  Using the determinantal formula of Biane, Bougerol, and O'Connell, we give
multitime joint probability densities to the noncolliding Brownian motion with
drift, where the number of particles is finite. We study a special case such
that the initial positions of particles are equidistant with a period $a$ and
the values of drift coefficients are well-ordered with a scale $\sigma$. We
show that, at each time $t >0$, the single-time probability density of particle
system is exactly transformed to the biorthogonal Stieltjes-Wigert matrix model
in the Chern-Simons theory introduced by Dolivet and Tierz. Here one-parameter
extensions ($\theta$-extensions) of the Stieltjes-Wigert polynomials, which are
themselves $q$-extensions of the Hermite polynomials, play an essential role.
The two parameters $a$ and $\sigma$ of the process combined with time $t$ are
mapped to the parameters $q$ and $\theta$ of the biorthogonal polynomials. By
the transformation of normalization factor of our probability density, the
partition function of the Chern-Simons matrix model is readily obtained. We
study the determinantal structure of the matrix model and prove that, at each
time $t >0$, the present noncolliding Brownian motion with drift is a
determinantal point process, in the sense that any correlation function is
given by a determinant governed by a single integral kernel called the
correlation kernel. Using the obtained correlation kernel, we study time
evolution of the noncolliding Brownian motion with drift.
