velocity of the difference between the deterministic and stochastic motion on the manifold.
