##
**Approximation of stochastic invariant manifolds. Stochastic manifolds for nonlinear SPDEs I.**
*(English)*
Zbl 1319.60002

SpringerBriefs in Mathematics. Cham: Springer (ISBN 978-3-319-12495-7/pbk; 978-3-319-12496-4/ebook). xv, 127 p. (2015).

The book under review is the first in a two-volume series and deals with approximation of stochastic manifolds that are invariant for dynamics of a parabolic Stratonovich SPDE driven by a one-dimensional Wiener process. The interest is focused on a derivation of a leading-order approximation of stochastic invariant manifolds and the topic is approached via the theory of random dynamical systems.

The authors consider a Stratonovich stochastic equation \[ du=(L_\lambda u+F(u))\,dt+\sigma u\circ dW \] in a Hilbert space \(\mathscr H\), where \(L_\lambda=-A+B_\lambda\), \(A\) is a sectorial operator with a compact resolvent and such that the spectrum of \(-A\) belongs to the real negative complex half-plane (\(\operatorname{Re}z<0\)), the \(B_\lambda\) are linear operators defined on a domain of \(A^\gamma\) for some \(\gamma\in[0,1)\) and depend continuously on \(\lambda\), \(F\) is a nonlinear, Lipschitz continuous or smooth (depending on the particular case) operator on a domain of \(A^\alpha\) for some \(\alpha\in[0,1)\) with values in \(\mathscr H\) satisfying \(F(0)=0\) and, if \(F\) is \(C^1\)-smooth, also \(F^\prime(0)=0\), \(\sigma\) is a positive constant and \(W\) is a two-sided one-dimensional Wiener process. The spectrum of \(L_\lambda\) is ordered lexicographically, the first \(m\) eigenvalues are assumed to be uniformly separated from the left from the rest of the spectrum by a vertical belt in the complex plane independent of the parameter \(\lambda\), and according to this uniform decomposition of the spectrum, the Hilbert space decomposes to an \(m\)-dimensional subspace \(\mathscr H^c(\lambda)\) corresponding to the first \(m\) eigenvalues of \(L_\lambda\) and its complement \(\mathscr H^s(\lambda)\) in \(\mathscr H\) or the complement \(\mathscr H^s_\alpha(\lambda)\) of \(\mathscr H^c(\lambda)\) in the domain of \(A^\alpha\), all spaces being \(L_\lambda\)-invariant.

The Stratonovich stochastic equation is transformed to a random partial differential equation that defines a random dynamical system cohomologous via a random \(C^\infty\)-diffeomorphism with the random dynamical system for the Stratonovich stochastic equation.

It is shown that there exists a global stochastic invariant \(m\)-dimensional Lipschitz or smooth manifold (depending on the hypotheses) that can be described as the graph of a random function \(h_\lambda\) with the domain \(\mathscr H^c(\lambda)\) and with values in \(\mathscr H^s_\alpha(\lambda)\), where the dependence on \(\lambda\) in \(h_\lambda\) is Lipschitz continuous or smooth. The high modes of the solutions are thus parametrized by the low modes.

Under additional assumptions, it is proved that the stochastic invariant manifold is both a pullback and a forward stochastic inertial manifold with a certain critical attraction rate.

In the next section, under suitably modified assumptions and dropping the assumption on global Lipschitzness of \(F\), the above results are shown in a local setting, i.e., the invariant manifold exists in a random neighbourhood of the origin.

In Chapter 6, the authors focus on a situation where the control parameter \(\lambda\) varies in an interval that contains a critical value \(\lambda_c\) at which the trivial steady state changes its linear stability (principle of exchange of stabilities) and where the leading term in \(F\) has a \(k\)-monomial form \(u^k\). It is proved that there exists an \(m\)-dimensional local stochastic critical (invariant) smooth manifold and its local critical (invariant) manifold function \(h_\lambda\) can be approximated to the leading order \(k\) (i.e., with an \(o(\|\xi\|^k_\alpha)\) error) in a random neighbourhood of the origin by an explicitly defined Lyapunov-Perron integral. In the case where the operators \(L_\lambda\) are selfadjoint, the Lyapunov-Perron integral approximation can be written as random homogeneous polynomials of order \(k\) in the critical state variable and thus constitute genuine leading-order Taylor approximations of the corresponding local critical manifolds.

In Chapter 7, the results of Chapter 6 are demonstrated in the case when the subspace \(\mathscr H^c\) contains not only critical modes, which lose their stability as the control parameter \(\lambda\) varies, but also modes that remain stable as \(\lambda\) varies (this assumption means that the corresponding local stochastic invariant manifold is hyperbolic for supercritical \(\lambda\)) and the condition of the principle of exchange of stabilities is replaced by a rather different condition.

