  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.
