at $\nu\in\mathcal{M}$ is the process $\{\nu\boxplus\gamma_t:t>0\}$, where $\gamma_t$ is the centered semicircular distribution of variance $t$. The connection between this process and the map $\mathbb{B}_t$ is via the identity $E_{\mathbb{B}_t(\mu)}=G_{\Phi(\mu)\boxplus\gamma_t}$. These authors also studied the regularity of measures in $\mathbb{B}_t(\mathcal{M})$. In \[\[BN2\]\], the same authors studied the map $\mathbb{B}_t$ on the space $\mathcal{D}_c(k)$ of distributions of $k$-tuples of self-adjoint elements in a $C^*$-probability space based on moments and combinatorics. As in \[\[BN1\]\], they showed that $\mathbb{B}_1$ is the multi-variable Boolean Bercovic-Pata bijection and investigated the relation between $\mathbb{B}_t$ and free Brownian motion. Later, for measures $\mu,\nu\in\mathcal{D}_c(k)$, Nica \[\[Nica\]\] studied the so-called subordination distribution of $\mu\boxplus\nu$ with respect to $\nu$, in which a property related to the present paper is that $(\mu^{\boxplus p})^{\uplus(p-1)/p}$ is $\boxplus$-infinitely divisible for any $p>1$. For other further developments on $\mathbb{B}_t$ and the $\boxplus$-divisibility indicator of the measure $(\mu^{\boxplus p})^{\uplus q}$, we refer the reader to \[\[Japan\]\].

In the present paper, we mainly use the subordination functions for the $\boxplus$-convolution powers to study $\boxplus$-infinitely divisible measures. We show that measures of the form $(\mu^{\boxplus p})^{\uplus q}$ are $\boxplus$-infinitely divisible for $\mu\in\mathcal{M}$, $p>1$, and $0<q\leq(p-1)/p$. We also provide explicit formulas for the Voiculescu transforms of $\boxplus$-infinitely divisible measures. 