Topics in Occupation Times and Gaussian Free Fields doc

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Alain-Sol Sznitman

Topics in
Occupation Times and
Gaussian Free Fields
Author:
Alain-Sol Sznitman
Departement Mathematik
ETH Zürich
Rämistrasse 101
8092 Zürich
Switzerland
2010 Mathematics Subject Classification: 60K35, 60J27, 60G15, 82B41
Key words: occupation times, Gaussian free field, Markovian loop, random interlacements
ISBN 978-3-03719-109-5
The Swiss National Library lists this publication in The Swiss Book, the Swiss national bibliography,
and the detailed bibliographic data are available on the Internet at .
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©
2012 European Mathematical Society
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9 8 7 6 5 4 3 2 1
Preface
The following notes grew out of the graduate course “Special topics in probability”,
which I gave at ETH Zurich during the Spring term 2011. One of the objectives was
to explore the links between occupation times, Gaussian free fields, Poisson gases
of Markovian loops, and random interlacements. The stimulating atmosphere during
the live lectures was an encouragement to write a fleshed-out version of the hand-
written notes, which were handed out during the course. I am immensely grateful to
Pierre-François Rodriguez, Artëm Sapozhnikov, Balázs Ráth, Alexander Drewitz,
and David Belius, for their numerous comments on the successive versions of these
notes. Support by the European Research Council through grant ERC-2009-AdG
245728-RWPERCRI is thankfully acknowledged.

Contents
Preface v
Introduction 1
1 Generalities 5
1.1 The set-up 5
1.2 The Markov chain X
:
(with jump rate 1) 7
1.3 Some potential theory 10
1.4 Feynman–Kac formula 23
1.5 Local times 25
1.6 The Markov chain

x
X
:
(with variable jump rate) 26
2 Isomorphism theorems 31
2.1 The Gaussian free field 31
2.2 The measures P
x;y
35
2.3 Isomorphism theorems 41
2.4 Generalized Ray–Knight theorems 45
3 The Markovian loop 61
3.1 Rooted loops and the measure 
r
on rooted loops 61
3.2 Pointed loops and the measure 
p
on pointed loops 70
3.3 Restriction property 74
3.4 Local times 75
3.5 Unrooted loops and the measure 

on unrooted loops 82
4 Poisson gas of Markovian loops 85
4.1 Poisson point measures on unrooted loops 85
4.2 Occupation field 87
4.3 Symanzik’s representation formula 91
4.4 Some identities 95
4.5 Some links between Markovian loops and random interlacements . . 100
References 111

Index 113

Introduction
This set of notes explores some of the links between occupation times and Gaussian
processes. Notably they bring into play certain isomorphism theorems going back to
Dynkin [4], [5] as well as certain Poisson point processes of Markovian loops, which
originated in physics through the work of Symanzik [26]. More recently such Poisson
gases of Markovian loops have reappeared in the context of the “Brownian loop soup”
of Lawler and Werner [16] and are related to the so-called “random interlacements”,
see Sznitman [27]. In particular they have been extensively investigated by Le Jan
[17], [18].
A convenient set-up to develop this circle of ideas consists in the consideration
of a finite connected graph E endowed with positive weights and a non-degenerate
killing measure. One can then associate to these data a continuous-time Markov chain
x
X
t
, t  0,onE, with variable jump rates, which dies after a finite time due to the
killing measure, as well as
the Green density g.x; y/, x; y 2 E, (0.1)
(which is positive and symmetric),
the local times
x
L
x
t
D
Z
t
0

1f
x
X
s
D xgds; t  0, x 2 E. (0.2)
In fact g.; / is a positive definite function on E  E, and one can define a centered
Gaussian process '
x
, x 2 E, such that
cov.'
x
;'
y
/.D EŒ'
x
'
y
/ D g.x; y/; for x; y 2 E. (0.3)
This is the so-called Gaussian free field.
It turns out that
1
2
'
2
z
, z 2 E, and
x
L
z
1

