Absolutely continuous flows generated by Sobolev class vector fields in finite and infinite dimensions. (English) Zbl 0956.60079
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 sufficiently smooth measure \(\mu\) is proved in the case where \(\nabla A\) and the divergence of \(A\) with respect to \(\mu\) are exponentially integrable. The uniqueness theorems for these cases are also obtained. Among the examples of measures for which the developed theory works there are symmetric invariant measures of infinite-dimensional diffusions and Gibbs measures (which are typically essentially non-Gaussian). Moreover, the flows whose values are not necessarily in Cameron-Martin space are also considered.
Reviewer: Serguei Zelik (Moskva)
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