Stochastic inertial manifold. (English) Zbl 0854.60059

Bensoussan, Alain; Flandoli, Franco

Stochastics Stochastics Rep. 53, No. 1-2, 13-39 (1995).

Summary: A nonlinear stochastic evolution equation in Hilbert space with generalized additive white noise is considered. A concept of stochastic inertial manifold is introduced, defined as a random manifold depending on time, which is finite-dimensional, invariant for the dynamic, and attracts exponentially fast all the trajectories as \(t \to \infty\). Under the classical spectral gap condition of the deterministic theory, the existence of a stochastic inertial manifold is proved. It is obtained as the solution of a stochastic partial differential equation of degenerate parabolic type, studied by a variant of Bernstein method. A result of existence and uniqueness of a stationary inertial manifold is also proved; the stationary inertial manifold contains the random attractor, introduced in previous works.

MSC:

60H15

Stochastic partial differential equations (aspects of stochastic analysis)

Keywords:

stochastic evolution equation; stochastic partial differential equation; Bernstein method