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Some fractal properties of the image measure of an \(\alpha\)-stable process. (Chinese. English summary) Zbl 0964.60047
Summary: Let \(X(t)\), \(t\in \mathbb{R}^+\), be an \(\alpha\)-stable process on \(\mathbb{R}^d\) (\(0<\alpha\leq 2\), \(\alpha\neq 1\)) and \(\mu\) be a finite positive Borel measure. We discuss the fractal properties of the imague measure \(\mu_X\), get the upper and lower Hausdorff dimension of \(\mu_X\), and we use the dimension of \(\mu\) to give a condition such that \(\mu_X\) is absolutely continuous with respect to \(\lambda_d\) (the Lebesgue measure on \(\mathbb{R}^d\)).
MSC:
60G17 Sample path properties
28A78 Hausdorff and packing measures
60G52 Stable stochastic processes
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