Almost sure existence of global solutions for general initial value problems

This article is concerned with the almost sure existence of global solutions for initial value problems of the form (center dot) gamma ( t ) = v ( t, gamma ( t )) on separable dual Banach spaces. We prove a general result stating that whenever there exists ( mu t ) t is an element of R a family of probability measures satisfying a related statistical Liouville equation, there exist global solutions to the initial value problem for mu 0-almost all initial data, possibly without uniqueness. The main assumption is a mild integrability condition of the vector field v with respect to ( mu t ) t is an element of R . As a notable application, we obtain from the above principle that Gibbs and Gaussian measures yield low regularity global solutions for several nonlinear dispersive PDEs as well as fluid mechanics equations including the Hartree, Klein-Gordon, NLS, Euler and modified surface quasi-geostrophic equations. In this regard, our result generalizes Bourgain's method [14] as well as Albeverio & Cruzeiro's method [2] of constructing low regularity global solutions, without the need for local wellposedness analysis. (c) 2024 The Author(s). Published by Elsevier Inc. This is an open access article under the CC BY license (http:// creativecommons .org /licenses /by /4 .0/).