Absolutely Continuous Flows Generated by Sobolev Class Vector Fields in Finite and Infinite Dimensions

Abstract

We prove the existence of the global flow {Ut} generated by a vector field A from a Sobolev class W1, 1(μ) on a finite- or infinite-dimensional space X with a measure μ, provided μ is sufficiently smooth and that a ∇A and |δμA| (where δμA is the divergence with respect to μ) are exponentially integrable. In addition, the measure μ is shown to be quasi-invariant under {Ut}. In the case X=Rn and μ=pdx, where p∈W1, 1loc(Rn) is a locally uniformly positive probability density, a sufficient condition is exp(c‖∇A‖)+exp(c|(A, (∇p/p))|)∈L1(μ) for all c. In the infinite-dimensional case we get analogous results for measures differentiable along sufficiently many directions. Examples of measures which fit our framework, important for applications, are symmetric invariant measures of infinite-dimensional diffusions and Gibbs measures. Typically, in both cases such measures are essentially non-Gaussian. Our result in infinite dimensions significantly extends previously studied cases where μ was a Gaussian measure. Finally, we study flows generated by vector fields whose values are not necessarily in the Cameron–Martin space.