  This article is concerned with the almost sure existence of global solutions
for initial value problems of the form $\dot{\gamma}(t)= v(t,\gamma(t))$ on
separable dual Banach spaces. We prove a general result stating that whenever
there exists $(\mu_t)_{t\in \mathbb{R}}$ 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 $(\mu_t)_{t\in \mathbb{R}}$.
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 as well as Albeverio &
Cruzeiro's method of constructing low regularity global solutions, without the
need for local well-posedness analysis.
