Transport equations and quasi-invariant flows on the Wiener space

Fang, Shizan; Luo, Dejun

doi:10.1016/j.bulsci.2009.01.001

Cited by 25 publications

(39 citation statements)

References 10 publications

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“…Ambrosio [2] generalized the results to the case where the coefficients have only bounded variation regularity by considering the continuity equation. These results have recently been generalized into different settings, [3] and [10] for infinite dimensional spaces, [11] and [20] for generalizations to transport-diffusion equations and its associated stochastic differential equations.…”

Section: Introductionmentioning

confidence: 99%

Lp-solutions of the stochastic transport equation

Catuogno

Olivera

2013

Random Operators and Stochastic Equations

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We consider the stochastic transport linear equation and we prove existence and uniqueness of weak L p −solutions. Moreover, we obtain a representation of the general solution and a Wong-Zakai principle for this equation. We make only minimal assumptions, similar to the deterministic problem. The proof is supported on the generalized Itô-Ventzel-Kunita formula (see [15]) and the theory of Lions-DiPerna on transport linear equation (see [9]).Key words: Stochastic perturbation, Transport equation, Itô formula. MSC2000 subject classification: 60H10 , 60H15 . IntroductionIn this article we establish global existence and uniqueness of solution of the transport linear equation with a stochastic perturbation. Namely, we consider the following equation:

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Section: Introductionmentioning

confidence: 99%

Lp-solutions of the stochastic transport equation

Catuogno

Olivera

2013

Random Operators and Stochastic Equations

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“…It is at this point that, in [2] and [11], exponential integrability is necessary; for our range condition we do not need it. Concerning the problem of proving a range condition itself, this is usually done by means of gradient estimates on solutions, which is a difficult problem; here we have the gradient estimate for free, see (2.8), because it holds for the P -regularized solution.…”

Section: Du(t X) Y = U (T X)(y) Y ∈ Hmentioning

confidence: 99%

“…This choice of a reference measure was proposed in [2] and they proved existence and uniqueness of solutions to (1.2) under certain conditions on the weak derivative and exponential μ-integrability conditions on its μ-divergence. (see [2,Theorem 3.1], see also [11] for improvements of the results on the corresponding transport equation in [2]). …”

Section: Du(t X) Y = U (T X)(y) Y ∈ Hmentioning

confidence: 99%

“…More precisely, we prove a suitable rank condition for the Kolmogorov operator in (1.3). But to implement this idea we have to regularize with a much more smoothing Ornstein-Uhlenbeck semi-group than the one in [2,11] (see Sect. 2 below).…”

Section: Du(t X) Y = U (T X)(y) Y ∈ Hmentioning

confidence: 99%

“…Choosing a reference measure as in [2,11] we also have to restrict to a sub-class of solutions μ t , t ∈ [0, ∞), to (1.2), namely those satisfying…”

Section: Du(t X) Y = U (T X)(y) Y ∈ Hmentioning

confidence: 99%

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Uniqueness for continuity equations in Hilbert spaces with weakly differentiable drift

Prato

Flandoli

Röckner³

2014

Stoch PDE: Anal Comp

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We prove uniqueness for continuity equations in Hilbert spaces H . The corresponding drift F is assumed to be in a first order Sobolev space with respect to some Gaussian measure. As in previous work on the subject, the proof is based on commutator estimates which are infinite dimensional analogues to the classical ones due to DiPerna-Lions. Our general approach is, however, quite different since, instead of considering renormalized solutions, we prove a dense range condition implying uniqueness. In addition, compared to known results by Ambrosio-Figalli and Fang-Luo, we use a different approximation procedure, based on a more regularizing OrnsteinUhlenbeck semigroup and consider Sobolev spaces of vector fields taking values in H rather than the Cameron-Martin space of the Gaussian measure. This leads to different conditions on the derivative of F, which are incompatible with previous work on the subject. Furthermore, we can drop the usual exponential integrability conditions on the Gaussian divergence of F, thus improving known uniqueness results in this respect.

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Horizontal Diffusion in C 1 Path Space

Arnaudon

Coulibaly

Thalmaier

2010

Séminaire De Probabilités XLIII

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We define horizontal diffusion in C 1 path space over a Riemannian manifold and prove its existence. If the metric on the manifold is developing under the forward Ricci flow, horizontal diffusion along Brownian motion turns out to be length preserving. As application, we prove contraction properties in the Monge-Kantorovich minimization problem for probability measures evolving along the heat flow. For constant rank diffusions, differentiating a family of coupled diffusions gives a derivative process with a covariant derivative of finite variation. This construction provides an alternative method to filtering out redundant noise.

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Transport equations and quasi-invariant flows on the Wiener space

Cited by 25 publications

References 10 publications

Lp-solutions of the stochastic transport equation

Lp-solutions of the stochastic transport equation

Uniqueness for continuity equations in Hilbert spaces with weakly differentiable drift

Horizontal Diffusion in C 1 Path Space

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