Bifractional Brownian Motions on Metric Spaces

Abstract

Fractional and bifractional Brownian motions can be defined on a metric space if the associated metric or distance function is conditionally negative definite (or of negative type). This paper introduces several forms of scalar or vector bifractional Brownian motions on various metric spaces and presents their properties. A metric space of particular interest is the arccos-quasi-quadratic metric space over a subset of $R^ {d+1}$ such as an ellipsoidal surface, an ellipsoid, or a simplex, whose metric is the composition of arccosine and quasi-quadratic functions. Such a metric is not only conditionally negative definite but also a measure definite kernel, and the metric space incorporates several important cases in a unified framework so that it enables us to study (bi, tri, quadri)fractional Brownian motions on various metric spaces in a unified manner. The vector fractional Brownian motion on the arccos-quasi-quadratic metric space enjoys an infinite series expansion in terms of spherical harmonics, and its covariance matrix function admits an ultraspherical polynomial expansion. We establish the property of strong local nondeterminism of fractional and bi(tri, quadri)fractional Brownian motions on the arccos-quasi-quadratic metric space.