  Two families of stochastic interacting particle systems, the interacting
Brownian motions and Bessel processes, are defined as extensions of Dyson's
Brownian motion models and the eigenvalue processes of the Wishart and Laguerre
processes where the parameter $\beta$ from random matrix theory is taken as a
real positive number. These are systems where several particles evolve as
individual Brownian motions and Bessel processes while repelling mutually
through a logarithmic potential. These systems are also special cases of Dunkl
processes, a broad family of multivariate stochastic processes defined by using
Dunkl operators. In this thesis, the steady state under an appropriate scaling
and and the freezing ($\beta\to\infty$) regime of the interacting Brownian
motions and Bessel processes are studied, and it is proved that the scaled
steady-state distributions of these processes converge in finite time to the
eigenvalue distributions of the $\beta$-Hermite and $\beta$-Laguerre ensembles
of random matrices. It is also shown that the scaled final positions of the
particles in these processes are fixed at the zeroes of the Hermite and
Laguerre polynomials in the freezing limit. These results are obtained as the
consequence of two more general results proved in this thesis. The first is
that Dunkl processes in general converge in finite time to a scaled
steady-state distribution that only depends on the type of Dunkl process
considered. The second is that in the freezing limit, their scaled final
position is fixed to a set of points called the peak set, which is the set of
points which maximizes their steady-state distribution.
