On R^d-valued multi-self-similar Markov processes

An R d-valued Markov process X (x) t = (X 1,x 1 t ,. .. , X d,x d t), t ≥ 0, x ∈ R d is said to be multi-self-similar with index (α 1 ,. .. , α d) ∈ [0, ∞) d if the identity in law (c i X i,x i /c i t ; i = 1,. .. , d) t≥0 (d) = (X (x) ct) t≥0 , where c = d i=1 c α i i , is satisfied for all c 1 ,. .. , c d > 0 and all starting point x. Multi-self-similar Markov processes were introduced by Jacobsen and Yor [11] in the aim of extending the Lamperti transformation of positive self-similar Markov processes to R d +-valued processes. This paper aims at giving a complete description of all R d-valued multi-self-similar Markov processes. We show that their state space is always a union of open orthants with 0 as the only absorbing state and that there is no finite entrance law at 0 for these processes. We give conditions for these processes to satisfy the Feller property. Then we show that a Lamperti-type representation is also valid for R d-valued multi-self-similar Markov processes. In particular, we obtain a one-to-one relationship between this set of processes and the set of Markov additive processes with values in {−1, 1} d × R d. We then apply this representation to study the almost sure asymptotic behavior of multi-self-similar Markov processes