Gaussian sample functions and the Hausdorff dimension of level crossings

SummaryLet Xt be a real Gaussian process with stationary increments, mean 0, σt2=E[(Xs+t−Xs)2] If σ σt2 behaves like tα as t ↺0, 0<α<1, the graph of a.e. sample function will have Hausdorff dimension 2 -α. This leads one to feel that the set of zeros of Xt should have Hausdorff dimension 1 -α. This is shown to be true provided the process is stationary and satisfies additional assumptions.