Absolutely Continuous Flows Generated by Sobolev Class Vector Fields in Finite and Infinite Dimensions
✍ Scribed by Vladimir Bogachev; Eduardo Mayer-Wolf
- Publisher
- Elsevier Science
- Year
- 1999
- Tongue
- English
- Weight
- 434 KB
- Volume
- 167
- Category
- Article
- ISSN
- 0022-1236
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✦ Synopsis
We prove the existence of the global flow [U t ] 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 that a {A and |$ + A| (where $ + A is the divergence with respect to +) are exponentially integrable. In addition, the measure + is shown to be quasi-invariant under [U t ]. In the case X=R n and += p dx, where p # W 1, 1 loc (R n ) is a locally uniformly positive probability density, a sufficient condition is exp(c &{A&)+exp(c |(A, ({p p))| ) # L 1 (+) for all c. In the infinitedimensional 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 + was a Gaussian measure. Finally, we study flows generated by vector fields whose values are not necessarily in the Cameron Martin space.