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

Abstract

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 nonGaussian. 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. 1999Academic Press

Existence, Uniqueness, Stability and Differentiability Properties of the Flow Associated to Weakly Differentiable Vector Fields

“…No general existence result for Sobolev (or even BV ) vector fields seems to be known in the infinite-dimensional case: the only reference we are aware of is [24], dealing with vector-fields having an exponentially integrable derivative, extending previous results in [49,50,51]. Also the investigation of nonEuclidean geometries, e.g.…”

The divergence of Banach space valued random variables on Wiener space

“…Thus 1 ∀n , so that by Lemma 3.4 (obviously F n → F weakly in L p (µ)) we conclude that F ∈ dom p δ and that δF L p (µ) ≤ C F p,1 .…”

“…Differentiation of random variables is generalized by stipulating differentiability subspaces other than H , smaller or larger, yielding Sobolev spaces which respectively contain or are contained in the standard ones D p, 1 .…”

“…For a fixed p > 1 denote s = {t ∈ I, |t − s| ≤ p−1 2p θ}. Next, let D t = η [1] t −η [0] t and for every λ ∈ [0, 1] consider the interpolated vector field η [λ] t = λη [1] …”

Elliptic equations for measures on infinite dimensional spaces and applications

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