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Flow of diffeomorphisms for SDEs with unbounded Hölder continuous drift

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TLDR
In this article, the existence of a global flow of diffeomorphisms by means of a special transformation of the drift of Ito-Tanaka type was proved by using non-standard elliptic estimates in Holder spaces.
Abstract
We consider a SDE with a smooth multiplicative non-degenerate noise and a possibly unbounded Holder continuous drift term. We prove the existence of a global flow of diffeomorphisms by means of a special transformation of the drift of Ito–Tanaka type. The proof requires non-standard elliptic estimates in Holder spaces. As an application of the stochastic flow, we obtain a Bismut–Elworthy–Li type formula for the first derivatives of the associated diffusion semigroup.

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Stochastic Homeomorphism Flows of SDEs with Singular Drifts and Sobolev Diffusion Coefficients

TL;DR: In this article, the stochastic homeomorphism flow property and the strong Feller property for deterministic differential equations with sigular time dependent drifts and Sobolev diffusion coefficients were proved.
Journal ArticleDOI

Gradient estimates and applications for SDEs in Hilbert space with multiplicative noise and Dini continuous drift

TL;DR: In this paper, the strong Feller property of the associated Markov semigroup is proved for the stochastic evolution equation in a separable Hilbert space H with a nice multiplicative noise and a locally Dini continuous drift.
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Wellposedness for stochastic continuity equations with Ladyzhenskaya-Prodi-Serrin condition

TL;DR: In this article, the authors considered the stochastic divergence-free continuity equations with the Ladyzhenskaya-Prodi-Serrin condition and proved that strong uniqueness may fail for the deterministic PDE.
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Wellposedness for stochastic continuity equations with Ladyzhenskaya-Prodi-Serrin condition

TL;DR: In this paper, the authors considered the stochastic divergence-free continuity equations with the Ladyzhenskaya-Prodi-Serrin condition and proved that uniqueness may fail for the deterministic PDE.
Journal ArticleDOI

Multiscale Stuart-Landau Emulators: Application to Wind-Driven Ocean Gyres

TL;DR: This paper demonstrates how the data-adaptive harmonic (DAH) decomposition and inverse stochastic modeling techniques introduced in (Chekroun and Kondrashov, (2017), Chaos, 27), allow for reproducing with high fidelity the main statistical properties of multiscale variability in a coarse-grained eddy-resolving ocean flow.
References
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Book

Continuous martingales and Brownian motion

Daniel Revuz, +1 more
TL;DR: In this article, the authors present a comprehensive survey of the literature on limit theorems in distribution in function spaces, including Girsanov's Theorem, Bessel Processes, and Ray-Knight Theorem.
Book

Stochastic integration and differential equations

TL;DR: In this article, the authors propose a method for general stochastic integration and local times, which they call Stochastic Differential Equations (SDEs), and expand the expansion of Filtrations.
MonographDOI

Ergodicity for infinite dimensional systems

TL;DR: In this article, the authors studied the asymptotic properties of solutions of stochastic evolution equations in infinite dimensional spaces, and special attention was paid to the invariant measures and ergodicity.
Book

Lectures on Elliptic and Parabolic Equations in Holder Spaces

TL;DR: Second-order elliptic equations with VMO coefficients in spaces with mixed norms were given in this article, where Sobolev embedding embedding theorems for these equations were obtained in the spaces of the small Fourier transform and the elliptic operator.
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