Analytical and geometrical problems involving non linear diffusion processes

这项研究项目的目的是发展对一些由非线性积分-微分方程式模拟的科学现象的数学理解。该项目的主旨集中在解决涉及反常(特别是积分)扩散过程的非线性问题，例如相变、流体动力学、博弈论和最优控制。正在研究的现象类型的具体例子包括，在连续介质力学方面，分析模拟表面现象的方程，如海洋-大气相互作用(准地转方程)或平面位错传播，以及多尺度上的扩散，如聚合物或湍流，而在概率方面，博弈论和金融学中的最优控制问题，涉及在变化的环境中不连续跳跃的随机变量(Levy过程或Levy游动)。从某种意义上说，本研究项目要研究的问题具有一定的普遍性，即相同的范式从几何和分析重现到流体动力学和材料科学，再到金融数学和最近的生物学和随机几何。拟议的研究计划将改善这些领域中建模和现象之间的相互作用。了解它们的基本结构将促进跨学科研究。

The purpose of this research project is to develop a mathematical understanding of a number of scientific phenomena that are modeled by nonlinear integro-differential equations. The main thrust of the project concentrates on solutions of nonlinear problems involving anomalous (in particular integral) diffusion processes arising, for instance, in phase transitions, fluid dynamics, game theory and optimal control. Specific examples of the type of phenomena being studied include, in the continuum mechanics side, the analysis of equations modeling surface phenomena, like ocean-atmosphere interaction (the quasigeostrophic equation) or planar dislocation propagation, and diffusion at multiple scales, like polymers or turbulent flow, while on the probability side, problems of optimal control in game theory and finance, involving random variables that jump discontinuously (Levy processes or Levy walks) in changing environments.The issues to be investigated in this research project have a certain universality in the sense that the same paradigm reappears from geometry and analysis, to fluid dynamics and material sciences, to financial mathematics and more recently biology and stochastic geometry. The proposed research program will improve the interplay between modeling and phenomena in these areas. Understanding their basic structure will promote interdisciplinary research.