Let X be a solution to the martingale problem for A with initial condition
L
X 0 = µ. By Lemma 3.3.3 below, there exists a sequence of functions
p
n
: → [0, ∞ such that
i
p
n
i = 1 for each n and
lim
n →∞
k
| p
n
i − k − p
n
j − k| = 0
∀i, j ∈ . 3.2.10
Let X
n
be a sequence of processes with sample paths in
D
K
[0, ∞ with law
L
X
n
=
k
p
n
k
L T
k
X . 3.2.11
Then each X
n
solves the martingale problem for A with initial condition
k
p
n
kµ ◦ T
−1 k
= µ, where we use that µ is shift-invariant. By Lemma 3.2.2 we can find a subsequence X
nm
and a solution X
∞
to the martingale problem for A such that X
nm
⇒ X
∞
. Clearly X
∞
has initial condition
L
X
∞
= µ and for any bounded continuous real function f on
D
K
[0, ∞ we have
|E[ f
T
j
X
nm
] − E[ f X
nm
] |
=
k
p
nm
kE[ f
T
j
T
k
X ] −
i
p
nm
kE[ f
T
k
X ] =
k
p
nm
k − jE[ f
T
k
X ] −
i
p
nm
kE[ f
T
k
X ] ≤
k
| p
nm
k − j − p
nm
k | f
∞
. 3.2.12
By 3.2.10 it follows that
T
j
X
∞
and X
∞
have the same distribution as a proba- bility measure on
D
K
[0, ∞, which implies that their finite-dimensional distribu-
tions agree. Hence X
∞
is shift-invariant.
3.2.2 Proof of Theorem 3.1.2
Define a normalized interaction kernel ˜a and a normalizing constant Z by
Z : =
i
ai ˜ai := Z
−1
ai . 3.2.13
For each M 1 there exist [38] strictly positive numbers γ
i i
∈
such that
i
γ
i
∞ and
i
˜a j − iγ
i
≤ Mγ
j
j ∈ .
3.2.14
Let L
2
γ be the Hilbert space
L
2
γ :
= {x ∈
R
d
:
i
γ
i
|x
i
|
2
∞} 3.2.15
with inner product x, y
γ
: =
i
γ
i
x
i
· y
i
, 3.2.16
where · denotes the standard inner product on
R
d
. Clearly, K ⊂ L
2
γ and
the topology on K coincides with the topology on L
2
γ . We write
x
γ
: =
x, x
γ
for the Hilbert norm on L
2
γ .
Set t : = X t − ˜Xt, where X and ˜X are solutions to 3.1.2, starting in
X 0 = ˜X0 and adapted to the same set of Brownian motions. Then
d
α i
t = Z
j
˜a j − i
α j
t −
α i
tdt +
β
σ
αβ
X
i
t − σ
αβ
˜ X
i
td B
β i
t. 3.2.17
By Itˆo’s formula we see that E
T
2 γ
=
T
E 2
i α
γ
i α
i
tZ
j
˜a j − i
α j
t −
α i
t +
i
γ
i αβ
σ
αβ
X
i
t − σ
αβ
˜ X
i
t
2
dt. 3.2.18
By the Lipschitz property of σ we have
αβ
σ
αβ
x − σ
αβ
y
2
1 2
≤ L|x − y| x, y
∈ K 3.2.19
for some L ∞. With ˜a t
α i
: =
j
˜a j − i
α j
t it follows that
E T
2 γ
≤
T
E 2Z
t, ˜a t − t
γ
+ L
2
t
2 γ
dt ≤
T
E 2Z
t
γ
˜a t
γ
− t
2 γ
+ L
2
t
2 γ
dt ≤
T
2Z M
1 2
− 1 + L
2
E t
2 γ
dt, 3.2.20
where we used Cauchy-Schwarz and the fact that, by Jensen’s inequality and by 3.2.14,
˜ax
2 γ
=
i
γ
i j
˜a j − ix
j 2
≤
i j
γ
i
˜a j − i|x
j
|
2
≤
j
Mγ
j
|x
j
|
2
= Mx
2 γ
. 3.2.21
The result now follows from Gronwall’s lemma.
3.3 Proofs of Theorem 3.1.3 and Lemma 3.1.4
3.3.1 Proof of Lemma 3.1.4
Note that, since any solution X to 3.1.2 solves the martingale problem for the operator A in 3.1.10, we have for any f
∈
C
2 fin
K E[ f X t]
− E[ f X 0] =
t
E[ A f X s]ds. 3.3.1
Using the continuity of A f , the continuity of the sample paths of X , and bounded convergence, we see that the function t
→ E[A f X t] is continuous. It follows that the function t
→ E[ f X t] is continuously differentiable and satisfies
∂ ∂
t
E[ f X t] = E[A f X t].
3.3.2 Applying the remarks above to the function f x
= x
α i
and using bounded conver- gence to interchange an infinite sum and expectation, we see that
∂ ∂
t
E[X
α i
t] =
j
a j − iE[X
α j
] − E[X
α i
]. 3.3.3
When X is shift-invariant, there clearly exist functions θ : [0, ∞ → K and C :
[0, ∞ × →
R
such that E[X
α i
t] = θ
α
t CovX
i
t, X
j
t = C
t
j − i
t ≥ 0, i, j ∈ , α = 1, . . . , d. 3.3.4
Applying this to 3.3.3, we see that
∂ ∂
t
θ t
= 0 and hence E[X
i
t] = θ
t ≥ 0, i ∈
3.3.5 for some θ
∈ K .
Let us put ˜ X
i
: = X
i
− θ. Applying 3.3.2 to the function f x =
α
x
α i
− θ
α
x
α j
− θ
α
, using bounded convergence to interchange an infinite sum and ex- pectation, we get
∂ ∂
t
CovX
i
t, X
j
t =
k,l
ak − lE
α
˜ X
α k
t − ˜X
α l
tδ
il
˜X
α j
t + δ
jl
˜X
α i
t +2δ
i j
E[tr wX t]. 3.3.6
Inserting 3.3.4 we get
∂ ∂
t
C
t
j − i =
k
ak − iC
t
j − k − C
t
j − i
+
k
ak − jC
t
k − i − C
t
j − i
+2δ
i j
E[tr wX t]. 3.3.7
Substituting ˜ı := j − i, ˜ := k − i and ˜k := j − k and reordering the summations,
we find that
∂ ∂
t
C
t
˜ı =
˜
a ˜C
t
˜ı − ˜ − C
t
˜ı +
˜k
a − ˜kC
t
˜ı − ˜k − C
t
˜ı +2δ
˜ı0
E[tr wX t]. 3.3.8
This shows that formula 3.1.34 holds.
3.3.2 Random walk representations
Let B be the Banach space of bounded real functions on , equipped with the supremum norm. The operator G in 3.1.35 is a bounded linear operator on B.
We define a Feller semigroup on B by
P
t
f : = e
t G
f, 3.3.9
where e
t G
: =
∞ n
=0 1
n
t G
n
. This semigroup corresponds to a continuous-time random walk I
t t
≥0
on that jumps from i to j with rate a
S
j − i. By shift-
invariance there exists a function P : [0, ∞ × →
R
such that P
t
j − i = P
i
[I
t
= j]. 3.3.10
We can consider P
t
j − i as the i, j-th element of the matrix of the operator P
t
in 3.3.9, in the following sense P
t
f i =
j
P
t
j − i f j.
3.3.11