, z 2 E, have intricate relationships. For
instance Dynkin’s isomorphism theorem states in our context that for any x;y 2 E,

x
L
z
1
C
1
2
'
2
z

z2E
under P
x;y
˝ P
G
(0.4)
has the “same law” as
1
2
.'
2
z
/
z2E
under '
x

'
y
P
G
, (0.5)
where P
x;y
stands for the (non-normalized) h-transform of our basic Markov chain,
with the choice h./ D g.;y/, starting from the point x, and P
G
for the law of the
Gaussian field '
z
, z 2 E.
2 Introduction
Eisenbaum’s isomorphism theorem, which appeared in [7], does not involve h-
transforms and states in our context that for any x 2 E, s 6D 0,

x
L
z
1
C
1
2
.'
z
C s/
2


z2E
under P
x
˝ P
G
(0.6)
has the “same law” as

1
2
.'
z
C s/
2
Á
z2E
under

1 C
'
x
s
Á
P
G
. (0.7)
The above isomorphism theorems are also closely linked to the topic of theorems of
Ray–Knight type, see Eisenbaum [6], and Chapters 2 and 8 of Marcus–Rosen [19].
Originally, see [13], [21], such theorems came as a description of the Markovian
character in the space variable of Brownian local times evaluated at certain random

times. More recently, the Gaussian aspects and the relation with the isomorphism
theorems have gained prominence, see [8], and [19].
Interestingly, Dynkin’s isomorphism theorem has its roots in mathematical physics.
It grew out of the investigation by Dynkin in [4] of a probabilistic representation for-
mula for the moments of certain random fields in terms of a Poissonian gas of loops
interacting with Markovian paths, which appeared in Brydges–Fröhlich–Spencer [2],
and was based on the work of Symanzik [26].
The Poisson point gas of loops in question is a Poisson point process on the state
space of loops on E modulo time-shift. Its intensity measure is a multiple ˛

of the
image 

of a certain measure 
rooted
, under the canonical map for the equivalence
relation identifying rooted loops  that only differ by a time-shift. This measure

rooted
is the -finite measure on rooted loops defined by

rooted
.d/ D
P
x2E
Z
1
0
Q
t

x;x
.d/
dt
t
; (0.8)
where Q
t
x;x
is the image of 1fX
t
D xgP
x
under .X
s
/
0ÄsÄt
,ifX
:
stands for the
Markov chain on E with jump rates equal to 1 attached to the weights and killing
measure we have chosen on E.
The random fields on E alluded to above, are motivated by models of Euclidean
quantum field theory, see [11], and are for instance of the following kind:
hF.'/iD
Z
R
E
F.'/e

1

2
E.';'/
Q
x2E
h

'
2
x
2
Á
d'
x
.
Z
R
E
e

1
2
E.';'/
Q
x2E
h

'
2
x
2

Á
d'
x
(0.9)
with
h.u/ D
Z
1
0
e
vu
d.v/, u  0, with  a probability distribution on R
C
;
Introduction 3
and E.'; '/ the energy of the function ' corresponding to the weights and killing
measure on E (the matrix E.1
x
;1
y
/, x;y 2 E is the inverse of the matrix g.x; y/,
x; y 2 E in (0.3)).
x
2
x
3
y
1
w
2

w
3
w
1
y
2
x
1
y
3
Figure 0.1. The paths w
1
;:::;w
k
in E interact with the gas of loops through the random
potentials.
The typical representation formula for the moments of the random field in (0.9) looks
like this: for k  1, z
1
;:::;z
2k
2 E,
h'
z
1
:::'
z
2k
iD
P

pairings
of z
1
;:::;z
2k
P
x
1
;y
1
˝˝P
x
k
;y
k
˝ Q

e

P
x2E
v
x
.L
x
C
x
L
x
1

.w
1
/CC
x
L
x
1
.w
k
//

Q

e

P
x2E
v
x
L
x

;
(0.10)
where the sum runs over the (non-ordered) pairings (i.e. partitions) of the symbols
z
1
;z
2
;:::;z

