Differential equations on infinite-dimensional Lie groups

无限维李群G上的两类微分方程在理论和应用上都有特别的意义，它们的解以及它们的参数依赖性：（a）G上时变右不变向量场的积分曲线（G的“正则性”）;（B）G上右不变弱黎曼度量的测地线。该项目调查了这两种情况的普遍性。在（a）中，代替光滑性或连续性，我们将仅假设向量场X_t对时间变量t的L^1依赖性（即，切空间T_eG中向量X_t（e）在G的中性元e处对t的L^1-依赖性。如果存在从e开始的绝对连续解并且光滑地依赖于向量场，则G称为L^1-正则的。我们将证明新的无限维李群类的L^1-正则性，并开发新的证明工具。我们还将探索L^1正则性的进一步加强（使用合适的向量测度代替L^1函数）。至于（B），我们感兴趣的是测地线的随机扰动和欧拉-拉格朗日方程的随机扰动，特别是与流体动力学相关的复同态李群（如保体积复同态或量子同态）及其Sobolev类似物。

Two types of differential equations on an infinite-dimensional Lie group G are of particular interest for theory and applications, as well as their solutions and the parameter- dependence of the latter: (a) Integral curves for time-dependent right-invariant vector fields on G ("regularity" of G); (b) Geodesics for right-invariant weak Riemannian metrics on G. The project investigates generalizations of both situations. In (a), instead of smoothness or continuity we shall only assume L^1-dependence of the vector field X_t on the time variable t (i.e., L^1-dependence on t for the vector X_t(e) in the tangent space T_eG at the neutral element e of G). If absolutely continuous solutions starting at e exist and depend smoothly on the vector field, then G is called L^1-regular. We shall prove L^1-regularity for new classes of infinite-dimensional Lie groups and develop new tools for the proof. We shall also explore further strengthenings of L^1-regularity (using suitable vector measures in place of L^1-functions). As to (b), stochastic perturbations of geodesics are of interest and stochastic perturbations of Euler-Lagrange equations, notably for Lie groups of diffeomorphisms related to fluid dynamics (like volume-preserving diffeomorphisms or quantomorphisms) and their Sobolev analogues.