The book is aimed at readers interested in stochastic partial differential equations and random dynamical systems.

The authors consider a Stratonovich stochastic equation \[ du=(L_\lambda u+F(u))\,dt+\sigma u\circ dW \] in a Hilbert space \(\mathscr H\), where \(L_\lambda=-A+B_\lambda\), \(A\) is a sectorial operator with a compact resolvent and such that the spectrum of \(-A\) belongs to the real negative complex half-plane (\(\operatorname{Re}z<0\)), the \(B_\lambda\) are linear operators defined on a domain of \(A^\gamma\) for some \(\gamma\in[0,1)\) and depend continuously on \(\lambda\), \(F\) is a nonlinear, Lipschitz continuous or smooth (depending on the particular case) operator on a domain of \(A^\alpha\) for some \(\alpha\in[0,1)\) with values in \(\mathscr H\) satisfying \(F(0)=0\) and, if \(F\) is \(C^1\)-smooth, also \(F^\prime(0)=0\), \(\sigma\) is a positive constant and \(W\) is a two-sided one-dimensional Wiener process. The spectrum of \(L_\lambda\) is ordered lexicographically, the first \(m\) eigenvalues are assumed to be uniformly separated from the left from the rest of the spectrum by a vertical belt in the complex plane independent of the parameter \(\lambda\), and according to this uniform decomposition of the spectrum, the Hilbert space decomposes to an \(m\)-dimensional subspace \(\mathscr H^c(\lambda)\) corresponding to the first \(m\) eigenvalues of \(L_\lambda\) and its complement \(\mathscr H^s(\lambda)\) in \(\mathscr H\) or the complement \(\mathscr H^s_\alpha(\lambda)\) of \(\mathscr H^c(\lambda)\) in the domain of \(A^\alpha\), all spaces being \(L_\lambda\)-invariant.

The Stratonovich stochastic equation is transformed to a random partial differential equation that defines a random dynamical system cohomologous via a random \(C^\infty\)-diffeomorphism with the random dynamical system for the Stratonovich stochastic equation.

It is shown that there exists a global stochastic invariant \(m\)-dimensional Lipschitz or smooth manifold (depending on the hypotheses) that can be described as the graph of a random function \(h_\lambda\) with the domain \(\mathscr H^c(\lambda)\) and with values in \(\mathscr H^s_\alpha(\lambda)\), where the dependence on \(\lambda\) in \(h_\lambda\) is Lipschitz continuous or smooth. The high modes of the solutions are thus parametrized by the low modes.

Under additional assumptions, it is proved that the stochastic invariant manifold is both a pullback and a forward stochastic inertial manifold with a certain critical attraction rate.

In the next section, under suitably modified assumptions and dropping the assumption on global Lipschitzness of \(F\), the above results are shown in a local setting, i.e., the invariant manifold exists in a random neighbourhood of the origin.

In Chapter 6, the authors focus on a situation where the control parameter \(\lambda\) varies in an interval that contains a critical value \(\lambda_c\) at which the trivial steady state changes its linear stability (principle of exchange of stabilities) and where the leading term in \(F\) has a \(k\)-monomial form \(u^k\). It is proved that there exists an \(m\)-dimensional local stochastic critical (invariant) smooth manifold and its local critical (invariant) manifold function \(h_\lambda\) can be approximated to the leading order \(k\) (i.e., with an \(o(\|\xi\|^k_\alpha)\) error) in a random neighbourhood of the origin by an explicitly defined Lyapunov-Perron integral. In the case where the operators \(L_\lambda\) are selfadjoint, the Lyapunov-Perron integral approximation can be written as random homogeneous polynomials of order \(k\) in the critical state variable and thus constitute genuine leading-order Taylor approximations of the corresponding local critical manifolds.

In Chapter 7, the results of Chapter 6 are demonstrated in the case when the subspace \(\mathscr H^c\) contains not only critical modes, which lose their stability as the control parameter \(\lambda\) varies, but also modes that remain stable as \(\lambda\) varies (this assumption means that the corresponding local stochastic invariant manifold is hyperbolic for supercritical \(\lambda\)) and the condition of the principle of exchange of stabilities is replaced by a rather different condition.

The book is aimed at readers interested in stochastic partial differential equations and random dynamical systems.

Reviewer: Martin Ondreját (Praha)

### MSC:

60-02 | Research exposition (monographs, survey articles) pertaining to probability theory |

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

60H30 | Applications of stochastic analysis (to PDEs, etc.) |

35R60 | PDEs with randomness, stochastic partial differential equations |

37A50 | Dynamical systems and their relations with probability theory and stochastic processes |