Subordinators and the actions of permutations with quasi-invariant measure

Abstract

We introduce a class of probability measures in the space of virtual permutations associated with subordinators (i.e., processes with stationary positive independent increments). We prove that these measures are quasiinvariant under both left and right actions of the countable symmetric group\(\mathfrak{S}^\infty \), and a simple formula for the corresponding cocycle is obtained. In case of a stable subordinator, we find the value of the spherical function of a constant vector on the class of transpositions. Bibliography: 19 titles.