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

[2020499] viens PI关于抛物型随机偏微分方程（SPDEs）行为的研究计划主要集中在四个主题：局部行为，长期行为，粒子表示和分数布朗运动（fBm）。目标是展示局部和长期现象之间的强烈相互依存关系，并开发可实现的粒子表示。PI考虑热方程的随机扰动，以及依赖于时间和空间的加性或乘性噪声。在时间上的依赖是白噪声型的，而在空间上的依赖可以跨越大范围的行为。微分算子是实线或圆（有时是高维）上的标准拉普拉斯算子，包含一个小的扩散参数“kappa”。在乘性噪声情况下，可能包含非线性。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中继承。要考虑的局部行为是解在空间变量中的连续性模。我们感兴趣的长期行为，具体到线性乘法的情况，是解的大时间指数渐近性（Lyapunov“指数”问题）。PI通过势本身的空间连续模量来表征这两种行为，将一些结果扩展到由fBm驱动的方程中，并设计了一种新的使用相互作用粒子系统求解的数值方法。结果表明，该解的Lyapunov指数为kappa的alpha/(1+alpha)次幂，其中alpha为该解的几乎确定空间Holder指数。还表明，当且仅当势的不定积分也成立时，解具有空间Holder指数。还考虑了使用分数噪声代替白噪声时这些结果如何变化的调查。对于非线性方程，构造了一个分支相互作用的粒子系统作为数值模拟方法的基础。证明了该数值格式在适当的缓和下收敛于连续函数值“cadlag”过程的Skorohod空间解。粒子相互作用是因为，当它们以均匀间隔的短间隔分支时，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种“适应度”计算的，并且明确取决于势的非线性。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在长期内可以表现出尺寸或能量的指数级增长，并且可以通过观察物理环境的短期空间变异性来精确预测指数级增长的速率。这在流体动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。这就是说，通过控制流体的介观湍流水平，可以精确地增强磁性流体中的磁场。该研究具有重要的教育意义。博士生和在PI的指导下，本科生和硕士生进行与粒子方法相关的计算机模拟，重点是设计有效的算法。这些模拟将作为课堂项目整合到一个强调计算金融和其他应用的新课程中，帮助学生为以技术为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，该研究汇集了当前概率论中非常有趣的几个领域：无限维随机分析，分支和相互作用粒子系统和数值方法，高斯正则性理论，李亚普诺夫指数。

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.