RANDOM FIELDS AND THEIR GEOMETRY

18 1. Random fields
independent of t. We call such a measure ν left relatively invariant under
G. It is easy to see that D(g) is a C ∞ homomorphism from G into the
multiplicative group of positive real numbers, i.e. D(g1g2) = D(g1)D(g2).
We say that ν is left invariant with respect to G if, and only if, it is left
relatively-invariant and D ≡ 1.
Here, finally, is the result.
Lemma 1.4.2 Suppose G acts smoothly on a smooth manifold T and ν
is left relatively invariant under G. Let D be as in (1.4.15) and let W be
Gaussian ν-noise on T . Then, for any F ∈ L 2 (T, ν),
f(g) =
1
� D(g) W (F ◦ θ g −1),
is a left stationary Gaussian random field on G.
Proof. We must prove that
E {f(g1)f(g2)} = C(g −1
1 g2)
for some C : G → R. From the definition of W , we have,
E {f(g1)f(g2)} =
�
1
� � � �
� F θ −1
g (t) F θ −1
1
g (t) ν(dt)
2
D(g1)D(g2) T
� �
�
=
F (θg2(t)) F (t) θg2∗(ν)(dt)
=
=
1
� D(g1)D(g2)
D(g2)
� D(g1)D(g2)
�
T
T
�
D(g −1
1 g2)
�
F
M
∆
= C(g −1
1 g2)
F
θ g −1
1
�
θ −1
g1 g2(t)
�
F (t) ν(dt)
�
θ g −1
1 g2(t)
�
F (t) ν(dt)
This completes the proof. ✷
It is easy to find simple examples to which Lemma 1.4.2 applies. The most
natural generic example of a Lie group acting on a manifold is its action on
itself. In particular, any right Haar measure is left relatively-invariant, and
this is a way to generate stationary processes. To apply Lemma 1.4.2 in
this setting one needs only to start with a Gaussian noise based on a Haar
measure on G. In fact, this is the example (1.4.12) with which we started
this Section.
A richer but still concrete example of a group G acting on a manifold T
is given by G = GL(N, R) × R N acting 19 on T = R N . For g = (A, t) and
19 Recall that GL(N, R) is the (general linear) group of transformations of R N by
rotation.