Ergodic theory of affine isometric actions on Hilbert spaces

The classical Gaussian functor associates to every orthogonal representation of a locally compact group G probability measure preserving action called action. In this paper, we generalize construction by associating affine isometric on Hilbert space, one-parameter family nonsingular actions whose ergodic properties are related in very subtle way the geometry original We show that these exhibit phase transition phenomenon and relate it new quantitative invariants for actions. use Patterson-Sullivan theory as well Lyons-Pemantle work tree-indexed random walks order give precise description groups acting trees. also without property (T) admits is free, weakly mixing stable type $$\mathrm{III}_1$$ .