Particles, Stochastic Partial Differential Equations, Random Fields

小行星9703648 研究人员将致力于推导出物理上有意义的随机 从扩散和相互作用的粒子系统的偏微分方程（SPDE）。 与传统的N扩散颗粒方法的主要区别在于 波动将由“相关”布朗运动表示（表示 波动力），其中相关性的强度取决于两个 给定粒子和一个参数，即所谓的关联长度。因此，SPDE 将取决于作为参数的相关长度。宏观和中心极限 定理是预期的，因为相关长度趋于零。此外，调查员 将研究SPDE的定性性质。均匀、各向同性和稳定 随机场应作为SPDE的解导出， 随机场的SPDE的系数进行。 从数学上讲，拟议的工作将在4个不同领域之间建立一座桥梁， 数学，即粒子系统，SPDE，随机场和偏微分 方程 许多物理、生物和经济现象都是由大量的 件.组成部分的大量和快速变化使得精确的 描述所有组件的时间演变非常困难，如果不是不可能的话。 例如湍流、化学反应、扩散、疾病传播（病毒， 细菌），人口增长，环境污染，气候，天气，金融市场 为了建立这些现象的数学模型，原子的真实的世界系统， 分子、病毒等被随机粒子系统所取代， 作为真实的世界系统的良好预测器。一个更传统的模型， 同样的现象是宏观（确定性）偏微分方程，许多 在世纪以前，它成为了不同领域的理论工具， 物理学和工程学，以及最近在生物科学的某些领域， 经济学这样的方程通常描述“在一个特定的时间点上（某种物质）有多少质量”。 在给定的时间给定的地点。“从数学上讲，大多数宏观方程都可以 从粒子系统的极限程序，假设粒子的数量 它们的质量变得非常小， 粒子总是统计上不相关的。这个推导也解释了为什么 宏观方程通常不能正确地预测真实的世界的时间演化 系统，但从长远来看，它们似乎是许多观察的平均值。 然而，宏观方程比粒子系统更容易计算。的 研究者将宏观偏微分方程的模型扩展到一个模型 偏微分方程（Partial Differential Equations）介观方程应 从粒子系统中导出的宏观方程相同的方式，但根据 对粒子系统的更现实的假设，即粒子的运动 在统计学上是相关的，当它们彼此接近时。 因此， 方程应该比宏观方程更符合观测结果。 方程，同时保持后者的计算简单性。而且 宏观方程将作为介观方程的极限情况出现。 介观方程在物理化学、生物科学、 流体力学和金融市场。

9703648 Kotelenez The investigator will work on the derivation of physically meaningful stochastic partial differential equations (SPDEs) from systems of diffusing and interacting particles. The main difference from the traditional approach for N diffusing particles is that the fluctuations will be represented by "correlated" Brownian motions (representing the fluctuation forces), where the strength of the correlations depends on the distance of two given particles and a parameter, the so-called correlation length. Consequently, the SPDEs will depend on the correlation length as a parameter. Macroscopic and central limit theorems are expected, as the correlation length tends to zero. Further, the investigator will study the qualitative properties of the SPDEs. Homogeneous, isotropic and stationary random fields shall be derived as solutions of the SPDEs, and the spectral analysis of the random fields in terms of the coefficients of the SPDEs shall be conducted. Mathematically, the proposed work will build a bridge between 4 different areas of mathematics, namely particle systems, SPDEs, random fields and partial differential equations. Many physical, biological and economic phenomena consist of a large number of components. The large number and the rapid changes of the components make an exact description of the time evolution of all components very difficult , if not impossible. Examples are turbulence, chemical reactions, diffusion, spreading of diseases (viruses, bacteria), population growth, environmental pollution, climate, weather, financial markets etc. To build a mathematical model of those phenomena, the real world systems of atoms, molecules, viruses etc. are replaced by random particle systems whose time evolution can serve as a good predictor for the real world systems. A more traditional model for the same phenomena are macroscopic (deterministic) partial differential equations, many of which, more than a century ago, became the ma in theoretical tools in different areas of physics and engineering and also, more recently, in some areas of the biosciences and economics. Such an equation typically describes "how much mass (of some matter) is at a given place at a given time." Mathematically, most of those macroscopic equations can be derived from particle systems by a limit procedure, assuming that the number of particles becomes infinite and their respective masses become very small, and that the motion of particles is always statistically uncorrelated. This derivation also explains why the macroscopic equations often do not correctly predict the time evolution of the real world systems, but that in the long run they appear to be an average of many observations. Macroscopic equations, however, are easier to compute than particle systems. The investigator will extend the model of macroscopic partial differential equations to a model of mezoscopic (stochastic) partial differential equations. The mezoscopic equations shall be derived from particle systems in the same way as the macroscopic equations, but under more realistic assumptions on the particle systems, namely that the motion of particles is statistically correlated, when they are close to one another. As a result the mezoscopic equations should be in better agreement with observations than the macroscopic equations, while preserving the computational simplicity of the latter ones. Moreover, the macroscopic equations will appear as the limiting case of the mezoscopic equations. Mezoscopic equations will have diverse applications in physical chemistry, the biosciences, fluid mechanics and financial markets.