It is well known that Brownian motion (BM) has a Hausdorff dimension (ie fractal dimension) of 2, for topological dimension >= 2. In other words, BM always "behaves like" a plane surface, no matter the dimensionality of the BM process itself (so long as the BM is topological dimension >= 2). This is related to the transience of BM for topological dimensions > 2, and recurrence of BM for topological dimension <= 2. I am trying to find out if there has been any work on the Hausdorff dimension of other multidimensional stochastic processes, including the Ornstein-Uhlenbeck process but also others as well. Are there any examples of the calculation of Hausdorff dimensions for stochastic processes other than BM? Please point me to any literature on this topic.
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2You might have much better luck with this on math.se. – jld Dec 14 '16 at 13:53