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

摘要

We prove the existence of the global flow {U_t} generated by a vector field A from a Sobolev class W~(1, 1)(#mu#) on a finite- or infinite-dimensional space X with a measure #mu#, provided #mu# is sufficiently smooth and that a nabla A and |#delta#_(#mu#)A| (where #delta#_(#mu#)A is the divergence with respect to #mu#) are exponentially integrable. In addition, the measure #mu# is shown to be quasi-invariant under {U_t}. In the case X=R~n and #mu#=p dx, where p implied by W~(1, 1)_(loc)(R~n) is a locally uniformly positive probability density, a sufficient condition is exp(c || nabla A ||)+exp(c |(A,(nabla p/p))|) implied by L~1(#mu#) 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 #mu# was a Gaussian measure. finally, we study flows generated by vector fields whose values are not necessarily in the Cameron-Martin space.