On independent times and positions for Brownian motions. (Q1394485)

Let \((B_t\); \(t\geq 0)\) (resp. \(((X_t,Y_t)\); \(t\geq 0)\))) be a one- (resp. two-) dimensional Brownian motion started at 0. Let \(T\) be a stopping time such that \((B_{t\wedge T}\); \(t\geq 0)\) (resp. \((X_{t\wedge T}\); \(t\geq 0)\); \((Y_{t\wedge T}\); \(t\geq 0)\)) is uniformly integrable. The authors look for some additional assumptions on the joint law of \((B_T,T)\) under which one hopes that the law of \(T\) is determined from the law \(\mu\) of \(B_T\). The main result of the paper states that: (1) if \(T\) and \(B_T\) are independent and \(T\) has all exponential moments, then \(T\) is constant; (2) if \(X_T\) and \(Y_T\) are independent and have all exponential moments, then \(X_T\) and \(Y_T\) are Gaussian. The proposed approach is rather general and allows also to prove similar results, when the Brownian motion \((B_t\); \(t\geq 0)\) is replaced by Brownian motion with drift, Ornstein-Uhlenbeck or Bessel processes. A number of examples of stopping times \(T\) (obtained by iteration or from intertwinings) with only some exponential moments, such that \(T\) and \(B_T\) are independent (and similarly for \(X_T\) and \(Y_T\)), are presented. The authors also obtain a negative answer on Tortrat's conjecture and exhibit a class of non-constant stopping times such that \(X_T\) and \(Y_T\) are independent and Gaussian. In the Appendix the relation of the obtained results with Skorokhod embedding problem for space-time Brownian motion \(((t,B_t)\); \(t\geq 0)\) is discussed.