Large deviations and mixing for dissipative PDEs with unbounded random kicks

作者：Jaksic V^*; Nersesyan V; Pillet C A; Shirikyan A

来源：Nonlinearity, 2018, 31(2): 540-596.

DOI：10.1088/1361-6544/aa99a7

• dissipative PDEs
• Navier-Stokes system
• Ginzburg-Landau equation
• unbounded kicks
• large deviations
• occupation measures

We study the problem of exponential mixing and large deviations for discrete-time Markov processes associated with a class of random dynamical 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 deviations principle (LDP) and establish its validity for the occupation measures of the Markov processes in question. The proof is based on Kifer's criterion for non-compact spaces, a result on large-time asymptotics for generalised Markov semigroup, and a coupling argument. These tools combined together constitute a new approach to LDP for infinite-dimensional processes without strong Feller property in a non-compact space. The results obtained can be applied to the two-dimensional Navier-Stokes system in a bounded domain and to the complex Ginzburg-Landau equation.

• 出版日期2018-2
• 单位McGill