  We begin with the study of some properties of the radial Dunkl process
associated to a reduced root system $R$. It is shown that this diffusion is the
unique strong solution for all $t \geq 0$ of a SDE with singular drift. Then,
we study $T_0$, the first hitting time of the positive Weyl chamber : we prove,
via stochastic calculus, a result already obtained by Chybiryakov on the
finiteness of $T_0$. The second and new part deals with the law of $T_0$ for
which we compute the tail distribution, as well as some insight via stochastic
calculus on how root systems are connected with eigenvalues of standard
matrix-valued processes. This gives rise to the so-called $\beta$-processes.
The ultraspherical $\beta$-Jacobi case still involves a reduced root system
while the general case is closely connected to a non reduced one. This process
lives in a convex bounded domain known as principal Weyl alcove and the strong
uniqueness result remains valid. The last part deals with the first hitting
time of the alcove's boundary and the semi group density which enables us to
answer some open questions.
