On flows associated to Sobolev vector fields in Wiener spaces: an approach \`a la DiPerna-Lions

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In this paper we extend the DiPerna-Lions theory on ODEs with Sobolev vector fields to the setting of abstract Wiener spaces.

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Well-posedness and large deviation for degenerate SDEs with Sobolev coefficients (2010, background): “In [14], Figalli proved the well-posedness of martingale soluti ns for SDE (1.2) with Sobolev coefficients by studying the associated Fokker-Planck equations.”
Continuity equations and ODE flows with non-smooth velocity* (2014, background): “A few existence result for Sobolev vector fields seem to be known in the infinite-dimensional case, see [31] and the more recent paper [22].”
The Flow Associated to Weakly Differentiable Vector Fields: Recent Results and Open Problems (2011, background): “the result in [ 14 ] seem closer to the finite-dimensional theory and are perfectly consistent with it (changing the reference measure from L d to γd in the same spirit of Section 3). Flows in P(R d ). As I anticipated, we may view (5.1) as an infinitedimensional ODE in P(R d ) and try to obtain exi”
Existence and uniqueness of solutions for Fokker–Planck equations on Hilbert spaces (2009, unknown): “, [3], [10]).”
Well posedness of Lagrangian flows and continuity equations in metric measure spaces (2014, methodology): “Given an H-valued field b = ∑ i b i hi , we introduce the derivation f 7→ b( f ) = ∑ i b i ∂ f/∂hi and we briefly compare our well-posedness results for the continuity equation with those contained in [Ambrosio and Figalli 2009].”