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