Grey noise. (English) Zbl 0761.60036

Ideas and methods in mathematical analysis, stochastics and applications. In memory of R. Høegh-Krohn, Vol. 1, 261-282 (1992).
Summary: The Mittag-Leffler function \(E_ \alpha\) is completely monotonic on \({\mathbf R}_ -\) for \(0<\alpha\leq 1\). This remarkable fact is exploited to define a probability measure \(\tau_ \alpha\) on a Hilbert triple \(K_ \alpha\subset H_ \alpha\subset K_ \alpha'\). 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 \(\alpha\).
For the entire collection see [Zbl 0764.00003].


60G20 Generalized stochastic processes
60G17 Sample path properties