Multi-self-similar Markov processes on Rn+ and their Lamperti representations

A classical result, due to Lamperti, establishes a one-to-one correspondence between a class of strictly positive Markov processes that are self-similar, and the class of one-dimensional Lévy processes. This correspondence is obtained by suitably time-changing the exponential of the Lévy process. In this paper we generalise Lampertiʹs result to processes in n dimensions. For the representation we obtain, it is essential that the same time-change be applied to all coordinates of the processes involved. Also for the statement of the main result we need the proper concept of self-similarity in higher dimensions, referred to as multi-self-similarity in the paper. The special case where the Lévy process (xi) is standard Brownian motion in n dimensions is studied in detail. There are also specific comments on the case where (xi) is an n-dimensional compound Poisson process with drift. Finally, we present some results concerning moment sequences, obtained by studying the multi-self-similar processes that correspond to n-dimensional subordinators.