On the Ergodic Theory of Equations of Mathematical Physics

Linear evolution equations of mathematical physics admitting an invariant in the form of a positive quadratic form are considered. In particular, this includes the string vibration equation, the Liouville kinetic equation, the Maxwell system of equations and the Schrodinger equation. Conditions for the existence of an invariant Gaussian measure are indicated, which makes it possible to apply well-known results of ergodic theory (Poincare’s recurrence theorem, Birkhoff–Khinchin ergodic theorem, etc.). We discuss the Hamiltonian property of such systems and conditions for their complete integrability. The ergodic properties of Kronecker flows on infinite-dimensional tori are studied. A general theorem on the averaging of quadratic forms is established.