An analogue of the L\'{e}vy-Hin\v{c}in formula for bi-free infinitely divisible distributions

TL;DR

This paper extends the classical Lévy-Hinčin formula to the bi-free probability setting, characterizing bi-free infinitely divisible distributions and constructing related processes.

Contribution

It introduces a bi-free Lévy-Hinčin formula, provides examples, and constructs bi-free Lévy processes and convolution semigroups.

Findings

Derived the bi-free Lévy-Hinčin formula for compactly supported measures

Provided explicit examples of bi-free infinitely divisible distributions

Constructed bi-free Lévy processes and convolution semigroups

Abstract

In this paper, we derive the bi-free analogue of the L\'{e}vy-Hin\v{c}in formula for compactly supported planar probability measures which are infinitely divisible with respect to the additive bi-free convolution introduced by Voiculescu. We also provide examples of bi-free infinitely divisible distributions with their bi-free L\'{e}vy-Hin\v{c}in representations. Furthermore, we construct the bi-free L\'{e}vy processes and the additive bi-free convolution semigroups generated by compactly supported planar probability measures.