Large deviations and mixing for dissipative PDEs with unbounded random kicks

Vojkan Jaksic (1) , Vahagn Nersesyan (2) , Claude-Alain Pillet (3, 4, 5) , Armen Shirikyan (6)

1 Department of Mathematics and Statistics [Montréal]
2 LMV - Laboratoire de Mathématiques de Versailles
3 FRUMAM - Fédération de Recherche des Unités de MAthématiques de Marseille
4 CPT - E5 Physique statistique et systèmes complexes
5 CPT - Centre de Physique Théorique - UMR 7332
6 AGM - UMR 8088 - Analyse, Géométrie et Modélisation

We study the problem of exponential mixing and large deviations for discrete-time Markov processes associated with a class of random dynam-ical systems. Under some dissipativity and regularisation hypotheses for the underlying deterministic dynamics and a non-degeneracy condition for the driving random force, we discuss the existence and uniqueness of a stationary measure and its exponential stability in the Kantorovich–Wasserstein metric. We next turn to the large deviation principle and establish its validity for the occupation measures of the Markov processes in question. The obtained results extend those established earlier for the case of the bounded noise and can be applied to the 2D Navier–Stokes system in a bounded domain and to the complex Ginzburg–Landau equation.

Mots clés: Dissipative PDE's; Navier–Stokes system; Ginzburg–Landau equation; unbounded kicks; large deviations; occupation measures

Domaines: Equations aux dérivées partielles [math.AP]; Probabilités [math.PR]; Physique mathématique [math-ph]

Vojkan Jaksic, Vahagn Nersesyan, Claude-Alain Pillet, Armen Shirikyan. Large deviations and mixing for dissipative PDEs with unbounded random kicks. Nonlinearity, 2018, 31 (2), pp.540-596. ⟨10.1088/1361-6544/aa99a7⟩. ⟨hal-01076897⟩