Generation of interfaces for multi-dimensional stochastic Allen–Cahn equation with a noise smooth in space
Generation of interfaces for multi-dimensional stochastic Allen–Cahn equation with a noise smooth in space
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空间噪声平滑的多维随机 Allen–Cahn 方程接口的生成
DOI：
10.1080/17442508.2018.1426586
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发表时间：
2016
期刊：
影响因子：
0.9
通讯作者：
Kai Lee
中科院分区：
文献类型：
--
作者：
Kai Lee
摘要本文研究了具有一般初值的随机Allen-Cahn方程在外部噪声由Q-布朗运动给出的多维情形下的界面生成问题。证明了带标度参数的d维随机Allen-Cahn方程的界面在阶时刻产生。特别地，在一维情况下，我们给出了比我们以前的结果更详细的生成界面的估计和形状。假设Q-布朗运动在空间变量上是光滑的，为了证明生成性，我们将偏微分方程的一个比较定理推广到单点微分方程。此外，我们将生成的界面与界面的一维运动联系起来。在这种情况下，我们考虑仅在时间上乘以的白色噪声作为噪声项，其中a是具有紧支集的平滑函数。这是Q-布朗运动的特例。为了研究界面的运动，我们采用了阶的时间尺度。
Abstract In this paper, we study the generation of interfaces for a stochastic Allen–Cahn equation with general initial value in the multi-dimensional case that external noise is given by Q-Brownian motion. We prove that interfaces, for d-dimensional stochastic Allen–Cahn equation with scaling parameter , are generated at the time of order . Especially, in one-dimensional case, we give more detailed estimate and shape of the generated interface than that obtained in our previous result. Assuming that the Q-Brownian motion is smooth in space variable, we extend a comparison theorem for PDE to SPDE’s in order to prove the generation. Moreover, we connect the generated interface to the motion of interface in one-dimension. In this case, we consider the white noise only in time multiplied by as the noise term, where a is a smooth function which has a compact support. This is the special case of Q-Brownian motion. We take the time scale of order for studying the motion of interface.