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Zbl 0761.60036
Schneider, W.R.
Grey noise. (English)
[A] Ideas and methods in mathematical analysis, stochastics and applications. In memory of R. Høegh-Krohn, Vol. 1, 261-282 (1992). ISBN 0-521-41929-8

Summary: The Mittag-Leffler function $E\sb \alpha$ is completely monotonic on ${\bold R}\sb -$ for $0<\alpha\le 1$. This remarkable fact is exploited to define a probability measure $\tau\sb \alpha$ on a Hilbert triple $K\sb \alpha\subset H\sb \alpha\subset K\sb \alpha'$. This measure is called grey noise. It reduces to white noise for $\alpha=1$. Mimicking the construction of Brownian motion yields its grey variant. The sample paths of grey Brownian motion are Hölder continuous with index arbitrarily close to $\alpha/2$. The well-known relation between Brownian motion and diffusion carries over to grey Brownian motion and fractional diffusion with time derivative of order $\alpha$.

MSC 2000:: *60G20 Generalized stochastic processes
60G17 Sample path properties

Keywords: grey noise; sample paths of grey Brownian motion; fractional diffusion

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