Existence of smooth stable manifolds for a class of parabolic SPDEs with fractional noise

Article Mathematics

Mean-square invariant manifolds for ill-posed stochastic evolution equations driven by nonlinear noise

Zonghao Li et al.

Summary: This paper investigates the invariant manifold of a class of ill-posed stochastic evolution equations driven by a nonlinear multiplicative noise. The existence of mean-square random unstable invariant manifold and only mean-square stable invariant set is established. A modified variation of constants formula is constructed by the resolvent operator due to the lack of the Hille-Yosida condition. The Lyapunov-Perron method is utilized with an unusual condition involving a non-decreasing map to derive the required estimates. The existence of mean-square random stable sets is alternatively established due to the loss of invariance in the forward time for the Lyapunov-Perron map caused by the adaptedness.

JOURNAL OF DIFFERENTIAL EQUATIONS (2022)

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Article Mathematics, Applied

CENTER MANIFOLDS FOR ILL-POSED STOCHASTIC EVOLUTION EQUATIONS

Zonghao Li et al.

Summary: The aim of this paper is to develop a center manifold theory for a class of stochastic partial differential equations with a non-dense domain using the Lyapunov-Perron method and the resolvent operator to construct a novel variation of constants formula. An additional condition involving a non-decreasing map is imposed to deduce the required estimate, as Young's convolution inequality is not applicable. As an application, a stochastic parabolic equation is presented to illustrate the obtained results.

DISCRETE AND CONTINUOUS DYNAMICAL SYSTEMS-SERIES B (2022)

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Article Mathematics, Applied

ROUGH CENTER MANIFOLDS

Alexandra Neamtu et al.

Summary: The paper analyzes invariant manifolds for rough differential equations (RDEs) and proves the existence and regularity of local center manifolds for such systems using a discretized Lyapunov-Perron-type method. The method works directly with RDEs and utilizes rough paths estimates to obtain relevant contraction properties of the Lyapunov-Perron map.

SIAM JOURNAL ON MATHEMATICAL ANALYSIS (2021)

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Article Mathematics, Applied

MEAN-SQUARE RANDOM INVARIANT MANIFOLDS FOR STOCHASTIC DIFFERENTIAL EQUATIONS

Bixiang Wang

Summary: We have developed a theory of mean-square random invariant manifolds for mean-square random dynamical systems generated by stochastic differential equations. The existence of mean-square random invariant unstable manifolds has been proved, but the existence of mean-square random stable invariant manifolds remains open.

DISCRETE AND CONTINUOUS DYNAMICAL SYSTEMS (2021)

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Article Mathematics