By the compactness of K , the maps
x →
∂ ∂
x
α i
f x
α =1,...,d
i ∈
x →
∂
2
∂ x
α i
∂ x
β j
f x
α,β =1,...,d
i, j ∈
3.4.29 are uniformly continous with respect to the norm on the spaces l
1
{1, . . . , d} × and l
1
{1, . . . , d}
2
×
2
. This implies that lim
n →∞
sup
x ∈K
i,α ∂
∂ x
α i
f π
n
x −
∂ ∂
x
α i
f x =
0, 3.4.30
and similarly for second derivatives. Finally, by Dini’s theorem see 3.4.21 lim
n →∞
sup
x ∈K
i n,α ∂
∂ x
α i
f x =
0, 3.4.31
and similarly for second derivatives, so A f
n
→ A
′
f .
3.4.3 Proof of Lemma 3.1.6
The model with zero diffusion: In the special case where w
= 0, the system of stochastic differential equations 3.1.2 reduces to
d X
i
t =
j ∈
a j − iX
j
t − X
i
tdt i
∈ , t ≥ 0. 3.4.32
By Theorems 3.1.1 and 3.1.2 this system of equations has a unique solution. We can write down the solution of 3.4.32 explicitly in terms of the random walk on
that jumps from i to j with rate a j − i. Let P
t
j − i denote the probability
that this random walk, starting in i at time 0, is in j at time t. Then the unique solution of 3.4.32 is given by see Lemma 3.3.1
X
i
t =
j
P
t
j − iX
j
0. 3.4.33
Let P
t t
≥0
be the semigroup on B associated with the random walk with kernel a see section 3.3.2. Let us denote by R
t t
≥0
the Feller semigroup on
C
K associated with the process in 3.4.32:
R
t
f x : = E[ f X
x
t] = f P
t
x. 3.4.34
Applying Lemma 3.4.5 to the case w = 0, we see that the generator of R
t t
≥0
is an extension of the operator
B f x : =
i,α j
a j − ix
j
− x
i ∂
∂ x
α i
f x 3.4.35
with domain
D
B : =
C
2 sum
K .
Evolution of harmonic functions: We now set out to prove Lemma 3.1.6. We start with the case f
∈
C
2
K ∩ H . Fix i ∈ , and let h ∈
C
2 fin
K be given by
hx : = f x
i
. 3.4.36
In the language above, we want to show that E[hX t]
= E[R
t
hX 0]. 3.4.37
It is not hard to see that R
t
h ∈
C
2 sum
K for each t
≥ 0, where by 3.4.34
∂ ∂
x
α k
R
t
hx = P
t
k − i
∂ ∂
x
α
f
j
P
t
j − ix
j ∂
2
∂ x
α k
∂ x
β l
R
t
hx = P
t
k − iP
t
l − i
∂
2
∂ x
α
∂ x
β
f
j
P
t
j − ix
j
. 3.4.38
General theory see [16], chapter 1 now tells us that t → R
t
h is continuously differentiable in
C
K and
∂ ∂
t
R
t
h = B R
t
h, 3.4.39
with B the operator in 3.4.35. By Lemma 3.4.5, X solves the martingale problem for the operator
A
′
: = B + C,
3.4.40 where
C f x : =
i,αβ
w
αβ
x
i ∂
2
∂ x
α i
∂ x
β i
f x 3.4.41
and
D
A
′
=
C
2 sum
K . It follows that
E[R hX T ]
− E[R
T
hX 0] = E
T
B + C +
∂ ∂
t
R
T −t
hX tdt = E
T
C R
T −t
hX tdt. 3.4.42
By 3.4.38 we have, for any x ∈ K
C R
T −t
hx =
j
P
T −t
j − i
2 αβ
w
αβ
x
j ∂
2
∂ x
α
∂ x
β
f
j
P
t
j − ix
j
. 3.4.43
Using Lemma 3.4.2 it is not hard to see that the stability of the boundary distribu- tion against a linear drift is equivalent to formula 3.1.46. The semigroup T
θ, t
t ≥0
maps differentiable functions into differentiable functions, and hence T
θ, t
C
2
K ∩ H ⊂
C
2
K ∩ H
∀θ ∈ K , t ≥ 0. 3.4.44
This means that GT
θ, t
f = 0 for all f ∈
C
2
K ∩ H and θ ∈ K , t ≥ 0, which says
that for any x ∈ K
αβ
w
αβ
x
∂
2
∂ x
α
∂ x
β
f θ + x − θe
−t
= e
−2t αβ
w
αβ
x
∂
2
∂ x
α
∂ x
β
f e
−t
x + 1 − e
−t
θ = 0
∀ f ∈
C
2
K ∩ H, θ ∈ K , t ≥ 0.
