The stable manifold theorem for semilinear stochastic evolution equations and stochastic partial differential equations.

*(English)*Zbl 1169.60014
Mem. Am. Math. Soc. 917, 105 p. (2008).

This monograph presents a detailed construction of stochastic semiflows mainly generated by semilinear evolution equations and stochastic partial differential equations with multiplicative noise. Due to the lack of an analogue of Kolmogorov’s continuity theorem in infinite dimensions it was unclear for a long time how to establish the existence of a local stable and unstable manifolds along a hyperbolic equlibrium. This work closes this gap in the literature and establishes the existence of a random stable and unstable local manifolds close to hyperbolic stationary solutions in infinite dimensions. The results are motivated and illustrated by several classical equations, like stochastic dissipative reaction diffusion equations and the stochastic Burgers equation.

In part 1 of this monography the authors show the existence of smooth locally compact perfect cocycles for mild solutions of semilinear stochastic evolution equations (SEEs) and stochastic partial differential equations (SPDEs) with purely multiplicative noise. The methods used are first develloped in the context of an abstract evolution equation with linear drift and linear noise part and rely on a chaos type representation of the flow of the linear parts as iterated stochastic integrals. With this technique the authors are able to bypass the lack of Kolmogorov’s continuity theorem in infinite dimensions, usually a necessary ingredient for respective results in finite dimensions. Now the authors extend these results to Lipschitz continuous drift by the representation of the cocycle as a convolution with respect to the above mentioned linear stochastic flow. These results are first applied to SPDEs with Lipschitz drift, where the linear stochastic flow is represented directly as operator-valued analogon to the geometric Brownian motion. In the last chapter of the first part it is applied to representative examples for semilinear SPDEs with non-Lipschitz nonlinerarity: stochastic reaction diffusion equations with dissipative nonlinearities and the Burgers equation with additive noise

Part 2 is based on the study of hyperbolic stationary solutions of a smooth cocycle. These are stationary solutions for which the linearized cocycle has a non-vanishing Lyapunov spectrum. After the introduction of the basic concepts the authors state the existence result of stable and unstable subspaces and the respective invariance and exponential dichotomies for linear cocycles, that will serve as tangent spaces for the nonlinear version. In the second subchapter the main theorem is finally stated. It basically guarantees for perfect \(C^{k, \epsilon}\) cocycles with a hyperbolic stationary point under the respective moment conditions the existence of a cocycle-invariant, stochastic stable and unstable manifold. These manifolds enjoy the same smoothness of the cocycle and are mutually transversal. Furthermore holds a local versions of the exponential dichotomy for a unique so-called “stochastic history process” of a deterministic starting point. The reason for this complication of the concept of exponential dichontomy lies in the fact that the stationary solution under the cocycle is not necessarily adapted as it usually depends on the entire driving path. The proof consists of three major steps. First the authors refer to their previous result that the Oseledec spaces are invariant under the linearized continuous time cocycle. Adapting a discrete time analysis by Ruelle and subsequent interpolation they establish the existence of a local stable and unstable manifold. In order to obtain asymptotic invariance of the stable manifold they first refine a perfect subadditive ergodic theorem by D. Ruelle [Ann. Math. (2) 115, 243–290 (1982; Zbl 0493.58015)], establish a serie of estimates involving the integrability condition of the cocycle and rather difficult arguments inspired by proofs by Ruelle. For the respective result for the unstable manifold they introduce the aforementioned “stochastic history process” and use perfection arguments as before. The last subsection is devoted to the application of these results in the case of semilinear SEEs with additive and linear noise, parabolic spdes with Lipschitz nonlinearities, stochastic dissipative reaction diffusion equations and the stochastic Burgers equation with additive noise.

Despite of the highly sophisticated techniciality and the complexity of the subject, the authors make a great effort to keep the presentation intuitive and lively. The proofs are structured clearly and well explained throughout. Numerous and detailed references in the text guide the reader’s way safely through the literature.

In part 1 of this monography the authors show the existence of smooth locally compact perfect cocycles for mild solutions of semilinear stochastic evolution equations (SEEs) and stochastic partial differential equations (SPDEs) with purely multiplicative noise. The methods used are first develloped in the context of an abstract evolution equation with linear drift and linear noise part and rely on a chaos type representation of the flow of the linear parts as iterated stochastic integrals. With this technique the authors are able to bypass the lack of Kolmogorov’s continuity theorem in infinite dimensions, usually a necessary ingredient for respective results in finite dimensions. Now the authors extend these results to Lipschitz continuous drift by the representation of the cocycle as a convolution with respect to the above mentioned linear stochastic flow. These results are first applied to SPDEs with Lipschitz drift, where the linear stochastic flow is represented directly as operator-valued analogon to the geometric Brownian motion. In the last chapter of the first part it is applied to representative examples for semilinear SPDEs with non-Lipschitz nonlinerarity: stochastic reaction diffusion equations with dissipative nonlinearities and the Burgers equation with additive noise

Part 2 is based on the study of hyperbolic stationary solutions of a smooth cocycle. These are stationary solutions for which the linearized cocycle has a non-vanishing Lyapunov spectrum. After the introduction of the basic concepts the authors state the existence result of stable and unstable subspaces and the respective invariance and exponential dichotomies for linear cocycles, that will serve as tangent spaces for the nonlinear version. In the second subchapter the main theorem is finally stated. It basically guarantees for perfect \(C^{k, \epsilon}\) cocycles with a hyperbolic stationary point under the respective moment conditions the existence of a cocycle-invariant, stochastic stable and unstable manifold. These manifolds enjoy the same smoothness of the cocycle and are mutually transversal. Furthermore holds a local versions of the exponential dichotomy for a unique so-called “stochastic history process” of a deterministic starting point. The reason for this complication of the concept of exponential dichontomy lies in the fact that the stationary solution under the cocycle is not necessarily adapted as it usually depends on the entire driving path. The proof consists of three major steps. First the authors refer to their previous result that the Oseledec spaces are invariant under the linearized continuous time cocycle. Adapting a discrete time analysis by Ruelle and subsequent interpolation they establish the existence of a local stable and unstable manifold. In order to obtain asymptotic invariance of the stable manifold they first refine a perfect subadditive ergodic theorem by D. Ruelle [Ann. Math. (2) 115, 243–290 (1982; Zbl 0493.58015)], establish a serie of estimates involving the integrability condition of the cocycle and rather difficult arguments inspired by proofs by Ruelle. For the respective result for the unstable manifold they introduce the aforementioned “stochastic history process” and use perfection arguments as before. The last subsection is devoted to the application of these results in the case of semilinear SEEs with additive and linear noise, parabolic spdes with Lipschitz nonlinearities, stochastic dissipative reaction diffusion equations and the stochastic Burgers equation with additive noise.

Despite of the highly sophisticated techniciality and the complexity of the subject, the authors make a great effort to keep the presentation intuitive and lively. The proofs are structured clearly and well explained throughout. Numerous and detailed references in the text guide the reader’s way safely through the literature.

Reviewer: Michael HĂ¶gele (Berlin)

##### MSC:

60H15 | Stochastic partial differential equations (aspects of stochastic analysis) |

60D05 | Geometric probability and stochastic geometry |

37L45 | Hyperbolicity; Lyapunov functions for infinite-dimensional dissipative dynamical systems |

37L55 | Infinite-dimensional random dynamical systems; stochastic equations |