2018/09/06 by Chaumont, Loïc, Lamine, Salem
#60J45 #FOS: Mathematics #Probability (math.PR)
paper · doi:10.48550/arxiv.1809.02085
An ℝd-valued Markov process X(x)t=(X1,x1t,…,Xd,xdt), t≥0,x∈ℝd is said to be multi-self-similar with index (α1,…,αd)∈[0,∞)d if the identity in law (ciXti,xi/ci;i=1,…,d)t≥0\ed(Xct(x))t≥0 , where c=∏i=1dciαi, is satisfied for all c1,…,cd>0 and all starting point x. Multi-self-similar Markov processes were introduced by Jacobsen and Yor \citejy in the aim of extending the Lamperti transformation of positive self-similar Markov processes to ℝd+-valued processes. This paper aims at giving a complete description of all ℝd-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 ℝd-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×ℝd. We then apply this representation to study the almost sure asymptotic behavior of multi-self-similar Markov processes.