Let us assume that for some i = 0 we have P
i
[τ ∞] = 1. Then by the
symmetry of the random walk, also P [τ
i
∞] = 1. But this implies that the random walk starting in 0 visits 0 infinitely often, which contradicts our assumption
that it is transient. It follows that P
i
[τ
j
∞] 1 for all i = j. Combining 3.3.43 and 3.3.41 we can conclude that
C
∞
i C
∞
∀i = 0. 3.3.44
Now Cauchy-Schwarz in 3.3.35 implies that P[X
i
∞ = X
j
∞] 1 for all i
= j.
3.4 Proofs of Theorem 3.1.5, Lemma 3.1.6
and Corollary 3.1.7
3.4.1 Potential theory
In this section we collect some elementary facts about w-harmonic functions from potential theory. We assume that for each x
∈ K , the non-interacting equation 3.1.19 has a unique weak solution X
x
with initial condition X
x
= x. We denote its last element by X
x
∞. We denote the semigroup on BK associated with 3.1.19 by S
t t
≥0
and we add a last element S
∞
as in 3.1.41. Restricted to the smaller domain
C
K ⊂ BK , the S
t t
≥0
form a Feller semigroup, whose generator we denote by G. Note that
D
G ⊂
C
K .
Lemma 3.4.1 For each solution X to 3.1.19
P[X ∞ ∈ ∂
w
K ] = 1.
3.4.1
Proof of Lemma 3.4.1: Since X is a bounded martingale, it converges. Now the lemma is just a special case of Theorem 3.1.3.
Lemma 3.4.2 Assume that S
∞
C
K ⊂
C
K . Consider sets H, H
′
, H
′′
, H
′′′
defined as H :
= { f ∈
D
G : G f = 0}
H
′
: = { f ∈
C
K : S
t
f = f ∀t ∈ [0, ∞]}
H
′′
: = { f ∈
C
K : S
∞
f = f }
H
′′′
: = {S
∞
φ : φ
∈
C
K }
3.4.2
Then H = H
′
= H
′′
= H
′′′
. For each φ ∈
C
K there exists a unique f ∈ H such
that f x
= φx x
∈ ∂
w
K 3.4.3
and this f is given by f
= S
∞
φ. 3.4.4
Proof of Lemma 3.4.2: It is easy to see that H ⊂ H
′
⊂ H
′′
⊂ H
′′′
. To see that H
′′′
⊂ H , note that φX t → φX ∞ almost surely, so bounded con- vergence implies that E
x
[φX t] → E
x
[φX ∞] for each x ∈ K . Since
|E
x
[φX t] | ≤ φ
∞
∞, it follows that S
t
φ → S
∞
φ as t
→ ∞ in the sense of bounded pointwise convergence. Therefore
S
t
S
∞
φ x
= lim
s →∞
S
t
S
s
φ x
= S
∞
φ x x
∈ K . 3.4.5
It follows that t
−1
S
t
− 1S
∞
φ = 0 for all t, so
lim
t →0
t
−1
S
t
− 1S
∞
φ = 0
3.4.6 in the topology on
C
K and this proves that H
′′′
⊂ H . By 3.4.2, S
∞
φ ∈ H for each φ ∈
C
K . To see that f : = S
∞
φ solves 3.4.3
it suffices to note that for each x ∈ ∂
w
K the process X t :
= x 3.4.7
solves 3.1.19. To see that f is the unique w-harmonic function satisfying 3.4.3, suppose that ˜
f ∈ H is another one. Then by 3.4.2 and by Lemma 3.4.1
˜f = S
∞
˜f = S
∞
φ = f.
3.4.8
In the proof of Theorem 3.1.5 we will make use of the function v
∗
, given by v
∗
x = trw
∗
, 3.4.9
where w
∗
is the special diffusion matrix mentioned in 3.1.52. The following lemma collects some elementary facts about v
∗
.