2k
into fx
1
;y
1
g;:::;fx
k
;y
k
g. Under Q the v
x
;x 2 E, are i.i.d. -
distributed (random potentials), independent of the L
x
, x 2 E, which are distributed
as the total occupation times (properly scaled to take account of the weights and
killing measure) of the gas of loops with intensity
1
2
, and the P
x
i
;y
i
, 1 Ä i Ä k are
defined just as below (0.4), (0.5).
The Poisson point process of Markovian loops has many interesting properties.
We will for instance see that when ˛ D
1
2

(i.e. the intensity measure equals
1
2
),
.L
x
/
x2E
has the same distribution as
1
2
.'
2
x
/
x2E
, where
.'
x
/
x2E
stands for the Gaussian free field in (0.3).
(0.11)
The Poisson gas of Markovian loops is also related to the model of random interlace-
ments [27], which loosely speaking corresponds to “loops going through infinity”. It
4 Introduction
appears as well in the recent developments concerning conformally invariant scaling
limits, see Lawler–Werner [16], Sheffield–Werner [24]. As for random interlace-
ments, interestingly, in place of (0.11), they satisfy an isomorphism theorem in the
spirit of the generalized second Ray–Knight theorem, see [28].

1 Generalities
In this chapter we describe the general framework we will use for the most part of
these notes. We introduce finite weighted graphs with killing and the associated
continuous-type Markov chains X
:
, with constant jump rate equal to 1, and
x
X
:
, with
variable jump rate. We also recall various notions related to Dirichlet forms and
potential theory.
1.1 The set-up
We introduce in this section the general set-up, which we will use in the sequel,
and recall some classical facts. We also refer to [14] and [10], where the theory is
developed in a more general framework. We assume that
E is a finite non-empty set (1.1)
endowed with non-negative weights
c
x;y
D c
y;x
 0, for x; y 2 E, and c
x;x
D 0, for x 2 E, (1.2)
so that
E, endowed with the edge set consisting of the pairs fx;yg such that
c
x;y
>0, is a connected graph.

(1.3)
We also suppose that there is a killing measure on E,
Ä
x
 0; x 2 E; (1.4)
and that
Ä
x
6D 0, for at least some x 2 E. (1.5)
We also consider a
cemetery state  not in E (1.6)
(we can think of Ä
x
as c
x;
).
With these data we can define a measure on E:

x
D
P
y2E
c
x;y
C Ä
x
;x2 E (note that 
x
>0, due to (1.2)–(1.5)): (1.7)
6 1 Generalities

We can also introduce the energy of a function on E,orDirichlet form
E.f; f / D
1
2
P
x;y2E
c
x;y
.f .y/  f.x//
2
C
P
x2E
Ä
x
f
2
.x/; (1.8)
for f W E ! R.
Note that .c
x;y
/
x;y2E
and .Ä
x
/
x2E
determine the Dirichlet form. Conversely,
the Dirichlet form determines .c
x;y

/
x;y2E
and .Ä
x
/
x2E
. Indeed, one defines, by
polarization, for f; g W E ! R,
E.f; g/ D
1
4
ŒE.f Cg; f C g/  E.f g; f g/
D
1
2
P
x;y2E
c
x;y
.f .y/  f .x//.g.y/ g.x// C
P
x2E
Ä
x
f.x/g.x/;
(1.9)
and one notes that
E.1
x
;1

y
/ Dc
x;y
; for x 6D y in E;
E.1
x
;1
x
/ D
P
y2E
c
x;y
C Ä
x
D 
x
; for x 2 E;
(1.10)
so that the Dirichlet form uniquely determines the weights .c
x;y
/
x;y2E
and the killing
measure .Ä
x
/
x2E
. Observe also that by (1.3), (1.5), (1.8), (1.9), the Dirichlet form
defines a positive definite quadratic form on the space F of functions from E to R,

see also (1.39) below.
We denote by .; /

the scalar product in L
2
.d/:
.f; g/

D
P
x2E
f.x/g.x/
x
; for f; g W E ! R: (1.11)
The weights and the killing measure induce a sub-Markovian transition probability
on E,
p
x;y
D
c
x;y

x
; for x; y 2 E; (1.12)
which is -reversible:

x
p
x;y
D 

y
p
y;x
; for all x; y 2 E: (1.13)
One then extends p
x;y
;x;y 2 E to a transition probability on E [fg by setting
p
x;
D
Ä
x

