Loop measures without transition probabilities

0.19 MB·English

by  Pat Fitzsimmons

Loop measures without transition probabilities
5
1
0 Pat Fitzsimmons Yves Le Jan Jay Rosen ∗
2

n February 3, 2015
a
J

1
3
Abstract

]
R
P
.
h
1 Introduction
t
a
m
To the best of our knowledge, loop measures first appeared in the work of
[
Symanzik on Euclidean quantum field theory, [11], where they are referred to

1
as ‘blob measures’. They next appear in the work of Lawler and Werner, [6].
v
8 In both works, the loop measure is that associated with Brownian motion. In
4 [7] loop measures associated with a large class of Markov processes are defined
1
and studied. In all these cases it is assumed that the underlying Markov
0
0 process has transition densities. The goal of this paper is to define and study
2. loop measures for Markov processes without transition densities.
0 Let X=(Ω,F ,X ,θ ,Px)beatransient Borel right process [10]with state
t t t
5
space S, which we assume to be locally compact with a countable base. We
1
: use the canonical representation of X in which Ω is the set of right continuous
v
paths paths ω : [0,∞) → S = S ∪∆ with ∆ ∈/ S, and is such that ω(t) = ∆
i ∆
X
for all t ≥ ζ = inf{t > 0|ω(t) = ∆}. Set X (ω) = ω(t).
t
r Let mbeaBorelmeasureonS whichis finiteon compactsets. Weassume
a
that with respect to m, X has strictly positive potential densities uα(x,y),
α ≥ 0, which satisfy the resolvent equations. We set u(x,y) = u0(x,y), and
assume that u(x,y) is excessive in x for each fixed y.
Let h (x) = u(x,z). If we assume that u is finite, then the h −transform
z z
of X is a right process on S, see [10, Section 62], with probabilities Px/hz. Let
∗ResearchofJ.RosenwaspartiallysupportedbygrantsfromtheNationalScienceFoun-
dation and PSC CUNY.
0 Key words and phrases: loop soups, Markov processes, intersection local times.
0 AMS 2000 subject classification: Primary 60K99, 60J55; Secondary 60G17.
1
Qz,z = u(z,z)Pz/hz. We can then define the loop measure as
F
µ(F) = Qz,z dm(z), (1.1)
ζ
Z (cid:18) (cid:19)
for any F measurable function F. Loop measures for processes with finite
potential densities but without transition densities are discussed in [4]. In the
present paper we assume that the potential densities u(x,y) are infinite on
the diagonal, but finite off the diagonal. In this case the construction of the
measures Pz/hz given in [10] breaks down. Assuming that all points are polar,
we show how to construct a family of measures Qz,z,z ∈ S, which generalize
the measures Qz,z = u(z,z)Pz/hz in the case of finite u(x,y).
After constructing Qz,z,z ∈ S and definingtheloop measureµ using(1.1),
we show how to calculate some important moments. We assume that
sup (u(x,y)+u(y,x))2 dm(y) <∞ (1.2)
x ZK
for any compact K ⊆ S. For exponentially killed Brownian motion in Rd this
means that d≤ 3.
Theorem 1.1 For any k ≥ 2, and bounded measurable functions f ,...,f
1 k
with compact support
k ∞
µ f (X )dt (1.3)
 j tj j
j=1Z0
Y
  k
= u(x ,x )u(x ,x )···u(x ,x ) f (x )dm(x ),
1 2 2 3 k 1 πj j j
πX∈Pk⊙Z jY=1
where P⊙ denotes the set of permutations of [1,k] on the circle. (For example,
k
(1,2,3), (3,1,2) and (2,3,1) are considered to be one permutation π ∈ P⊙.)
