Lagrangian flows driven by $$BV$$ fields in Wiener spaces

Trevisan, Dario

doi:10.1007/s00440-014-0589-1

Cited by 6 publications

(7 citation statements)

References 30 publications

(45 reference statements)

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“…Section 5 is devoted to the proof of uniqueness of solutions to the continuity equation. The classical proof in [DL89] is based on a smoothing scheme that, in our context, is played by the semigroup P (an approach already proved to be successful in [AF09], [Tre13], in Wiener spaces). For t fixed, one has to estimate carefully the so-called commutator C α (b t , u t ) := div((P α u t )b t ) − P α (div(u t b t ))…”

Section: Introductionmentioning

confidence: 99%

Well-posedness of Lagrangian flows and continuity equations in metric measure spaces

2014

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We establish, in a rather general setting, an analogue of DiPerna-Lions theory on wellposedness of flows of ODE's associated to Sobolev vector fields. Key results are a wellposedness result for the continuity equation associated to suitably defined Sobolev vector fields, via a commutator estimate, and an abstract superposition principle in (possibly extended) metric measure spaces, via an embedding into R ∞ . When specialized to the setting of Euclidean or infinite dimensional (e.g. Gaussian) spaces, large parts of previously known results are recovered at once. Moreover, the class of RCD(K, ∞) metric measure spaces introduced in [AGS14b] and object of extensive recent research fits into our framework. Therefore we provide, for the first time, wellposedness results for ODE's under low regularity assumptions on the velocity and in a nonsmooth context.

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Section: Introductionmentioning

confidence: 99%

Well-posedness of Lagrangian flows and continuity equations in metric measure spaces

2014

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“…[18], [28], [24], [25]; recently, in [23], properties of sets with nite Gaussian perimeter have been linked to some application in information technology. We point out also [26], for an application of BV functions to Lagrangian ows in Wiener spaces.…”

Section: Introductionmentioning

confidence: 99%

Some Fine Properties of BV Functions on Wiener Spaces

Ambrosio

Miranda

Pallara

2015

Analysis and Geometry in Metric Spaces

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Abstract:In this paper we de ne jump set and approximate limits for BV functions on Wiener spaces and show that the weak gradient admits a decomposition similar to the nite dimensional case. We also de ne the SBV class of functions of special bounded variation and give a characterisation of SBV via a chain rule and a closure theorem. We also provide a characterisation of BV functions in terms of the short-time behaviour of the Ornstein-Uhlenbeck semigroup following an approach due to Ledoux.

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“…We recover the uniqueness results for flows in Wiener spaces [8,Thm. 3.1], with the exception of the case p = 1 (and of the BV case in [34]). In Da Prato's setting, we obtain a result that quantitatively improves [21,Thm.…”

Section: Introductionmentioning

confidence: 99%