Lemma 3.4.3 Assume that S
∞
C
K ⊂
C
K . Then there exists a unique func- tion v
∗
∈
D
G such that −
1 2
Gv
∗
x = trwx
x ∈ K
v
∗
x = 0
x ∈ ∂
w
K . 3.4.10
This function v
∗
satisfies v
∗
x ≥ 0
x ∈ K
v
∗
x = 0 ⇔ x ∈ ∂
w
K , 3.4.11
and is given by the formula v
∗
x = VarX
x
∞. 3.4.12
Proof of Lemma 3.4.3: We write x
2
for the function x → |x|
2
. Then G x
2
= 2trw. 3.4.13
Thus, 3.4.10 can be rewritten as Gv
∗
+ x
2
= 0 on K
v
∗
+ x
2
= x
2
on ∂
w
K . 3.4.14
Lemma 3.4.2 shows that v
∗
+ x
2
can be uniquely solved from these equations. Using the fact that X
x
solves the martingale problem for G we see with the help of 3.4.10 and 3.4.13 that
E X
x
∞ − x
2
= 2
∞
E[tr wX t]dt = −E[v
∗
X
x
∞] + v
∗
x. 3.4.15
Lemma 3.4.1 implies that E[v
∗
X
x
∞] = 0 and so we see that 3.4.12 holds. Formula 3.4.12 immediately implies that v
∗
x ≥ 0. Finally v
∗
x = 0 implies
VarX
x
∞ = 0 so that X
x
∞ = x, and by Lemma 3.4.1 this in turn implies that x
∈ ∂
w
K .
3.4.2 Infinite-dimensional differentiation
We will need to extend the domain of the operator A in 3.1.10 to include functions depending on infinitely many coordinates. In order to do this properly, we intro-
duce the space
C
2 sum
K of functions with summable continuous second deriva-
tives. For a function f :
R
d
→
R
we define
∂ ∂
x
α i
f x in the usual way.
C
2
R
d
is the class of functions for which all zeroth, first and second order derivatives are continuous functions on
R
d
.
C
2
K is the set of funtions on
K that can be extended to functions in
C
2
R
d
.
C
2 sum
K , finally, is the space of
functions in
C
2
K for which
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 ∈
3.4.16
are continuous maps from K into l
1
{1, . . . , d} × and l
1
{1, . . . , d}
2
×
2
, respectively, spaces of absolutely summable sequences, equipped with the l
1
-norm.
Lemma 3.4.4 For i, j
∈ and α, β = 1, . . . , d, let b
i,α
and a
i j,αβ
be functions in
C
K satisfying uniform bounds
b
i,α ∞
≤ M
1
∀i ∈ , α = 1, . . . , d a
i j,αβ ∞
≤ M
2
∀i, j ∈ , α, β = 1, . . . , d. 3.4.17
Then for each f ∈
C
2 sum
K and for all finite
n
⊂ with
n
↑ , the limit lim
n →∞
i ∈
n
, α
b
i,α
x
∂ ∂
x
α i
+
i j ∈
n
, αβ
a
i j,αβ
x
∂
2
∂ x
α i
∂ x
β j
f x 3.4.18
exists in the topology on
C
K and does not depend on the choice of the
n
.
Proof of Lemma 3.4.4: We treat only the convergence of the first order derivatives; the argument for second order derivatives is then the same. Define operators A
n
by A
n
f x : =
i ∈
n
,α
b
i,α
x
∂ ∂
x
α i
f x. 3.4.19
For each n ≤ m and for each f ∈
C
2 sum
K we have
A
n
f − A
m
f
∞
= sup
x ∈K
i ∈
m
\
n
, α
b
i,α
x
∂ ∂
x
α i
f x ≤ M
1
sup
x ∈K
i ∈\
n
, α ∂
∂ x
α i
f x .
3.4.20
The functions g
n
: K → l
1
{1, . . . , d} × given by g
n
x : =
i ∈\
n
, α ∂
∂ x
α i
f x 3.4.21
are continuous functions decreasing to zero as n → ∞, and hence by Dini’s the-
orem g
n
→ 0 uniformly. Thus we see by 3.4.20 that the sequence A
n
f is a Cauchy sequence in the Banach space
C
K .
To see that the limit does not depend on the summation order, observe that for each x
∈ K
i,α
b
i,α
x
∂ ∂
x
α i
f x ≤
M
1 i,α
∂ ∂
x
α i
f x ∞.
3.4.22 This means that we can write
lim
n →∞
A
n
f x =
i,α
b
i,α
x
∂ ∂
x
α i
f x, 3.4.23
