  We propose new copulae to model the dependence between two Brownian motions
and to control the distribution of their difference. Our approach is based on
the copula between the Brownian motion and its reflection. We show that the
class of admissible copulae for the Brownian motions are not limited to the
class of Gaussian copulae and that it also contains asymmetric copulae. These
copulae allow for the survival function of the difference between two Brownian
motions to have higher value in the right tail than in the Gaussian copula
case. Considering two Brownian motions $B_t^1$ and $B_t^2$, the main result is
that the range of possible values for $\mathbb{P}\left(B_t^1-B^2_t \geq
\eta\right)$ with $\eta > 0$ is the same for Markovian pairs and all pairs of
Brownian motions, that is
$\left[0,2\Phi\left(\frac{-\eta}{2\sqrt{t}}\right)\right]$ with $\Phi$ being
the cumulative distribution function of a standard Gaussian random variable.
