Stochastic PDEs: Interdependence of Local and Long-term Behaviors, and Representation

随机偏微分方程：局部和长期行为的相互依赖性以及表示

• 批准号: 0204999
• 负责人: Frederi Viens
• 金额: $ 12.2万
• 依托单位: Purdue University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2002
• 资助国家: 美国
• 起止时间: 2002-09-01 至 2006-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0204999&HistoricalAwards=false
• 关键词: Stochastic; PDEs; Interdependence; Local; Long

0204999Viens The PI's research program on the behavior of parabolic stochastic partial differential equations (SPDEs) focuses on four topics: local behavior, long-term behavior, particle representation, and fractional Brownian motion (fBm). The goals are to exhibit a strong interdependence between local and long-term phenomena and to develop an implementable particle representation. The PI considers stochastic perturbations of the heat equation, with additive or multiplicative noise that depends on time and space. The dependence in time is of white-noise type, while the dependence in space can span a wide range of behaviors. The differential operator is the standard Laplacian on the real line or on the circle (or sometimes in higher dimensions), with a small diffusion parameter "kappa" included. In the multiplicative noise case, a nonlinearity may be included. The basic point of view is simple: an SPDE with a unique solution is an input-output system, and as such the solution's behavior must be inherited from the equation's coefficients. The local behavior to be considered is the modulus of continuity of the solution in the space variable. The long term behavior of interest, specific to the linear multiplicative case, is the large-time exponential asymptotics of the solution (Lyapunov "exponent" question). The PI characterizes both of these behaviors via the spatial modulus of continuity of the potential itself, extends some of the results to equations driven by fBm, and designs a new numerical method for the solution using a system of interacting particles. It is shown that the solution's Lyapunov exponent is of the order of kappa to the power of alpha/(1+alpha) where alpha is the almost-sure spatial Holder exponent of the solution. It is also shown that the solution has spatial Holder exponent alpha if and only if the same holds for the potential's antiderivative. An investigation of how these results change when one uses fractional noise instead of white noise is also considered. For the nonlinear equation, a branching and interacting particle system is constructed as the basis for a numerical method for simulating the solution. It is proved that the numerical scheme, when properly mollified, converges to the solution in the Skorohod space of continuous-function-valued "cadlag" processes. The particles interact because, as they branch at evenly spaced short intervals, their mean number of offspring is a random variable whose mean is calculated relative to a certain "fitness" of all the other particles, and depends explicitly on the non-linearity of the potential. The main physical meaning of the research is that a complex space-time-dependent system with significant random perturbations can exhibit an exponentially strong increase of size or energy in the long term, and that the exponential rate of increase can be predicted precisely by looking at the short-range spatial variability of the physical environment. This has potentially important applications in hydrodynamics and turbulence, including a local characterization of the so-called "fast dynamo effect." It would say that a magnetic field in a magnetic fluid can be enhanced precisely by controlling the fluid's mezoscopic level of turbulence. The research has important educational effects. Ph.D. students and, under the PI's supervision, undergraduates and MS students, conduct computer simulations in connection with the particle methods, with an emphasis on designing efficient algorithms. The simulations are being integrated as class projects in a new curriculum emphasis in computational finance and other applications, preparing the students for the technology-dominated job market and workplace. From the purely theoretical point of view, the research brings together several areas of great current interest in probability theory: infinite dimensional stochastic analysis, branching and interacting particle systems and numerical methods, Gaussian regularity theory, Lyapunov exponents.

0204999 Viens PI关于抛物型随机偏微分方程（SPDE）行为的研究计划集中在四个主题：局部行为，长期行为，粒子表示和分数布朗运动（fBM）。我们的目标是表现出强烈的相互依存关系之间的本地和长期的现象，并制定一个可实施的粒子表示。PI考虑热方程的随机扰动，具有取决于时间和空间的加性或乘性噪声。时间上的依赖是白噪声类型的，而空间上的依赖可以跨越广泛的行为。微分算子是在真实的直线上或圆上（有时在更高的维度上）的标准拉普拉斯算子，其中包括小的扩散参数“kappa”。在乘性噪声的情况下，可以包括非线性。基本观点很简单：具有唯一解的SPDE是一个输入输出系统，因此解的行为必须从方程的系数继承。要考虑的局部行为是空间变量中解的连续模。长期的利益行为，具体到线性乘法的情况下，是大时间指数渐近的解决方案（李雅普诺夫“指数”问题）。PI通过势本身的空间连续性模量来表征这两种行为，将一些结果扩展到由fBm驱动的方程，并设计了一种新的数值方法，用于使用相互作用粒子系统的解决方案。结果表明，该解的李雅普诺夫指数为kappa的alpha/（1+alpha）次方，其中alpha为解的几乎确定空间保持器指数.它还表明，该解决方案具有空间保持器指数α当且仅当同样适用于潜在的反导数。当使用分数噪声而不是白色噪声时，这些结果如何变化的调查也被考虑。对于非线性方程，一个分支和相互作用的粒子系统的基础上构造的数值方法模拟的解决方案。证明了该数值格式在适当的软化条件下收敛于连续函数值cadlag过程的Skorohod空间的解。粒子之间相互作用是因为，当它们以均匀的短间隔分支时，它们后代的平均数量是一个随机变量，其平均值是相对于所有其他粒子的某个“适合度”计算的，并且明确地取决于势的非线性。 该研究的主要物理意义在于，具有显著随机扰动的复杂时空依赖系统可以在长期内表现出指数级的规模或能量增长，并且可以通过观察物理环境的短期空间变化来精确预测指数增长率。这在流体力学和湍流中具有潜在的重要应用，包括所谓的“快速发电机效应”的局部表征。“它会说，磁性流体中的磁场可以通过控制流体的中等湍流水平来精确增强。该研究具有重要的教育效果。博士在PI的监督下，学生，本科生和MS学生，进行与粒子方法有关的计算机模拟，重点是设计有效的算法。这些模拟被整合为计算金融和其他应用的新课程重点中的课堂项目，为学生在技术主导的就业市场和工作场所做好准备。从纯理论的角度来看，该研究汇集了概率论中当前最感兴趣的几个领域：无限维随机分析，分支和相互作用的粒子系统和数值方法，高斯正则性理论，李雅普诺夫指数。