Random Perturbations of Ultraparabolic Partial Differential Equations under rescaling

这个建议是在非线性偏微分方程（PDE）的领域。更确切地说，我感兴趣的是证明严格收敛的解决方案的随机扰动非线性PDE的解决方案的有效确定性非线性PDE.I期待在不同的问题（一阶和二阶）的非线性偏微分方程，与适当的Hoermander向量场。Hoermander向量场（Carnot-Caratheodory空间）的几何是退化的，在这个意义上，某些方向的运动是禁止的（不可容许的）。一个向量场族被称为满足Hoermander条件（步长=k），如果这个向量场族的所有向量连同它们的直到某个k-1阶的所有分解子在任何一点生成整个切空间。如果Hoermander条件被满足，那么人们总是可以通过只沿着向量场方向上的路径（容许路径）去任何地方。与这些基本几何相关联的PDE问题的自然尺度是各向异性的。例如，考虑标准均匀椭圆/抛物PDE的齐次化，通常将极限取为取决于例如（x/y，y/z，z/z）的方程的趋于零（即小参数），其中（x，y，z）是三维欧几里得空间中的点。这意味着方程被各向同性地重新标度。另一方面，当考虑与Hoermander向量场相关的退化PDE时，重新缩放需要适应新的几何基础结构，例如，点（x，y，z）可以缩放为（x/ε，y/ε，z/ε ^2）。这些极限定理的研究中的挑战是找到不依赖于欧几里得结构的交换性或流形（点）和切空间（速度）之间的识别的方法。由于这种曲线的高度不规则性，测地线参数的使用有限，这使得问题更加复杂。因此，拟议的项目需要将分析、概率和几何的思想和技术复杂地结合起来。

This proposal is in the area of nonlinear partial differential equations (PDEs). More precisely I am interesting in proving rigorous convergence for solutions of a randomly perturbed nonlinear PDE to the solution of an effective deterministic nonlinear PDE.I look at different problems (both first-order and second-order) for nonlinear PDEs, associated to suitable Hoermander vector fields. The geometry of Hoermander vector fields (Carnot-Caratheodory spaces) is degenerate in the sense that some directions for the motion are forbidden (non admissible). A family of vector fields is said to satisfy the Hoermander condition (with step=k) if the vectors of the family together with all their commutators up to some order k-1 generate at any point the whole tangent space. If the Hoermander condition is satisfied, then one can always go everywhere by following only paths in the directions of the vector fields (admissible paths).The natural scaling for PDE problems associated to these underlying geometries is anisotropic. For example, thinking of homogenisation of a standard uniformly elliptic/parabolic PDE, one usually takes the limit as epsilon (i.e. a small parameter) tends to zero of an equation depending for example on (x/epsilon,y/epsilon,z/epsilon), where (x,y,z) is a point in the 3-dimensional Euclidean space. This means that the equation is isotropically rescaled. On the other end, when considering a degenerate PDE related to Hoermander vector fields, the rescaling needs to adapt to the new geometric underlying structure, e.g. a point (x,y,z) may scale as (x/epsilon,y/epsilon, z/epsilon^2). The challenge in the study of these limit theorems is to find approaches which do not rely on the commutativity of the Euclidean structure or on the identification between manifold (points) and tangent space (velocities). Further complications come from the limited use of geodesic arguments due to the highly irregular nature of such curves.Thus the proposed project requires an intricate combination of ideas and techniques from analysis, probability and geometry.