  We consider a pair of probability measures $\mu,\nu$ on the unit circle such
that $\Sigma_{\lambda}(\eta_{\nu}(z))=z/\eta_{\mu}(z)$. We prove that the same
type of equation holds for any $t\geq 0$ when we replace $\nu$ by
$\nu\boxtimes\lambda_t$ and $\mu$ by $\mathbb{M}_t(\mu)$, where $\lambda_t$ is
the free multiplicative analogue of the normal distribution on the unit circle
of $\mathbb{C}$ and $\mathbb{M}_t$ is the map defined by Arizmendi and Hasebe.
These equations are a multiplicative analogue of equations studied by Belinschi
and Nica. In order to achieve this result, we study infinite divisibility of
the measures associated with subordination functions in multiplicative free
Brownian motion and multiplicative free convolution semigroups. We use the
modified $\mathcal{S}$-transform introduced by Raj Rao and Speicher to deal
with the case that $\nu$ has mean zero. The same type of the result holds for
convolutions on the positive real line. We also obtain some regularity
properties for the free multiplicative analogue of the normal distributions.
