  Dunkl processes are generalizations of Brownian motion obtained by using the
differential-difference operators known as Dunkl operators as a replacement of
spatial partial derivatives in the heat equation. Special cases of these
processes include Dyson's Brownian motion model and the Wishart-Laguerre
eigenvalue processes, which are well-known in random matrix theory. It is known
that the dynamics of Dunkl processes is obtained by transforming the heat
kernel using Dunkl's intertwining operator. It is also known that, under an
appropriate scaling, their distribution function converges to a steady-state
distribution which depends only on the coupling parameter $\beta$ as the
process time $t$ tends to infinity. We study scaled Dunkl processes starting
from an arbitrary initial distribution, and we derive expressions for the
intertwining operator in order to calculate the asymptotics of the distribution
function in two limiting situations. In the first one, $\beta$ is fixed and $t$
tends to infinity (approach to the steady state), and in the second one, $t$ is
fixed and $\beta$ tends to infinity (strong-coupling limit). We obtain the
deviations from the limiting distributions in both of the above situations, and
we find that they are caused by the two different mechanisms which drive the
process, namely, the drift and exchange mechanisms. We find that the deviation
due to the drift mechanism decays as $t^{-1}$, while the deviation due to the
exchange mechanism decays as $t^{-1/2}$.
