HAUSDORFF MEASURE OF TRAJECTORIES OF MULTIPARAMETER FRACTIONAL, BROWNIAN-MOTION

Consider 0 < alpha < 1 and the Gaussian process Y(t) on R(N) with covariance E(Y(t)Y(s)) = \t\(2 alpha) + \s\(2 alpha) - \t - s\(2 alpha), where \t\ is the Euclidean norm of t. Consider independent copies X(1),...,X(d) Of Y and the process X(t) = (X(1)(t),...,X(d)(t)) valued in R(d). In the transient case (N < alpha d) we show that a.s. for each compact set L of R(N) with nonempty interior, we have 0 < mu(phi)(X(L)) < infinity, where mu(phi), denotes the Hausdorff measure associated with the function (phi(epsilon) = epsilon(N/alpha) log log(1/epsilon). This result extends work of A. Goldman in the case alpha = 1/2; the proofs are considerably simpler.