# On independent times and positions for Brownian motions

### Bernard de Meyer

Université Henri Poincaré, Vandoeuvre lès Nancy, France### Bernard Roynette

Université Henri Poincaré, Vandoeuvre lès Nancy, France### Pierre Vallois

Université Henri Poincaré, Vandoeuvre lès Nancy, France### Marc Yor

Université Paris VI, France

## Abstract

Let $(B_{t};t≥0)$, \( \big(\mbox{resp. }((X_t, Y_t) ; t \ge 0)\big) \) be a one (resp. two) dimensional Brownian motion started at 0. Let $T$ be a stopping time such that $(B_{t∧T};t≥0)$ \big(resp. $(X_{t∧T};t≥0);(Y_{t∧T};t≥0))$ is uniformly integrable. The main results obtained in the paper are: \begin{itemize} \item[1)] if $T$ and $B_{T}$ are independent and $T$ has all exponential moments, then $T$ is constant. \item[2)] If $X_{T}$ and $Y_{T}$ are independent and have all exponential moments, then $X_{T}$ and $Y_{T}$ are Gaussian. \end{itemize} We also give a number of examples of stopping times $T$, with only some exponential moments, such that $T$ and $B_{T}$ are independent, and similarly for $X_{T}$ and $Y_{T}$. We also exhibit bounded non-constant stopping times $T$ such that $X_{T}$ and $Y_{T}$ are independent and Gaussian.

## Cite this article

Bernard de Meyer, Bernard Roynette, Pierre Vallois, Marc Yor, On independent times and positions for Brownian motions. Rev. Mat. Iberoam. 18 (2002), no. 3, pp. 541–586

DOI 10.4171/RMI/328