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

dξ =

Rd

dξ

|ξ|d+2−β

t

0

1 − cos(2s|ξ|)

2

ds

=

Rd

dξ

|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

dξ ∞.

Proof. The change of variables ξ → sξ 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 kβ (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 H¨ older continuous functions with

exponent δ).