Processes : Existence , Uniqueness and Hitting Time

Abstract. We give shorter proofs of the following known results: the radial Dunkl process associated with a reduced system and a strictly positive multiplicity function is the unique strong solution for all times t of a stochastic differential equation with a singular drift (see [11] for the original proof and [4] for a proof under an additional restriction), the first hitting time of the Weyl chamber by a radial Dunkl process is finite almost surely for small values of the multiplicity function. Compared to the original proofs, ours give more information on the behaviour of the process. More precisely, the first proof allows to give a positive answer to a conjecture announced by Gallardo and Yor in [4] while the second one shows that the process hits almost surely the wall corresponding to the simple root with a small multiplicity value.