On Rd-valued multi-self-similar Markov processes

An R d -valued Markov process X t ( x ) = ( X t 1 , x 1 , … , X t d , x d ) , 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 t i , x i ∕ c i , t ≥ 0 ) 1 ≤ i ≤ d = ( d ) ( X c α t ( x ) , t ≥ 0 ) , where c α = ∏ i = 1 d c i α 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 (2003) 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.