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Dilated fractional stable motions. (English) Zbl 1055.60041
The authors present a deep and detailed analysis of dilated fractional stable motions, namely processes which are symmetric \(\alpha\)-stable, self similar, with stationary increments and associated with dissipative flows, assuming the stability exponent \(\alpha\) to be less than 2 so that the variance is infinite and the Gaussian case is excluded. They establish several theorems and give well-chosen illustrating examples.
The paper consists of eight sections. After an outline of the general concepts the authors investigate in Section 2 equivalent representations and connections to other processes. Then, in Sections 3 and 4, they describe the space of integrands and lay open the connection to Sobolev spaces. Section 5 is devoted to five examples, the first of these the telecom process. In Sections 6 and 7 questions of uniqueness or non-uniqueness are treated, in a more refined way in Section 7 (with four examples), the proofs being delegated to Section 8.

MSC:
60G52 Stable stochastic processes
60G51 Processes with independent increments; Lévy processes
60G18 Self-similar stochastic processes
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