x
; for x 2 E, and p
;
D 1; (1.14)
so the corresponding discrete-time Markov chain on E [fg is absorbed in the
cemetery state  once it reaches . We denote by
Z
n
;n 0, the canonical discrete Markov chain on the space of
discrete trajectories in E [fg, which after finitely many steps
reaches  and from then on remains at ,
(1.15)
1.2 The Markov chain X
:
(with jump rate 1) 7
and by
P

x
the law of the chain starting from x 2 E [fg. (1.16)
We will attach to the Dirichlet form (1.8) (or, equivalently, to the weights and the
killing measure), two continuous-time Markov chains on E [fg, which are time
change of each other, with discrete skeleton corresponding to Z
n
, n  0. The
first chain X
:
will have a unit jump rate, whereas the second chain
x
X
:
(defined in
Section 1.6) will have a variable jump rate governed by .
1.2 The Markov chain X
:
(with jump rate 1)
We introduce in this section the continuous-time Markov chain on E [fg(absorbed
in the cemetery state ), with discrete skeleton described by Z
n
, n  0, and expo-
nential holding times of parameter 1. We also bring into play some of the natural
objects attached to this Markov chains.
The canonical space D
E
for this Markov chain consists of right-continuous func-
tions with values in E [fg, with finitely many jumps, which after some time enter
 and from then on remain equal to . We denote by
X

t
;t  0, the canonical process on D
E
;
Â
t
;t  0, the canonical shift on D
E
: Â
t
.w/./ D w.Ct/, for w 2 D
E
;
P
x
the law on D
E
of the Markov chain starting at x 2 E [fg:
(1.17)
Remark 1.1. Whenever convenient we will tacitly enlarge the canonical space D
E
and work with a probability space on which (under P
x
) we can simultaneously con-
sider the discrete Markov chain Z
n
, n  0, with starting point a.s. equal to x, and an
independent sequence of positive variables T
n
, n  1, the “jump times”, increasing

to infinity, with increments T
nC1
 T
n
, n  0, i.i.d. exponential with parameter 1
(with the convention T
0
D 0). The continuous-time chain X
t
, t  0, will then be
expressed as
X
t
D Z
n
; for T
n
Ä t<T
nC1
, n  0:
Of course, once the discrete-time chain reaches the cemetery state , the subsequent
“jump times” T
n
are only fictitious “jumps” of the continuous time chain. 
Examples. 1) Simple random walk on the discrete torus killed at a constant rate
E D .Z=N Z/
d
, where N>1, d  1,
8 1 Generalities
endowed with the graph structure, where x, y are neighbors if exactly one of their

coordinates differs by ˙1, and the other coordinates are equal. We pick
c
x;y
D 1
fx;y are neighborsg
;x;y2 E;
Ä
x
D Ä>0:
So X
t
, t  0, is the simple random walk on .Z=N Z/
d
with exponential holding
times of parameter 1, killed at each step with probability
Ä
2dCÄ
, when N>2, and
probability
Ä
d CÄ
, when N D 2.
2) Simple random walk on Z
d
killed outside a finite connected subset of Z
d
, that
is:
E is a finite connected subset of Z
d

, d  1.
c
x;y
D 1
fjxyjD1g
; for x; y 2 E;
Ä
x
D
P
y2Z
d
nE
1
fjxyjD1g
; for x 2 E.
x
0
E
x
Ä
x
0
D 0
Ä
x
D 2
Figure 1.1
Then X
t

