ON ABSOLUTELY CONTINUOUS AND INVARIANT EVOLUTION OF SMOOTH MEASURE INHILBERT-SPACE

We consider an evolution of a smooth measure mu bar arrow pointing rig ht V(t)' mu = mu(t), where V(t) is a semigroup solving the Cauchy prob lem for a heat equation in finite or infinite dimensions; V(t)' is dua l to V(t) semigroup acting on measures. Conditions of absolute continu ity mu(t) < mu and formula for the density mu(t)(dx)/mu(dx) are obtain ed; we also get conditions for mu to be invariant under the evolution mentioned. We consider an evolution of a measure related to a hyperbol ic differential equation and get conditions of its invariance under th e corresponding transformation.