CHAPTER 2

The Fundamental Solution of the Wave Equation

and the Covariance Function

The first part of this chapter is devoted to introducing the smoothing of the

fundamental solution of the wave equation used throughout the paper. We prove

some of its properties as well as some of the properties of the fundamental solution

itself. In the second part, we obtain expressions for first and second order increments

of the covariance function. Informally, these express the covariance function as

a fractional integral of its fractional derivative; they are proved by applying the

semigroup property of the Riesz kernels.

2.1. Some Properties of the Fundamental Solution and Its

Regularisations

Let d ≥ 1 and ψ :

Rd

→ R+ be a function in

C∞(Rd)

with support included

in B1(0) and such that

Rd

ψ(x)dx = 1 (Br(x) denotes the open ball centered at

x ∈

Rd

with radius r ≥ 0). For any t ∈ ]0, 1] and n ≥ 1, we define

(2.1) ψn(t, x) =

n

t

d

ψ

n

t

x

and

(2.2) Gn(t, x) = (ψn(t, ·) ∗ G(t))(x),

where “∗ denotes the convolution operation in the spatial variable. Observe that

Rd

ψn(t, x)dx = 1 and

supp Gn(t, ·) ⊂ Bt(1+

1

n

)

(0).

The following elementary scaling property plays an important role in the study

of regularity properties in time of the stochastic integral. Its proof is included for

convenience of the reader.

Lemma 2.1. Let d = 3. For any s, t ∈ [0, T ] and v0 ∈ C(R3),

(2.3)

R3

G(s, du) v0(u) =

s

t

R3

G(t, du) v0

s

t

u ,

and for any x ∈

R3,

(2.4) Gn t,

t

s

x =

s

t

2

Gn(s, x).

Proof. The first equality follows from the fact that the transformation u →

s

t

u

maps G(t, ·) onto

t

s

G(s, ·).

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