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Random Perturbations of Ultraparabolic Partial Differential Equations under rescaling

重标度下超抛物型偏微分方程的随机扰动

基本信息

批准号：
EP/N003209/1
负责人：
Federica Dragoni
金额：
$ 12.73万
依托单位：
CARDIFF UNIVERSITY
依托单位国家：
英国
项目类别：
Research Grant
财政年份：
2015
资助国家：
英国
起止时间：
2015 至无数据
项目状态：
已结题

来源：
https://gtr.ukri.org/projects?ref=EP%2FN003209%2F1
关键词：
Random Perturbations Ultraparabolic Partial Differential

项目摘要

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.

这个建议是在非线性偏微分方程（PDEs）的领域。更确切地说，我感兴趣的是证明随机扰动非线性PDE的解对有效确定性非线性PDE的解的严格收敛性。我看不同的问题（一阶和二阶）非线性偏微分方程，相关的合适的Hoermander向量场。Hoermander向量场（Carnot-Caratheodory空间）的几何是简并的，即运动的某些方向是禁止的（不可接受的）。如果一组向量场的向量及其k-1阶以下的交换子在任意点生成整个切空间，则该向量场族满足Hoermander条件（步长=k）。如果满足Hoermander条件，则只要沿着向量场方向上的路径（可容许路径）就可以到达任何地方。与这些底层几何图形相关的偏微分方程问题的自然缩放是各向异性的。例如，考虑标准均匀椭圆/抛物PDE的均质化，人们通常将极限取为依赖于（x/epsilon,y/epsilon,z/epsilon）的方程的epsilon（即小参数）趋于零，其中（x,y,z）是三维欧几里得空间中的一个点。这意味着方程是各向同性重新标度的。另一方面，当考虑与Hoermander向量场相关的退化偏微分方程时，重新缩放需要适应新的几何底层结构，例如，点（x,y,z）可以缩放为（x/epsilon,y/epsilon, z/epsilon^2）。研究这些极限定理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，对测地线论证的使用受到限制，这就更加复杂了。因此，拟议的项目需要从分析，概率和几何的思想和技术的复杂组合。