Inversions of Infinitely Divisible Distributions and Conjugates of Stochastic Integral Mappings

摘要

The dual of an infinitely divisible distribution on ?~d without Gaussian part defined in Sato (ALEA Lat. Am. J. Probab. Math. Statist. 3:67-110, 2007) is renamed to the inversion. Properties and characterization of the inversion are given. A stochastic integral mapping is a mapping μ=Φ _f ρ of ρ to μ in the class of infinitely divisible distributions on ?~d, where μ is the distribution of an improper stochastic integral of a nonrandom function f with respect to a Lévy process on ?~d with distribution ρ at time 1. The concept of the conjugate is introduced for a class of stochastic integral mappings and its close connection with the inversion is shown. The domains and ranges of the conjugates of three two-parameter families of stochastic integral mappings are described. Applications to the study of the limits of the ranges of iterations of stochastic integral mappings are made.