Differential equations on infinite-dimensional Lie groups

无限维李群G上的两类微分方程及其解和参数依赖关系是非常有理论和应用价值的：(A)时间相关的右不变向量场在G上的积分曲线(“正则性”)；(B)G上右不变弱黎曼度量的测地线。在(A)中，我们只假定向量场X_t对时间变量t的L^1-依赖性，而不是光滑性或连续性(即，在G的中立元e处切空间T_e中的向量X_t(E)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，则称G为L^1-正则。我们将证明新的无限维李群的L^1-正则性，并开发新的证明工具。我们还将探索进一步加强L^1-正则性(用适当的向量度量代替L^1-函数)。关于(B)，测地线的随机扰动和欧拉-拉格朗日方程的随机扰动是有意义的，特别是与流体动力学有关的微分同胚李群(如保体积微分同态或量子同态)及其Sobolv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.