A generalisation of the fractional Brownian field based on non-Euclidean norms

Molchanov, Ilya; Ralchenko, Kostiantyn (2015). A generalisation of the fractional Brownian field based on non-Euclidean norms. Journal of mathematical analysis and applications, 430(1), pp. 262-278. Elsevier 10.1016/j.jmaa.2015.04.085

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We explore a generalisation of the L´evy fractional Brownian field on the
Euclidean space based on replacing the Euclidean norm with another norm.
A characterisation result for admissible norms yields a complete description
of all self-similar Gaussian random fields with stationary increments. Several
integral representations of the introduced random fields are derived. In a similar vein, several non-Euclidean variants of the fractional Poisson field are introduced and it is shown that they share the covariance structure with the fractional Brownian field and converge to it. The shape parameters of the Poisson and Brownian variants are related by convex geometry transforms, namely the radial pth mean body and the polar projection transforms.

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