, t  0, when starting in x 2 E, corresponds to the simple random
walk in Z
d
with exponential holding times of parameter 1 killed at the first time it
exits E. 
Our next step is to introduce some natural objects attached to the Markov chain
X
:
, such as the transition semi-group, and the Green function.
1.2 The Markov chain X
:
(with jump rate 1) 9
Transition semi-group and transition density
Unless otherwise specified, we will tacitly view real-valued functions on E, as func-
tions on E [fg, which vanish at the point .
The sub-Markovian transition semi-group of the chain X
t
, t  0,onE is defined
for t  0, f W E ! R and x 2 E,by
R
t
f.x/D E
x
Œf .X
t
/
D
P
n0
e

t
t
n
nŠ
E
x
Œf .Z
n
/
D
P
n0
e
t
t
n
nŠ
P
n
f.x/D e
t.PI/
f.x/;
(1.18)
where I denotes the identity map on R
E
, and for f W E ! R, x 2 E,
Pf .x/ D
P
y2E
p

x;y
f.y/
(1.15)
D E
x
Œf .Z
1
/: (1.19)
As a result of (1.13) and (1.18),
P and R
t
(for any t  0) are bounded self-adjoint operators on L
2
.d/; (1.20)
R
tCs
D R
t
R
s
, for t;s  0 (semi-group property): (1.21)
We then introduce the transition density
r
t
.x; y/ D .R
t
1
y
/.x/
1


y
; for t  0, x;y 2 E: (1.22)
It follows from the self-adjointness of R
t
, cf. (1.20), that
r
t
.x; y/ D r
t
.y; x/; for t  0, x;y 2 E (symmetry) (1.23)
and from the semi-group property, cf. (1.21), that for t;s  0, x;y 2 E,
r
tCs
.x; y/ D
P
z2E
r
t
.x;z/r
s
.z;y/
z
(Chapman–Kolmogorov equations). (1.24)
Moreover due to (1.3), (1.12), (1.18), we see that
r
t
.x;y/>0; for t>0, x;y 2 E: (1.25)
Green function
We define the Green function (or Green density):

g.x; y/ D
Z
1
0
r
t
.x;y/dt
(1.18), (1.22)
D
Fubini
E
x
h
Z
1
0
1fX
t
D ygdt
i
1

y
; for x; y 2 E:
(1.26)
10 1 Generalities
Lemma 1.2.
g.x; y/ 2 .0; 1/ is a symmetric function on E  E: (1.27)
Proof. By (1.23), (1.25) we see that g.; / is positive and symmetric. We now prove
that it is finite. By (1.1), (1.3), (1.5) we see that for some N  0, and ">0,

inf
x2E
P
x
ŒZ
n
D , for some n Ä N ">0: (1.28)
As a result of the simple Markov property at times, which are multiple of N , we find
that
.P
kN
1
E
/.x/ D P
x
ŒZ
n
6D ; for 0 Ä n Ä kN 
simple Markov
Ä
(1.28)
.1  "/
k
; for k  1:
It follows by a straightforward interpolation that with suitable c;c
0
>0,
sup
x2E
.P

n
1
E
/.x/ Ä ce
c
0
n
; for n  0: (1.29)
As a result inserting this bound in the last line of (1.18) gives:
sup
x2E
.R
t
1
E
/.x/ Ä ce
t
P
n0
t
n
nŠ
e
c
0
n
D c expft.1e
c
0
/g; (1.30)

so that
g.x; y/ Ä
1

y
Z
1
0
.R
t
1
E
/.x/ dt Ä
c

y
1
1  e
c
0
Ä c
00
< 1; (1.31)
whence (1.27). 
1.3 Some potential theory
In this section we introduce some natural objects from potential theory such as the
equilibrium measure, the equilibrium potential, and the capacity of a subset of E.We
also provide two variational characterizations for the capacity. We then describe the
orthogonal complement under the Dirichlet form of the space of functions vanishing
on a subset of K. This also naturally leads us to the notion of trace form (and network

reduction).
1.3 Some potential theory 11
The Green function gives rise to the potential operators
Qf .x/ D
P
y2E
g.x; y/f.y/
y
; for f W E ! R (a function); (1.32)
the potential of the function f , and
G.x/ D
P
y2E
g.x; y/
y
, for  W E ! R (a measure); (1.33)
the potential of the measure . We also write the duality bracket (between functions
and measures on E):
h; f iD
P
x

x
f.x/; for f W E ! R,  W E ! R: (1.34)
In the next proposition we collect several useful properties of the Green function and
the Dirichlet form.
Proposition 1.3.
E.;/
def
Dh; GiD

