Measurable regularity properties of infinite-dimensional Lie groups (1601.02568v1)

Abstract: We consider differential equations of the form y'(t)=f(t,y(t)) on a (possibly infinite-dimensional) Lie group G, for f : [0,1] x G -> TG a time-dependent left invariant vector field with measurable (but not necessarily continuous) dependence on t. If a solution Evol(c):=y on [0,1] starting at the neutral element e of G exists for each f corresponding to an L^1-curve c : [0,1] -> g in the Lie algebra g of G, and Evol is smooth as a map from L^1([0,1],g) to C([0,1],G), then G is called L^1-regular. We show that all Banach-Lie groups are L^1-regular, as well as all direct limits of finite-dimensional Lie groups and many further classical examples of infinite-dimensional Lie groups, like diffeomorphism groups of paracompact finite-dimensional smooth manifolds. If a Lie group G is L^1-regular, then the Trotter product formula and the commutator formula hold in G, in a strong sense. The same conclusion holds if G merely satisfies certain weaker measurable regularity properties (like L^p-regularity), which are discussed as well. As a tool, we study differentiability properties of certain non-linear mappings to vector-valued Lebesgue spaces and spaces of vector-valued absolutely continuous functions.