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

An ${\mathbb{R}}^d$-valued Markov process $X_t^{(x)}=(X_t^{1,x_1},\dots ,X_t^{d,x_d})$, $t\ge 0,x\in {\mathbb{R}}^d$ is said to be multi-self-similar with index $({\alpha }_1,\dots ,{\alpha }_d)\in [0,\mathrm{\infty }{)}^d$ if the identity in law $$(c_i{X}_t^{i,x_i\mathrm{/}{c}_i};i=1,\dots ,d{)}_{t\ge 0}\text{ed}(X_{ct}^{(x)}{)}_{t\ge 0},$$ where $c={\prod }_{i=1}^d{c}_i^{{\alpha }_i}$, 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.