2 2-dimensional Brownian motions are close to each other

Question

Suppose $B^1$ is a standard 2 dimensional Brownian motion and $B^2$ is a 2 dimensional Brownian motion with mean zero and covariance matrix $\Gamma = \begin{pmatrix} a & b \\ b & a \\ \end{pmatrix} $, both defined in the same probability space $(\Sigma, A, P)$. I would like to know how I can estimate $P(\sup\{ | B^1_{t} - B^2_{t} |, t \in [0,T] \} > \epsilon) $?