P
x;y2E

x
g.x; y/
y
, for ; W E ! R (1.35)
defines a positive definite, symmetric bilinear form.
Q D .I P/
1
(see (1.19), (1.32) for notation). (1.36)
G D .L/
1
; (1.37)
where
Lf .x/ D
P
y2E
c
x;y
f.y/ 
x
f.x/; for f W E ! R:
E.G; f / Dh; f i, for  W E ! R and f W E ! R: (1.38)
9>0, such that E.f; f /  kf k
2
L
2
.d/
, for all f W E ! R: (1.39)

GÄ D 1 (the killing measure Ä is also called an equilibrium measure of E): (1.40)
Proof.
 (1.35):
One can give a direct proof based on (1.23)–(1.26), but we will instead derive (1.35)
with the help of (1.37)–(1.39). The bilinear form in (1.35) is symmetric by (1.27).
Moreover, for  W E ! R,
0 Ä E.G; G/
(1.38)
Dh; GiDE.;/ (the energy of the measure ):
12 1 Generalities
By (1.39), 0 D E.G; G/ H) G D 0, and by (1.37) it follows that  D
.L/ G D 0. This proves (1.35) (assuming (1.37)–(1.39)).
 (1.36):
By (1.29):
Z
1
0
P
n0
e
t
t
n
nŠ
jP
n
f.x/jdt Ä c
Z
1
0

e
t
P
n0
t
n
nŠ
e
c
0
n
dt kf k
1
(1.30)
D ckf k
1
Z
1
0
e
t.1e
c
0
/
dt < 1:
By Lebesgue’s domination theorem, keeping in mind (1.18), (1.26),
Qf .x/
(1.32)
D
Z

1
0
R
t
f.x/dt
(1.18)
D
Z
1
0
P
n0
e
t
t
n
nŠ
P
n
f.x/dt
D
P
n0
Z
1
0
e
t
t
n

nŠ
dt P
n
f.x/D
P
n0
P
n
f.x/
(1.29)
D .I P/
1
f.x/; (1 is not in the spectrum of P by (1.29)).
This proves (1.36).
 (1.37):
Note that in view of (1.19),
L D .I  P/ (composition of .I  P/and the multiplication by 
:
,
i.e. .f /.x/ D 
x
f.x/for f W E ! R, and x 2 E):
(1.41)
Hence L is invertible and
.L/
1
D .I P/
1

1

(1.36)
D Q
1
(1.32)
D
(1.33)
G:
This proves (1.37).
 (1.38):
By (1.10) we find that
E.f; g/ D
P
x;y2E
f.x/g.y/E.1
x
;1
y
/
(1.10)
D
P
x2E

x
f.x/ g.x/ 
P
x;y2E
c
x;y
f.x/g.y/

Dhf; Lgi
(1.2)
DhLf; g i:
(1.42)
1.3 Some potential theory 13
As a result
E.G; f / DhLG; f i
(1.37)
Dh; f i; whence (1.38):
 (1.39):
Note that for x 2 E, f W E ! R
f.x/Dh1
x
;fi
(1.38)
D E.G1
x
;f/:
Now E.; / is a non-negative symmetric bilinear form. We can thus apply Cauchy–
Schwarz’s inequality to find that
f.x/
2
Ä E.G1
x
;G1
x
/ E.f; f /
(1.38)
Dh1
x

;G1
x
iE.f; f /
D g.x; x/E.f; f /:
As a result we find that
kf k
2
L
2
.d/
D
P
x2E
f.x/
2

x
Ä
P
x2E
g.x; x/
x
E.f; f /; (1.43)
and (1.39) follows with

1
D
P
x2E
g.x; x/

x
:
 (1.40):
By (1.39), E.; / is positive definite and by (1.9)
E.1; f /
(1.9)
D
P
x
Ä
x
f.x/DhÄ; f i
(1.38)
D E.GÄ; f /; for all f W E ! R:
It thus follows that 1 D GÄ, whence (1.40). 
Remark 1.4. Note that we have shown in (1.42) that for all f; g W E ! R,
E.f; g/ DhLf ; g iDhf; Lgi: (1.44)
Since L D .I  P/, we also find, see (1.11) for notation,
E.f; g/ D I  P/f;g/

