Particles, Stochastic Partial Differential Equations, Random Fields

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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.