Inversions of infinitely divisible distributions and conjugates of stochastic integral mappings

The dual of an infinitely divisible distribution on ${\mathbb{R}}^d$ without Gaussian part defined in Sato, ALEA {\bf 3} (2007), 67–110, is renamed to the inversion. Properties and characterization of the inversion are given. A stochastic integral mapping is a mapping $\mu ={\mathrm{\Phi }}_f\rho$ of $\rho$ to $\mu$ in the class of infinitely divisible distributions on ${\mathbb{R}}^d$, where $\mu$ is the distribution of an improper stochastic integral of a nonrandom function $f$ with respect to a L'{e}vy process on ${\mathbb{R}}^d$ with distribution $\rho$ 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.