Abstract
In this paper, we derive the bi-free analogue of the Lévy-Hinčin formula for compactly supported planar probability measures that 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 Lévy-Hincin representations. Furthermore, we construct the bi-free Levy processes and the additive bi-free convolution semigroups generated by compactly supported planar probability measures.