Large deviations and Gallavotti-Cohen principle for dissipative PDE's with rough noise

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 - Centre de Physique Théorique - UMR 7332
5 CPT - E5 Physique statistique et systèmes complexes
6 AGM - UMR 8088 - Analyse, Géométrie et Modélisation

We study a class of dissipative PDE's perturbed by an unbounded kick force. Under some natural assumptions, the restrictions of solutions to integer times form a homogeneous Markov process. Assuming that the noise is rough with respect to the space variables and has a non-degenerate law, we prove that the system in question satisfies a large deviation principle in tau-topology. Under some additional hypotheses, we establish a Gallavotti-Cohen type symmetry for the rate function of an entropy production functional and the strict positivity and finiteness of the mean entropy production in the stationary regime. The latter result is applicable to PDE's with strong nonlinear dissipation.

Mots clés: Burgers equation; Ginzburg-Landau equation; Navier-Stokes system; entropy production; Gallavotti-Cohen fluctuation relation; reaction-diffusion system; occupation measures; large deviation principle; dissipative PDE's

Domaines: Equations aux dérivées partielles [math.AP]; Physique mathématique [math-ph]; Probabilités [math.PR]; Mécanique statistique [cond-mat.stat-mech]

Vojkan Jaksic, Vahagn Nersesyan, Claude-Alain Pillet, Armen Shirikyan. Large deviations and Gallavotti-Cohen principle for dissipative PDE's with rough noise. Communications in Mathematical Physics, 2015, 336 (1), pp.131-170. ⟨10.1007/s00220-014-2279-3⟩. ⟨hal-00916917v3⟩