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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 saidto be multi-self-similar with index $(\alpha_1,\dots,\alpha_d)\in[0,\infty)^d$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$c=\prod_{i=1}^dc_i^{\alpha_i}$, is satisfied for all $c_1,\dots,c_d>0$ and allstarting point $x$. Multi-self-similar Markov processes were introduced byJacobsen and Yor \cite{jy} in the aim of extending the Lamperti transformationof positive self-similar Markov processes to $\mathbb{R}^d_+$-valued processes.This paper aims at giving a complete description of all $\mathbb{R}^d$-valuedmulti-self-similar Markov processes. We show that their state space is always aunion of open orthants with 0 as the only absorbing state and that there is nofinite entrance law at 0 for these processes. We give conditions for theseprocesses to satisfy the Feller property. Then we show that a Lamperti-typerepresentation is also valid for $\mathbb{R}^d$-valued multi-self-similarMarkov processes. In particular, we obtain a one-to-one relationship betweenthis 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 thealmost sure asymptotic behavior of multi-self-similar Markov processes.

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