Construction of processes with stationary independent increments in Lie groups

Abstract

It is shown that a process with stationary independent increments in an exponential Lie group can be considered as a time-homogeneous diffusion with jumps. In case of nilpotent Lie groups exact formula is given for construction. The formula involves stochastic integrals with respect to the corresponding process with stationary independent (additive) increments in the corresponding Lie algebra, and reflects the group law.