Lévy’s Brownian motion; Total positivity structure of M(t)-process and deterministic character

Abstract

Let X = {X(A); A ∈ Q} be a Lévy’s Brownian motion with the basic time parameter space Q, where Q is taken to be the n-dimensional metric space Qn,k of constant curvature (2 ≤ n ≤ ∞, — ∞ < k: < ∞), i.e., Q is one of(a) Euclidean space for k = 0, (b) sphere for k > 0 and(c) real hyperbolic space for K < 0.The increment X(A) — X(B) is, by definition, Gaussian in distribution and has mean 0 and variance d(A, B), the distance between A and B. The existence of such a Gaussian random field is well known ([3], [4], [16] and [23]).