Strong solutions of SDE's with generalized drift and multidimensional fractional Brownian initial noise

Baños, David R.; Ortiz-Latorre, Salvador; Proske, Frank

doi:10.48550/arxiv.1705.01616

Cited by 6 publications

(19 citation statements)

References 25 publications

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“…, p. We may then write In what follows we need an important estimate (see e.g. Proposition 3.3 in the second revision of [12] for the improved version of the result that we state below). In order to state this result, we need some notation.…”

Section: Technical Resultsmentioning

confidence: 99%

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Well-posedness of the Deterministic Transport Equation with Singular Velocity Field Perturbed along Fractional Brownian Paths

Amine¹,

Mansouri²,

Proske³

2020

Preprint

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In this article we prove path-by-path uniqueness in the sense of Davie [26] and Shaposhnikov [45] for SDE's driven by a fractional Brownian motion with a Hurst parameter H ∈ (0, 1 2 ), uniformly in the initial conditions, where the drift vector field is allowed to be merely bounded and measurable.Using this result, we construct weak unique regular solutions in W k,p loc [0, 1] × R d , p > d of the classical transport and continuity equations with singular velocity fields perturbed along fractional Brownian paths.The latter results provide a systematic way of producing examples of singular velocity fields, which cannot be treated by the regularity theory of DiPerna-Lyons [29], Ambrosio [2] or Crippa-De Lellis [24].Our approach is based on a priori estimates at the level of flows generated by a sequence of mollified vector fields, converging to the original vector field, and which are uniform with respect to the mollification parameter. In addition, we use a compactness criterion based on Malliavin calculus from [25] as well as supremum concentration inequalities.

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Section: Technical Resultsmentioning

confidence: 99%

“…with respect to a standard Brownian motion (Bm). Then, using techniques from Malliavin calculus and arguments of a local time variational calculus kind, as recently developed in the series of works [11], [12], [6], the following class of vector fields were investigated…”

Section: From Sde's To Random Ode'smentioning

confidence: 99%

Well-posedness of the Deterministic Transport Equation with Singular Velocity Field Perturbed along Fractional Brownian Paths

Amine¹,

Mansouri²,

Proske³

2020

Preprint

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“…Approximating double-sequence. Recall the truncation operator π d , d ≥ 1, defined in ( 6) and the change of basis operator τ defined in (7). We define the operator…”

Section: Definition 42mentioning

confidence: 99%

“…Applying Theorem B.1 we obtain the following crucial estimate (compare [1], [2], [6], and [7]): Proposition B.2 Let the functions f and κ be defined as in (42) and (43), respectively. Further, let 0 ≤ θ ′ < θ < t ≤ T and for some m ≥ 1…”

Section: Examplementioning

confidence: 99%

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Restoration of Well-Posedness of Infinite-dimensional Singular ODE's via Noise

Baños¹,

Bauer²,

Meyer‐Brandis³

et al. 2019

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In this paper we aim at generalizing the results of A. K. Zvonkin [41] and A. Y. Veretennikov [39] on the construction of unique strong solutions of stochastic differential equations with singular drift vector field and additive noise in the Euclidean space to the case of infinite-dimensional state spaces. The regularizing driving noise in our equation is chosen to be a locally non-Hölder continuous Hilbert space valued process of fractal nature, which does not allow for the use of classical construction techniques for strong solutions from PDE or semimartingale theory. Our approach, which does not resort to the Yamada-Watanabe principle for the verification of pathwise uniqueness of solutions, is based on Malliavin calculus.

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C-infinity-regularization by Noise of Singular ODE's

Amine,

Baños,

Proske

2017

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In this paper we construct a new type of noise of fractional nature that has a strong regularizing effect on differential equations. We consider an equation with this noise with a highly irregular coefficient. We employ a new method to prove existence and uniqueness of global strong solutions, where classical methods fail because of the "roughness" and non-Markovianity of the driving process. In addition, we prove the rather remarkable property that such solutions are infinitely many times classically differentiable with respect to the initial condition in spite of the vector field being discontinuous. The technique used in this article corresponds to the Nash-Moser principle combined with a new concept of "higher order averaging operators along highly fractal stochastic curves". This approach may provide a general principle for the study of regularization by noise effects in connection with important classes of partial differential equations.

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Strong solutions of SDE's with generalized drift and multidimensional fractional Brownian initial noise

Cited by 6 publications

References 25 publications

Well-posedness of the Deterministic Transport Equation with Singular Velocity Field Perturbed along Fractional Brownian Paths

Well-posedness of the Deterministic Transport Equation with Singular Velocity Field Perturbed along Fractional Brownian Paths

Restoration of Well-Posedness of Infinite-dimensional Singular ODE's via Noise

C-infinity-regularization by Noise of Singular ODE's

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