the set of $\boxplus$-infinitely divisible measures $\mu$. Recall that $\mu$ is $\boxplus$-infinitely divisible if for any $n\in\mathbb{N}$ there exists a measure $\mu_n\in\mathcal{M}$ such that $\mu_n^{\boxplus
n}=\mu$. Another operation of convolution is the Boolean convolution $\uplus$ introduced by Speicher and Woroudi \[\[Boolean\]\]. The connection among free, Boolean, and classical infinite divisibilities was thoroughly studied by Bercovici and Pata \[\[BP\]\]. An aspect of this connection between infinite divisibility with respect to $\boxplus$ and $\uplus$ is the Boolean Bercovici-Pata bijection $\mathbb{B}$.

Another map $\mathbb{B}_t:\mathcal{M}\to\mathcal{M}$ connecting free and Boolean convolutions is defined by $$\mathbb{B}_t(\mu)=\left(\mu^{\boxplus(t+1)}\right)^{\uplus\frac{1}{t+1}},\;\;\;\;\;t\geq0,\;\;\mu\in\mathcal{M}.$$ This map introduced by Belinshi and Nica \[\[BN1\]\] satisfies $\mathbb{B}_t\circ\mathbb{B}_s=\mathbb{B}_{t+s}$, $s,t\geq0$. More importantly, the map $\mathbb{B}_1$ coincides with the Boolean Bercovici-Pata bijection $\mathbb{B}$. As a result, $\mathbb{B}_t(\mu)$ is $\boxplus$-infinitely divisible for any $\mu\in\mathcal{M}$ and $t\geq1$. This led the authors to associate to each measure $\mu\in\mathcal{M}$ a nonnegative number $\mathrm{Ind}(\mu)$, which is called $\boxplus$-divisibility indicator. For instance, the semicircular and Cauchy distributions have $\boxplus$-divisibility indicators $1$ and $\infty$, respectively. It was also shown that $\mu$ is $\boxplus$-infinitely divisible if and only if $\mathrm{Ind}(\mu)\geq1$. For any measure $\mu\in\mathcal{M}$ with mean zero and unit variance, denote by $\Phi(\mu)$ the unique measure in $\mathcal{M}$ such that $E_\mu=G_{\Phi(\mu)}$. Recall that the free Brownian motion started 