Abstract
I study different types of statistical solutions (Hopf, Foias , Vishik-Fursikov) for nonlinear evolution equations. As a test equation, I use the nonlinear Schrödinger equation with power-like nonlinearity in the case where the proofs of uniqueness are not available. When there is no uniqueness in the original equation, statistical solutions are not unique. For autonomous differential equations, there is a formal semigroup property. I propose to look for statistical solutions with an analogous property. For statistical solutions, this should be the homogeneous Markov property. I call such solutions Markov statistical solutions. The proofs of the existence of the Markov statistical solutions rely on the Markov selection theorem. N.V. Krylov was the first to realize the importance of the Markov selection in the context of Markov processes. D.W. Stroock, and S.R.S. Varadhan re-framed Krylov’s selection in the context of solutions to the martingale problem. Recently, their results have been used by F. Flandoli and M. Romito, and Goldys et al., for the analysis of the Navier-Stokes equation with additive noise. I use the Markov selection theorem to prove the existence of Markov statistical solutions. I give a new proof of the Markov selection theorem. This proof has prompted me to look back at the set-valued solutions of deterministic equations, where the analog of the homogeneous Markov property should be the semigroup property. It turned out that no theorems of existence of selections satisfying the semigroup property were known. I state and prove a selection theorem for measurable selections with the semigroup property. Such result is important in its own right. I use it here to give a second proof of the existence of Vishik-Fursikov measures.