D .f; .I  P/g/

: (1.44
0
)

14 1 Generalities
As a next step we introduce some important random times for the continuous-time
Markov chain X
t

, t  0.GivenK Â E, we define
H
K
Dinfft  0IX
t
2 Kg; the entrance time in K;
z
H
K
Dinfft>0I X
t
2 K and there exists s 2 .0; t/ with X
s
6D X
0
g;
the hitting time of K;
T
K
Dinfft  0I X
t
… Kg; the exit time from K;
L
K
Dsupft>0I X
t
2 Kg; the time of last visit to K
(with the convention sup  D 0, inf  D1):
(1.45)
H

K
,
z
H
K
, T
K
are stopping times for the canonical filtration .F
t
/
t0
,onD
E
(i.e. a
Œ0; 1-valued map T on D
E
, see (1.17) above, such that fT Ä t g2F
t
def
D .X
s
;0Ä
s Ä t/, for each t  0). Of course L
K
is in general not a stopping time.
Given U Â E, the transition density killed outside U is
r
t;U
.x; y/ D P
x

ŒX
t
D y;t < T
U

1

y
Ä r
t
.x; y/; for t  0, x;y 2 E, (1.46)
and the Green function killed outside U is
g
U
.x; y/ D
Z
1
0
r
t;U
.x; y/dt Ä g.x; y/; for x;y 2 E: (1.47)
Remark 1.5. 1) When U is a connected (non-empty) subgraph of the graph in (1.3),
r
t;U
.x; y/, t  0, x; y 2 U , and g
U
.x; y/, x;y 2 U , simply correspond to the
transition density and the Green function in (1.22), (1.26), when one chooses on U
– the weights c
x;y

, x; y 2 U (i.e. restriction to U  U of the weights on E),
– the killing measure QÄ
x
D Ä
x
C
P
y2EnU
c
x;y
, x 2 U .
2) When U is not connected the above remark applies to each connected compo-
nent of U , and r
t;U
.x; y/ and g
U
.x; y/ vanish when x;y belong to different connected
components of U . 
Proposition 1.6 (U Â E;A D EnU ).
g
U
.x; y/ D g
U
.y; x/; for x;y 2 E: (1.48)
g.x; y/ D g
U
.x; y/ C E
x
ŒH
A

< 1;g.X
H
A
; y/; for x; y 2 E: (1.49)
E
x
ŒH
A
< 1;g.X
H
A
;y/D E
y
ŒH
A
< 1;g.X
H
A
; x/; for x;y 2 E
(Hunt’s switching identity).
(1.50)
1.3 Some potential theory 15
Proof.
 (1.48):
This is a direct consequence of the above remark and (1.27).
 (1.49):
g.x; y/
(1.26)
D E
x

h
Z
1
0
1fX
t
D ygdt
i
1

y
D E
x
h
Z
1
0
1fX
t
D y;t < T
U
gdt
i
1

y
C E
x
h
Z

1
T
U
1fX
t
D ygdt; T
U
< 1
i
1

y
Fubini
D
(1.46), (1.47)
g
U
.x; y/ C E
x
h
T
U
< 1;

Z
1
0
1fX
t
D ygdt

Á
B Â
T
U
i
1

y
strong Markov
D g
U
.x; y/ C E
x
h
T
U
< 1;E
X
T
U
h
Z
1
0
1fX
t
D ygdt
ii
1


y
T
U
D H
A
D
(1.26)
g
U
.x; y/ C E
x
ŒH
A
< 1;g.X
H
A
; y/:
This proves (1.49).
 (1.50):
This follows from (1.48), (1.49) and the fact that g.; / is symmetric, cf. (1.27). 
Example. Consider x
0
2 E.By(1.49) we find that for x 2 E (with A Dfx
0
g,
U D Enfx
0
g/
g.x; x
0

/ D 0 C P
x
ŒH
x
0
< 1g.x
0
;x
0
/;
writing H
x
0
for H
fx
0
g
, so that
P
x
ŒH
x
0
< 1 D
g.x; x
0
/
g.x
0
;x

