An analogue of the Lévy-Hinčin formula for bi-free infinitely divisible distributions

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.