Fractional Brownian Models for Vector Field Data

In this note we introduce a vector generalization of fractional Brownian motion. Our definition takes into account directional properties of vector fields—such as divergence, rotational behaviour, and interactions with coordinate transformations—that have no counterpart in the scalar setting. Apart from the Hurst exponent which dictates the scale-dependent structure of the field, additional parameters of the new model control the balance between solenoidal and irrotational behaviour. This level of versatility makes these random fields potentially interesting candidates for the stochastic modelling of physical phenomena in various fields of application such as fluid dynamics, field theory, and medical image processing.