Differential equations on infinite-dimensional Lie groups

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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.