2.2. THE COVARIANCE FUNCTION AND RIESZ KERNELS 9
Proof. By Fubini’s theorem, and using the change of variables w = t ξ, we see that
t
0
ds
Rd
|FG(s)(ξ)|2
|ξ|d−β
=
Rd

|ξ|d+2−β
t
0
1 cos(2s|ξ|)
2
ds
=
Rd

|ξ|d+2−β
t
2

sin(2t|ξ|
4|ξ|
=
t3−β
J,
where
J =
Rd
dw
|w|d+2−β
1
2

sin(2|w|)
4|w|
.
Note that J ∞. Indeed, J J1 + J2, where
J1 =
|w| 1
dw
|w|d+2−β
, J2 =
|w|≤1
dw
|w|d+2−β
1
2

sin(2|w|)
4|w|
.
Clearly, J1 ∞. For J2, since sin(2|w|) = 2|w|
23
3!
cos(ζ)|w|3,
with ζ ]0, |w|[,
J2 C
|w|≤1
dw
|w|d−β
∞.
This establishes (2.9).
We end this subsection with a result concerning the behavior near the origin
of s FG(s).
Lemma 2.3. For any b 0 and β ]0, 2[ such that β + b ]0, 3[,
(2.10) sup
t∈[0,T ]
t
0
ds
sb
Rd
|FG(s)(ξ)|2
|ξ|d−β
∞.
Proof. The change of variables ξ shows that the integral in (2.10) is equal to
t
0
ds
s2−(β+b)
Rd
sin2
|ξ|
|ξ|d+2−β
dξ.
The inner integral is finite for β ]0, 2[. Therefore (2.10) holds when β + b 3.
2.2. The Covariance Function and Riesz Kernels
We assume that the covariance measure of the noise
˙
F is absolutely continuous
with respect to Lebesgue measure, that is, Γ(dx) = f(x)dx. In addition, we suppose
that f satisfies the following assumption.
Assumption 2.4. There is β ]0, 2[ and δ ]0, 1] such that
f(x) = ϕ(x)kβ (x),
where (x) =
|x|−β,
x
Rd
\ {0}, and ϕ is bounded and positive, ϕ
C1(Rd)
and ∇ϕ
Cb(Rd) δ
(the space of bounded and older continuous functions with
exponent δ).
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