Smooth Stable and Unstable Manifolds for Stochastic Evolutionary Equations

By Jinqiao Duan, Kening Lu, B. Schmalfuβ. 2004

156 citations recorded by the source corpus.

Invariant manifolds are fundamental tools for describing and understanding nonlinear dynamics. In this paper, we present a theory of stable and unstable manifolds for infinite dimensional random dynamical systems generated by a class of stochastic partial differential equations. We first show the existence of Lipschitz continuous stable and unstable manifolds by the Lyapunov–Perron’s method. Then, we prove the smoothness of these invariant manifolds.

Subfields: Applied Mathematics, Control and Systems Engineering, Finance

View Paper: https://www.semanticscholar.org/paper/6a9a72d1f9e92f9d34e284b69e02f949b43a4390

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In Applied Mathematics

1. Chunrong Feng
2. Huaizhong Zhao
3. Kening Lu
4. Mickaël D. Chekroun
5. Michael Ghil
6. B. Schmalfuβ
7. Tomás Caraballo
8. Bixiang Wang

In Control and Systems Engineering

1. Chunrong Feng
2. Huaizhong Zhao
3. Bixiang Wang
4. Kening Lu
5. B. Schmalfuβ
6. Tomás Caraballo
7. Boya Zhou
8. Jinqiao Duany

In Finance

1. Chunrong Feng
2. Huaizhong Zhao
3. Kening Lu
4. B. Schmalfuβ
5. Tomás Caraballo
6. Boya Zhou
7. Bixiang Wang
8. Yue Wu

Citations in the corpus

• Pathwise Random Periodic Solutions of Stochastic Differential Equations (2011, background): “In deterministic and random dynamical systems, to find the existence of stationary solutions and random periodic solutions, and to construct local stable and unstable manifolds near a hyperbolic stationary point is a fundamental problem [1,7,8,11,12,17,22].”
• Climate dynamics and fluid mechanics: Natural variability and related uncertainties (2008, background): “Thus RDS concepts and tools are not restricted to finite-dimensional systems [104, 105, 106].”
• Random periodic solutions of SPDEs via integral equations and Wiener-Sobolev compact embedding (2012, background): “s about SPDEs, see Sinai [28], [29], Mattingly [21], E, Khanin, Mazel and Sinai [11], Caraballo, Kloeden and Schmalfuss [3], Liu and Zhao [20], Zhang and Zhao [32], [33], Duan, Lu and Schmalfuss [9], [10], Mohammed, Zhang and Zhao [22], Lian and Lu [19], though there are still many problems that nee”
• Invariant Manifolds for Random and Stochastic Partial Differential Equations (2009, background): “Stable, unstable, center, and inertial manifolds, as special random invariant structures, have been considered in the investigation of stochastic partial differential equations or stochastic evolutionary equations in infinite dimensional spaces [8, 12, 13, 19, 5, 15, 11, 27].”