The Hausdorff dimension of multivariate operator-self-similar Gaussian random fields

Let {X(t):t∈Rd} be a multivariate operator-self-similar random field with values in Rm. Such fields were introduced in [22] and satisfy the scaling property {X(cEt):t∈Rd}=d{cDX(t):t∈Rd} for all c>0, where E is a d×d real matrix and D is an m×m real matrix. We solve an open problem in [22] by calculating the Hausdorff dimension of the range and graph of a trajectory over the unit cube K=[0,1]d in the Gaussian case. In particular, we enlighten the property that the Hausdorff dimension is determined by the real parts of the eigenvalues of E and D as well as the multiplicity of the eigenvalues of E and D.