3
Our assumption (1.2) will guarantee that the right hand side of (1.3) is
finite. Note that if k = 1 our formula would give
∞
µ f(X )dt = u(x,x)f(x)dm(x) = ∞, (1.4)
t j
(cid:18)Z0 (cid:19) Z
for any f ≥ 0, by our assumption that the potentials u(x,y) are infinite on
the diagonal.
2
For f ,...,f as above consider more generally the multiple integral
1 k
Mf1,...,fk = f (X )··· f (X )dr ···dr , (1.5)
t π(1) r1 π(k) rk 1 k
πX∈Tk⊙Z0≤r1≤···≤rk≤t
where T⊙ denotes the set of translations π of [1,k] which are cyclic mod k,
k
that is, for some i, π(j) = j +i, mod k, for all j = 1,...,k. In the proof of
Theorem 1.1 we first show that
k
µ Mf1,...,fk = u(x ,x )u(x ,x )···u(x ,x ) f (x )dm(x ). (1.6)
∞ 1 2 2 3 k 1 j j j
(cid:16) (cid:17) Z jY=1
(1.3) will then follow since
k ∞
f (X )dt = Mfπ1,...,fπk. (1.7)
j tj j ∞
jY=1Z0 πX∈Pk⊙
Thereisarelated measurewhichweshallusewhichgives finitevalueseven
for k = 1. Set
ν(F) = Qz,z(F) dm(z). (1.8)
Z
Assume that for any α > 0, any compact K ⊆ S, and any K which is a
compact neighborhood of K
e
sup uα(z,x) < ∞. (1.9)
z∈Kc,x∈K
e
Theorem 1.2 For any k ≥ 1, α > 0, and bounded measurable functions
f ,...,f with compact support
1 k
k ∞
ν f (X )dt e−αζ (1.10)
 j tj j 
j=1Z0
Y
  k
= uα(z,x )uα(x ,x )···uα(x ,z) f (x )dm(x )dm(z),
1 1 2 k πj j j
πX∈PkZ jY=1
where P denotes the set of permutations of [1,k], and both sides are finite.
k
We call µ the loop measure of X because, when X has continuous paths,
µ is concentrated on the set of continuous loops with a distinguished starting
3
point (since Qx,x is carried by loops starting at x). Moreover, in the next
Theorem we show that it is shift invariant. More precisely, let ρ denote the
u
loop rotation defined by
ω(s+u mod ζ(ω)), if 0 ≤ s < ζ(ω)
ρ ω(s) =
u ∆, otherwise.
(cid:26)
Here, for two positive numbers a,b we define a mod b = a−mb for the unique
positive integer m such that 0≤ a−mb < b.
For the next Theorem we need an additional assumption: for any δ > 0
and compact K ⊆ S
P (z,dx)u(x,z)dm(z) < ∞. (1.11)
δ
ZK
Theorem 1.3 µ is invariant under ρ , for any u.
u
Note that if we have transition densities p (z,x) then
δ
P (z,dx)u(x,z)dm(z) = p (z,x)u(x,z)dm(x)dm(z)(1.12)
δ δ
ZK Z ZK
∞
= p (x,x)dt dm(x).
t
ZK(cid:18)Zδ (cid:19)
In our work on processes with transition densities, it was always assumed that
∞
sup p (x,x)dt < ∞ for any δ > 0, which indeed gives (1.11).
x δ t
For the next Theorem we assume that the measure m is excessive. With
R
this assumption there is always a dual process Xˆ (essentially uniquely deter-
mined), but in general it is a moderate Markov process. We assume that the
measures Uˆ(·,y) are absolutely continuous with respect to m for each y ∈S.
For CAF’s Lν1,...Lνk with Revuz measures ν ,...,ν , let
t t 1 k
Aν1,...,νk = dLνπ(1)··· dLνπ(k). (1.13)
t r1 rk
πX∈Tk⊙Z0≤r1≤···≤rk≤t
We refer to Aν1,...,νk as a multiple CAF.
t
Theorem 1.4 For any k ≥ 2, and any CAF’s Lν1,...Lνk with Revuz mea-
t t
sures ν ,...,ν ,
1 k
k
µ(Aν1,...,νk)= u(x ,x )u(x ,x )···u(x ,x ) dν (x ). (1.14)
∞ 1 2 2 3 k 1 j j
Z j=1
Y
4
and
k k
µ Lνj = u(x ,x )u(x ,x )···u(x ,x ) dν (x ). (1.15)
 ∞ 1 2 2 3 k 1 πj j
jY=1 πX∈Pk⊙Z jY=1
 
