The Stable Manifold Theorem for Semilinear Stochastic Evolution Equations and Stochastic Partial Dif

By Salah-Eldin A. Mohammed, Tusheng Zhang, Huaizhong Zhao. 2008

140 citations recorded by the source corpus.

The main objective of this paper is to characterize the pathwise local structure of solutions of semilinear stochastic evolution equations (see's) and stochastic partial differential equations (spde's) near stationary solutions. Such characterization is realized through the long-term behavior of the solution field near stationary points. The analysis falls in two parts 1, 2. In Part 1, we prove general existence and compactness theorems for Ck-cocycles of semilinear see's and spde's. Our results cover a large class of semilinear see's as well as certain semilinear spde's with Lipschitz and non-Lipschitz terms such as stochastic reaction diffusion equations and the stochastic Burgers equation with additive infinite- dimensional noise. In Part 2, stationary solutions are viewed as cocycle-invariant random points in the infinite-dimensional state space. The pathwise local structure of solutions of semilinear see's and spde's near stationary solutions is described in terms of the almost sure longtime behavior of trajectories of the equation in relation to the stationary solution. More specifically, we establish local stable manifold theorems for semilinear see's and spde's (Theorems 2.4.1-2.4.4). These results give smooth stable and unstable manifolds in the neighborhood of a hyperbolic stationary solution of the underlying stochastic equation. The stable and unstable manifolds are stationary, live in a stationary tubular neighborhood of the stationary solution and are asymptotically invariant under the stochastic semiflow of the see/spde. Furthermore, the local stable and unstable manifolds intersect transversally at the stationary point, and the unstable manifolds have fixed finite dimension. The proof uses infinite-dimensional multiplicative ergodic theory techniques, interpolation and perfection arguments (Theorem 2.2.1).

Subfields: Computational Theory and Mathematics, Control and Systems Engineering, Finance

View Paper: https://www.semanticscholar.org/paper/30b3c8b44ad854542cfce9abb3ae503785ff99a6

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In Computational Theory and Mathematics

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

In Control and Systems Engineering

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

In Finance

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

Citations in the corpus

• 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].”
• Random periodic solutions of random dynamical systems (2009, background): “H.Zhao@lboro.ac.uk · Institute of Applied Mathematics, Academy of Mathematics and Systems Sciences, Chinese Academy of Sciences, Beijing 100080, P. R. China. zhzheng@amt.ac.cn 1 However, in many real problems, such kinds of the deterministic equilibria actually don’t exist due to the presence of ran”
• Existence, Stability and Bifurcation of Random Complete and Periodic Solutions of Stochastic Parabolic Equations (2013, background): “For the existence of invariant manifolds for stochastic PDEs, we refer the reader to [20, 21, 36, 37] for details.”
• Pathwise Random Periodic Solutions of Stochastic Differential Equations (2011, background): “In fact, for the existence of stationary solutions, results are only known in very few cases [3,6,17,23,24,27].”
• Random periodic solutions of SPDEs via integral equations and Wiener-Sobolev compact embedding (2012, background): “Y. Liu, H. Z. Zhao, Representation of pathwise stationary solutions of stochastic Burgers equations, Stochastics and Dynamics, Vol. 9 (2009), No. 4, 613-634 21.”
• Existence and Upper Semicontinuity of Attractors for Stochastic Equations with Deterministic Non-autonomous Terms (2012, background): “The reader is also referred to [1, 17, 18, 29, 31] for the existence of random invariant manifolds.”
• Anticipating Random Periodic Solutions--I. SDEs with Multiplicative Linear Noise (2015, methodology): “In [16], applying the unstable manifold theorem ([23]), we can still deduce the IHSIEs in the infinite dimensional case.”
• Stationary solutions of SPDEs and infinite horizon BDSDEs with non-Lipschitz coefficients (2006, background): “S.-E. A. Mohammed, T. Zhang, H. Z. Zhao, The stable manifold theorem for semilinear stochastic evolution equations and stochastic partial differential equations.”