## 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].

### MSC:

 60G20 Generalized stochastic processes 60G17 Sample path properties