Gaussian Processes with Stationary Increments: Local Times and Sample Function Properties

Abstract

Let $X(t), 0 \leqq t \leqq 1$, be separable measurable Gaussian process with mean 0, stationary increments, and $\sigma^2(t) = E(X(t) - X(0))^2$. If $\sigma^2(t) \sim C|t|^\alpha, t \rightarrow 0$, for some $\alpha, 0 < \alpha < 2$, then the Hausdorff dimension of $\{s: X(t) = X(s)\}$ is equal to $1 - (\alpha/2)$ for almost all $t$, almost surely. Under further variations and refinements of this condition there is a jointly continuous local time for almost every sample function. This extends the author's previous results for stationary Gaussian processes and for continuity in the space variable alone. The result on joint continuity of the local time is used to prove that the sample function has an "approximate derivative" of infinite magnitude at each point (and so is nowhere differentiable); and that the set of values in the range of at most countable multiplicity is nowhere dense in the range.