On Rd-valued multi-self-similar Markov processes

An R-d-valued Markov process x(t)((x))= (x(t)(1,x1),...,X-t(d,xd)),t >= 0,x is an element of R-d is said to be multi-self-similar with index (alpha(1),..., alpha(d)) is an element of [0, infinity)(d) if the identity in law (ci X-t(i,xi/ci),t >= 0)(1 <= i <= d )=((d))(X-c alpha t((x)), t >= 0), where c(alpha) = Pi(d)(i)=1 c(i)(alpha 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) x R-d . We then apply this representation to study the almost sure asymptotic behavior of multi-self-similar Markov processes. (C) 2019 Elsevier B.V. All rights reserved.