Hausdorff Dimension and Gaussian Fields

Let $X(t)$ be a Gaussian process taking values in Rᵈ and with its parameter in RN. Then if Xⱼ has stationary increments and the function σ²(t) = E\|Xⱼ(s + t) - Xⱼ(s)|²\ behaves like |t|2α as |t| ↓ 0, 0 < α < 1, the graph of X has Hausdorff dimension min /α, N + d(1 - α)\ with probability one. If X is also ergodic and stationary, and if N - dα 0, then the dimension of the level sets of X is a.s. N - dα.