On independent times and positions for Brownian motions

Let (Bt ; t ≥ 0), ( resp. ((Xt, Yt) ; t ≥ 0) ) be a one (resp. two) dimensional Brownian motion started at 0. Let T be a stopping time such that (Bt∧T ; t ≥ 0) ( resp. (Xt∧T ; t ≥ 0) ; (Yt∧T ; t ≥ 0) ) is uniformly integrable. The main results obtained in the paper are: 1) if T and BT are independent and T has all exponential moments, then T is constant. 2) If XT and YT are independent and have all exponential moments, then XT and YT are Gaussian. We also give a number of examples of stopping times T , with only some exponential moments, such that T and BT are independent, and similarly for XT and YT . We also exhibit bounded non-constant stopping times T such that XT and YT are independent and Gaussian.