The measure preserving transformations of the multidimensional stable Lévy processes

Let $\xi(t)$, $t\in[0,1]$ be a $\alpha-$stable Lévy process in $\mathbb R^d$. Denote by $\mathcal P_\xi$ the measure generated by $\xi$ in Skorokhod space $\mathbb D([0,1],\mathbb R^d)$. Under some conditions on the spectral measure of the process $\xi$ we construct a group of the $\mathcal P_\xi-$preserving transformations of $\mathbb D([0,1]\mathbb R^d$.