2.2 Generating Functions
Our main tool will be the usage of generating functions, which we introduce now. The Green func- tions related to P
i
and P are given by G
i
x
i
, y
i
|z := X
n≥0
p
n i
x
i
, y
i
z
n
and Gx, y|z :=
X
n≥0
p
n
x, y z
n
, where z ∈ C, x
i
, y
i
∈ V
i
and x, y ∈ V . At this point we make the basic assumption that the radius of convergence R of G·, ·|z is strictly bigger than 1. This implies transience of our random walk on
V . Thus, we may exclude the case r = 2 = |V
1
| = |V
2
|, because we get recurrence in this case. For instance, if all P
i
govern reversible Markov chains, then R 1; see [29, Theorem 10.3]. Furthermore,
it is easy to see that R 1 holds also if there is some i ∈ I such that p
n i
o
i
, o
i
= 0 for all n ∈ N. The first visit generating functions related to P
i
and P are given by F
i
x
i
, y
i
|z := X
n≥0
P Y
i n
= y
i
, ∀m ≤ n − 1 : Y
i m
6= y
i
| Y
i
= x
i
z
n
and F x, y|z :=
X
n≥0
P X
n
= y, ∀m ≤ n − 1 : X
m
6= y | X = x
z
n
, where Y
i n
n∈N
describes a random walk on V
i
governed by P
i
. The stopping time of the first return to o is defined as T
o
:= inf{m ≥ 1 | X
m
= o}. For i ∈ I , define H
i
z := X
n≥1
P [T
o
= n, X
1
∈ V
× i
] z
n
and ξ
i
z := α
i
z 1 − H
i
z .
We write also ξ
i
:= ξ
i
1, ξ
min
:= min
i∈I
ξ
i
and ξ
max
:= max
i∈I
ξ
i
. Observe that ξ
i
1; see [11, Lemma 2.3]. We have F x
i
, y
i
|z = F
i
x
i
, y
i
|ξ
i
z for all x
i
, y
i
∈ V
i
; see Woess [29, Prop. 9.18c]. Thus,
ξ
i
z := α
i
z 1 −
P
j∈I \{i}
P
s∈V
j
α
j
p
j
o
j
, sz F
j
s, o
j
ξ
j
z .
For x
i
∈ V
i
and x ∈ V , define the stopping times T
i x
i
:= inf{m ≥ 1 | Y
i m
= x
i
} and T
x
:= inf{m ≥ 1 | X
m
= x}, which take both values in N ∪ {∞}. Then the last visit generating functions related to P
i
and P are defined as L
i
x
i
, y
i
|z := X
n≥0
P Y
i n
= y
i
, T
i x
i
n | Y
i
= x
i
z
n
, Lx, y|z :=
X
n≥0
P X
n
= y, T
x
n | X = x
z
n
. If x = x
1
. . . x
n
, y = x
1
. . . x
n
x
n+1
∈ V with τx
n+1
= i then Lx, y|z = L
i
o
i
, x
n+1
ξ
i
z ;
2.3 this equation is proved completely analogously to [29, Prop. 9.18c]. If all paths from x ∈ V to
w ∈ V have to pass through y ∈ V , then Lx, w|z = Lx, y|z · L y, w|z;
80
this can be easily checked by conditioning on the last visit of y when walking from x to w. We have the following important equations, which follow by conditioning on the last visits of x
i
and x, the first visits of y
i
and y respectively: G
i
x
i
, y
i
|z = G
i
x
i
, x
i
|z · L
i
x
i
, y
i
|z = F
i
x
i
, y
i
|z · G
i
y
i
, y
i
|z, Gx, y|z =
Gx, x|z · Lx, y|z = F x, y|z · G y, y|z. 2.4
Observe that the generating functions F ·, ·|z and L·, ·|z have also radii of convergence strictl bigger than 1.
3 The Asymptotic Entropy
3.1 Rate of Escape w.r.t. specific Length Function
Kategori