QUASI-INVARIANCE OF COMPLETELY RANDOM MEASURES

Let X be a locally compact Polish space. Let K(X) denote the space of discrete Radon measures on X. Let mu be a completely random discrete measure on X, i.e., mu is (the distribution of) a completely random measure on X that is concentrated on K(X). We consider the multiplicative (current) group C-0(X -> R+) consisting of functions on X that take values in R+ = (0, infinity) and are equal to 1 outside a compact set. Each element theta is an element of C-0(X -> R+) maps K(X) onto itself; more precisely, theta sends a discrete Radon measure Sigma(i) s(i)delta(xi) to Sigma(i) theta(s(i))s(i)delta(xi). Thus, elements of C-0(X -> R+) transform the weights of discrete Radon measures. We study conditions under which the measure mu is quasi-invariant under the action of the current group C-0(X -> R+) and consider several classes of examples. We further assume that X = R-d and consider the group of local diffeomorphisms Diff(0)(X). Elements of this group also map K(X) onto itself. More precisely, a diffeomorphism phi is an element of Diff(0)(X) sends a discrete Radon measure Sigma(i) s(i)delta(xi) to Sigma(i) s(i)delta(phi) (x(i)). Thus, diffeomorphisms from Diff(0)(X) transform the atoms of discrete Radon measures. We study quasiinvariance of mu under the action of Diff(0)(X). We finally consider the semidirect product G := Diff(0)(X) x C-0(X -> R+) and study conditions of quasi-invariance and partial quasi-invariance of mu under the action of G.