Hausdorff dimension of the boundary of bubbles of additive Brownian motion and of the Brownian sheet

We first consider the additive Brownian motion process (X(s(1), s(2)), (s(1), s(2)) is an element of R-2) defined by X(s(1), s(2)) = Z(1)(s(1)) - Z2(s2), where Z(1) and Z(2) are two independent (two-sided) Brownian motions. We show that with probability 1, the Hausdorff dimension of the boundary of any connected component of the random set {(s(1,) s(2)) is an element of R-2 : X(s(1), s(2)) > 0} is equal to