| The evaluation of fractal dimensions of some random sets, such as the range sets,graph sets or level sets associated with a Brownian motion, has been a problem ofgreat interest in the research of fractal analysis and stochastic processes. We discussthe development, extensions and variations of this problem, paying special attention toBrownian motions in R~d. At last, we focus on the Brownian motion on the Sierpinskigasket, with extending a recent result about the exact Hausdorff measure of graph sets.
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