3.4.45
For the x here we insert the x
i
in 3.4.43 and we fit θ and t such that e
−t
= P
t
and 1 − e
−t
θ =
j =i
P
t
j − ix
j
. Inserting this into 3.4.43 we see that each term in the sum over j there is zero, and therefore 3.4.42 gives
E[ f X
i
T ] = E
f
j
P
T
j − iX
j
. 3.4.46
To generalize this to arbitrary f ∈ H it suffices to note that the set of functions
f ∈ BK for which 3.4.46 holds is bp-closed.
3.4.4 Proof of Theorem 3.1.5
Comparison argument: The function x
→ trwx is continuous, takes only non-negative values, and satisfies
tr wx = 0 ⇔ x ∈ ∂
w
K . 3.4.47
The same is true for the function x → v
∗
x see Lemma 3.4.3 and therefore for each ε 0 we can find a λ 0 such that
tr wx ≥ λv
∗
x − ε
x ∈ K .
3.4.48 When we insert this inequality into formula 3.1.34 in Lemma 3.1.4 we see that
for all i ∈ , t ≥ 0
∂ ∂
t
C
t
i ≥
j
a
S
j − iC
t
j − C
t
i + 2λδ
i 0
E[v
∗
X t]
− ε. 3.4.49
We apply Lemma 3.4.3 to see that the function x
→ v
∗
x + |x − θ|
2
3.4.50 is w-harmonic. Lemma 3.1.6 therefore tells us that for all t
≥ 0 E[v
∗
X t]
+ VarX t
= E v
∗ j
P
t
j X
j
+ Var
j
P
t
j X
j
. 3.4.51
By Lemma 3.3.3, Lemma 3.3.2 and the spatial ergodicity of
L
X 0 this implies that
lim
t →∞
E[v
∗
X t]
+ C
t
= v
∗
θ . 3.4.52
Combining this with 3.4.49 we see there exists a T such that for all t ≥ T
∂ ∂
t
C
t
i ≥
j
a
S
j − iC
t
j − C
t
i + 2λδ
i 0
v
∗
θ − C
t
− 2ε. 3.4.53
Random walk representation: Let us define
D
t
i : = v
∗
θ − C
t
i − 2ε
i ∈ , t ≥ 0.
3.4.54 Then 3.4.53 can be rewritten as
∂ ∂
t
D
t
i ≤
j
a
S
j − iD
t
j − D
t
i − 2λδ
i 0
D
t
t ≥ T .
3.4.55 We note that since t
→ C
t
is continuously differentiable in B, so is t → D
t
. Arguing as in the proof of Lemma 3.3.1, we can represent solutions of the differen-
tial inequality 3.4.55 in terms of a contracting semigroup P
λ t
t ≥0
on B, with generator
G f i : =
j
a
S
j − i f j − f i − 2λδ
i 0
f 0 f
∈ B. 3.4.56 This semigroup is related to a random walk on that jumps from i to j with rate
a
S
j − i and that is killed at the origin with rate 2λ. When P
λ t
j, i denotes the probability that this random walk, starting from a point i , is in j at time t, then
P
λ t
f i =
j
P
λ t
j, i f j f
∈ B, 3.4.57
and for solutions of 3.4.55 we have the representation D
T +t
i ≤
j
P
λ t
j, i D
T
j t
≥ 0. 3.4.58