0
/
; for x 2 E: (1.51)
A second application of (1.49) now yields (with U D Enfx
0
g)
g
U
.x; y/ D g.x; y/ 
g.x; x
0
/g.x
0
;y/
g.x
0
;x
0
/
; for x; y 2 E: (1.52)

16 1 Generalities
Given A Â E, we introduce the equilibrium measure of A:
e
A
.x/ D P
x
Œ
z
H

A
D11
A
.x/ 
x
;x2 E: (1.53)
Its total mass is called the capacity of A (or the conductance of A):
cap.A/ D
P
x2A
P
x
Œ
z
H
A
D1
x
: (1.54)
Remark 1.7. As we will see below in the case of A D E the terminology in (1.53)
is consistent with the terminology in (1.40). There is an interpretation of the weights
.c
x;y
/ and the killing measures .Ä
x
/ on E as an electric network grounding E at the
cemetery point , which is implicit in the use of the above terms, see for instance
Doyle–Snell [3]. 
Before turning to the next proposition, we simply recall that given A Â E, by our
convention in (1.45)

fH
A
< 1gDfL
A
>0gDthe set of trajectories that enter A:
Also given a measure  on E, we write
P

D
P
x2E

x
P
x
and E

for the P

-integral (or “expectation”): (1.55)
Proposition 1.8 (A Â E).
P
x
ŒL
A
>0;X
L

A
D y D g.x; y/e

A
.y/; for x; y 2 E; (1.56)
.X
L

A
is the position of X
:
at the last visit to A, when L
A
>0):
h
A
.x/
def
D P
x
ŒH
A
< 1 D P
x
ŒL
A
>0D Ge
A
.x/; for x 2 E (1.57)
(the equilibrium potential of A).
When A 6D ,
e
A

is the unique measure  supported on A such that G D 1 on A: (1.58)
Let A Â B Â E then under P
e
B
the entrance “distribution” in A and the last exit
“distribution” of A coincide with e
A
:
P
e
B
ŒH
A
< 1;X
H
A
D y D P
e
B
ŒL
A
>0;X
L

A
D y D e
A
.y/; for y 2 E:
(1.59)
In particular when B D E,

under P
Ä
, the entrance distribution in A and the exit distribution of A
coincide with e
A
.
(1.60)
1.3 Some potential theory 17
Proof.
 (1.56):
Both members vanish when y … A. We thus assume y 2 A. Using the discrete-time
Markov chain Z
n
, n  0 (see (1.15)), we can write:
P
x
ŒL
A
>0;X
L

A
D y D P
x
h
S
n0
fZ
n
D y; and for all k>n;Z

k
… Ag
i
pairwise
disjoint
D
P
n0
P
x
ŒZ
n
D y; and for all k>n;Z
k
… A
Markov
property
D
P
n0
P
x
ŒZ
n
D yP
y
Πfor all k > 0; Z
k
… A
Fubini

D
(1.45)
E
x
h
P
n0
1fZ
n
D yg
i
P
y
Œ
z
H
A
D1
D E
x
h
Z
1
0
1fX
t
D ygdt
i
P
y

Œ
z
H
A
D1
(1.26)
D
(1.53)
g.x; y/e
A
.y/:
This proves (1.56).
 (1.57):
Summing (1.56) over y 2 A, we obtain
P
x
ŒH
A
< 1 D P
x
ŒL
A
>0D
P
y2A
g.x; y/e
A
.y/
(1.33)
D Ge

A
.x/; whence (1.57).
 (1.58):
Note that e
A
is supported on A and Ge
A
D 1 on A by (1.57). If  is another such
measure and  D  e
A
,
h; GiD0
because G D 0 on A, and  is supported on A.By(1.35) it follows that  D 0,
whence (1.58).
 (1.59), (1.60):
By (1.50) (Hunt’s switching identity): for y 2 E,
E
e
B
ŒH
A
< 1;g.X
H
A
;y/D E
y
ŒH
A
< 1;.Ge
B

/.X
H
A
/
(1.58)
D
AÂB
P
y
ŒH
A
< 1: