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On exponential convergence to a stationary measure for a class of random dynamical systems

Sergei B. Kuksin — 2001

Journées équations aux dérivées partielles

For a class of random dynamical systems which describe dissipative nonlinear PDEs perturbed by a bounded random kick-force, I propose a “direct proof” of the uniqueness of the stationary measure and exponential convergence of solutions to this measure, by showing that the transfer-operator, acting in the space of probability measures given the Kantorovich metric, defines a contraction of this space.

Weakly nonlinear stochastic CGL equations

Sergei B. Kuksin — 2013

Annales de l'I.H.P. Probabilités et statistiques

We consider the linear Schrödinger equation under periodic boundary conditions, driven by a random force and damped by a quasilinear damping: d d t u + i - Δ + V ( x ) u = ν Δ u - γ R | u | 2 p u - i γ I | u | 2 q u + ν η ( t , x ) . ( * ) The force η is white in time and smooth in x ; the potential V ( x ) is typical. We are concerned with the limiting, as ν → 0 , behaviour of solutions on long time-intervals 0 ≤ t ≤ ν - 1 T , and with behaviour of these solutions under the double limit t → ∞ and ν → 0 . We show that these two limiting behaviours may be described in terms of solutions for the( * ) which is a well posed semilinear...

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