Construction of processes with stationary independent increments in Lie groups
Lévy processes on exponential Lie groups are described as unique solutions of stochastic differential equations where these SDEs are given in terms of the generators of the associated convolution semigroups. This result is stated not for the Lévy process on the exponential Lie group but rather for the associated process on the Lie algebra. The proof is based on the well-posedness of some martingale problem due to Ph. Feinsilver. Closely related characterizations for functionals fo Lévy processes were recently derived by \textit{D. Applebaum} and \textit{H. Kunita} [J. Math. Kyoto Univ. 33, No. 4, 1103-1123 (1993; Zbl 0804.58057)]. In the end of this note, the main result is applied to simply connected nilpotent Lie groups in which case an inductive explicit description of Lévy processes is given.
scientific article; zbMATH DE number 1145406