FRACTIONAL BROWNIAN VECTOR FIELDS 1. Introduction. A one ...

can be shown as follows:
〈B H,ξ (σ·), f (·)〉 = 〈ε H [(−
FRACTIONAL BROWNIAN VECTOR FIELDS 15
= 〈σ H+ d 2 εH (−
2H+d
− 4 ´∆) W ](σ·), f (·)〉 by (3.4),
−ξ
= 〈σ H+ d 2 εH W (σ·),(−
= 〈σ H+ d 2 σ
− d 2 εH W (·),(−
= 〈σ H ε H (−
2H+d
− 4 ´∆) {W (σ·)}, f (·)〉 by (2.4),
−ξ
2H+d
− 4 `∆)
−ξ
2H+d
− 4 `∆)
−ξ
f (·)〉
by duality,
f (·)〉 by the homogeneity of W ,
2H+d
− 4 ´∆) {W (·)}, f (·)〉 by duality,
−ξ
= 〈σ H B H,ξ (·), f (·)〉 by (3.4).
4.2. Rotation invariance. For any orthogonal transformation matrix Ω, the random
fields B H,ξ and ΩB H,ξ (Ω T·) follow the same stochastic law. The demonstration is similar to
the previous one.
4.3. Non-stationarity. Vector fractional Brownian motion is non-stationary. Indeed,
2H+d
− 4
the operator (− `∆) is not translation-invariant, and consequently the random variables
−ξ
and
〈B H,ξ , f (·)〉 = 〈ε H W ,(−
〈B H,ξ , f (· + h)〉 = 〈ε H W ,(−
are not identically distributed in general.
2H+d
− 4 `∆) {f (·)}〉
−ξ
2H+d
− 4 `∆) {f (· + h)}〉
−ξ
4.4. Stationary n-th order increments. We shall now show that the increments of
order ⌊H⌋ + 1 of the field B H,ξ are stationary. In particular, for 0 < H < 1, B H,ξ has
stationary first-order increments, as is the case for standard fractional Brownian motion [53].
For this purpose, let us first define the n-th order symmetric difference operator D h1 ,...,h n
recursively by the relations
D h1
: f (·) → f (· + h 1
2 ) − f (· − h 1
2 ),
D h1 ,...,h n
:= D hn
D h1 ,...,h n−1
,
with h 1 ,...,h n ∈ d \{0}. The above operator is represented in the Fourier domain by the
expression
∏
1≤i≤n
2sin 〈h i ,ω〉
2
.
We have
THEOREM 4.1. The vector fBm field B H,ξ has stationary increments of order ⌊H⌋+1; that
is, the random field
D h1 ,...,h n
B H,ξ