Hausdorff Dimension and Gaussian Fields

Abstract

Let $X(t)$ be a Gaussian process taking values in $R^d$ and with its parameter in $R^N$. Then if $X_j$ has stationary increments and the function $\sigma^2(t) = E\{|X_j(s + t) - X_j(s)|^2\}$ behaves like $|t|^{2\alpha}$ as $|t| \downarrow 0, 0 < \alpha < 1$, the graph of $X$ has Hausdorff dimension $\min \{N/\alpha, N + d(1 - \alpha)\}$ with probability one. If $X$ is also ergodic and stationary, and if $N - d\alpha \geqq 0$, then the dimension of the level sets of $X$ is a.s. $N - d\alpha$.