It is well known that Brownian motion (BM) has a Hausdorff dimension of 2, for topological dimension >= 2. (I have been reading Morters and Peres' book on Brownian Motion.) As I understand it, 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, I believe, 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 multidimensional 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. Thankyou.
2026-03-29 06:08:01.1774764481
Hausdorff (fractal) Dimension of a Stochastic Process
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