Fractional Brownian Fields as Integrals of White Noise

A Fractional Brownian Field (FBF) of index pis a random field LP : n x Rd -+ R such that (i) Lp(O) = 0 a.s.(ii) For all x1, x2, ... , Xn E Rd, the random vector (Lp(x1), Lp(x2), ... , Lp(xn)) is gaussian with mean zero (iii) For all x,y E Rd,E((Lp(x)-Lp(Y)) 2 ) = llx-YIIP (iv) x-+ Lp(x,w) is continuous for almost all w.Lp exists for 0 < p < 2 (and even for p = 2 when the dimension dis one).Note that the covariance of Lp can be calculated from (i)-(iii):Since LP is gaussian, this tells us that its law is uniquely determined by (i)-(iv).When p = 1, LP is Levy-Brownian motion -one of several natural generalizations of Brownian motion to the multiparmeter case (the Brownian sheet is another one).FBFs are statistically self-similar in the following sense.IT x 0 E IRd and a E R+, then the random field iP defined by is also an FBF of index p.In the last twenty years, this fact has been used extensively to generate pictures of "fractal landscapes" and also to model fractal phenomena in natural and social sciences (see, e.g., [2], [5], [6], [7] and (10]).My own interest in FBFs grew out of a wish to understand some recent applications of these techniques to oil reservoir modelingThe purpose of the present note is to point out that FBFs can be conveniently represented as integrals of white noise.As a matter of fact, all that is required is a slight extension of Andreas Stoll's representation theorem for Levy-Brownian motion (see [1] and [9]).One way of phrasing Stoll's result is to say that ford> 1,