On $\mathbb{R}^d$-valued multi-self-similar Markov processes

Abstract

An $\mathbb{R}^d$-valued Markov process $X^{(x)}_t=(X^{1,x_1}_t,\dots,X^{d,x_d}_t)$, $t\ge0,x\in\mathbb{R}^d$ is said to be multi-self-similar with index $(α_1,\dots,α_d)\in[0,\infty)^d$ if the identity in law \[(c_iX_t^{i,x_i/c_i};i=1,\dots,d)_{t\ge0}\ed(X_{ct}^{(x)})_{t\ge0}\,,\] where $c=\prod_{i=1}^dc_i^{α_i}$, is satisfied for all $c_1,\dots,c_d>0$ and all starting point $x$. Multi-self-similar Markov processes were introduced by Jacobsen and Yor \cite{jy} in the aim of extending the Lamperti transformation of positive self-similar Markov processes to $\mathbb{R}^d_+$-valued processes. This paper aims at giving a complete description of all $\mathbb{R}^d$-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 $\mathbb{R}^d$-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 $\{-1,1\}^d\times\mathbb{R}^d$. We then apply this representation to study the almost sure asymptotic behavior of multi-self-similar Markov processes.