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, ·).
7
Previous Page Next Page