Absolutely continuous flows generated by Sobolev class vector fields in finite and infinite dimensions

We prove the existence of the global flow fU t g generated by a vector field A from a Sobolev class W 1;1 (¯) on a finite or infinite dimensional space X with a measure ¯, provided ¯ is sufficiently smooth, and rA and jffi ¯ Aj (where ffi ¯ A is the divergence with respect to ¯) are exponentially integrable. In addition, the measure ¯ is shown to be quasi-invariant under fU t g. In the case X = IR n and ¯ = pdx, where p 2 W 1;1 loc (IR n ) is a locally uniformly positive probability density, a sufficient condition is: exp(ckrAk)+ exp(cj(A; rp p )j) 2 L 1 (¯) 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 pr..