Kostiantyn Ralchenko: A generalisation of the fractional Brownian field based on non-Euclidean norms
Kostiantyn Ralchenko (Taras Shevchenko National University of Kyiv) gives a talk with the title: A generalisation of the fractional Brownian field based on non-Euclidean norms
We explore a generalisation of the Lévy 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.