The finiteness of the right hand side of (1.15) will depend on the potential
densities u(x,y) and the measures ν ,...,ν . For a more thorough discussion
1 k
see [8, (1.5)] and the paragraph there following (1.5).
With the results of this paper, most of the results of [4, 8, 9] on loop
measures, loop soups, CAF’s and intersection local times will carry over to
processes without transition densities.
2 Construction of Qz,z
Letusfixz ∈ S andconsidertheexcessive functionh (x) := u(x,z), finiteand
z
strictly positive on the subspace S := {x ∈ S : x 6= z}. Doob’s h-transform
z
theory yields the existence of laws Px,z, x ∈ S , on path space under which
z
the coordinate process is Markov with transition semigroup
h (y)
Pz(x,dy) := P (x,dy) z . (2.1)
t t h (x)
z
See, for example, [10, pp. 298–299]. Now consider the family of measures
ηz(dx) := P (z,dx)h (x). (2.2)
t t z
Since we assume that the singleton {z} is polar, the transition semigroup
(P ) will not charge {z}, so these may be viewed as measures on S or on S.
t z
Adopting the latter point of view, it is immediate that (ηz) is an entrance
t t>0
law for (Pz). There is a general theorem guaranteeing the existence of a right
t
processwithone-dimensionaldistributions(ηz)andtransitionsemigroup(Pz);
t t
see [5, Proposition (3.5)]. The law of this process is the desired Qz,z. Aside
from the entrance law identity ηzPz = ηz , their result only requires that
t s t+s
each of the measures ηz be σ-finite, which is clearly the case in the present
t
discussion.
With this we immediately obtain, for 0 < t < ··· < t ,
1 k
k k k
Qz,z f (X ) = ηz (dx ) Pz (x ,dx ) f (x ) (2.3)
 j tj  t1 1 tj−tj−1 j−1 j j j
j=1 j=2 j=1
Y Y Y
  k k
= P (x ,dx ) f (x )u(x ,z)
tj−tj−1 j−1 j j j k
j=1 j=1
Y Y
5
with t = 0 and x = z. Hence
0 0
k
Qz,z f (X )dt (2.4)
 j tj j
Z0<t1<···<tk<∞j=1
Y
 k
= u(z,x )u(x ,x )···u(x ,z) f (x )dm(x ),
1 1 2 k j j j
Z j=1
Y
so that
k ∞
Qz,z f (X )dt (2.5)
 j tj j
j=1Z0
Y
  k
= u(z,x )u(x ,x )···u(x ,z) f (x )dm(x ).
1 1 2 k πj j j
πX∈PkZ jY=1
Returning to (2.3) with 0 < t < ··· < t and using the fact that ζ > t
1 k k
implies that ζ = t +ζ ◦θ we have
k tk
k
Qz,z f (X )e−αζ (2.6)
 j tj 
j=1
Y
 
k
= Qz,z f (X )e−αtk e−αζ ◦θ
 j tj tk 
jY=1 (cid:16) (cid:17)
k  k 
= Pα (x ,dx ) f (x )h (x )Pxk,z e−αζ .
tj−tj−1 j−1 j j j z k
jY=1 jY=1 (cid:16) (cid:17)
Note that by (2.1) and the fact that X has α-potential densities for all
α ≥ 0
∞ ∞
Pxk,z e−αt1 (X )dt = e−αtPxk,z 1 (X ) dt (2.7)
{S} t {S} t
(cid:18)Z0 (cid:19) Z0
∞ (cid:0) h(cid:1)(y)
= e−αt P (x ,dy) z dt
t k
h (x )
Z0 ZS z k
h (y)
= uα(x ,y) z dm(y).
k
h (x )
z k
Z
6
Combining this with our assumption that the α-potential densities satisfy the
resolvent equation we see that
ζ
h (x )Pxk,z e−αζ = h (x )Pxk,z 1−α e−αtdt (2.8)
z k z k
(cid:16) (cid:17) (cid:18) Z0 (cid:19)
∞
= h (x )−αh (x )Pxk,z e−αt1 (X )dt
z k z k {S} t
(cid:18)Z0 (cid:19)
= u(x ,z)−α uα(x ,y)u(y,z)dm(y) = uα(x ,z).
k k k
Z
Using this in (2.6) we obtain
k
Qz,z f (X )e−αζ (2.9)
 j tj 
j=1
Y
k  k
= Pα (x ,dx )uα(x ,z) f (x ).
tj−tj−1 j−1 j k j j
j=1 j=1
Y Y
We then have
k
Qz,z e−αζ f (X )dt (2.10)
 j tj j
Z0<t1<···<tk<∞j=1
Y
 k 
= uα(z,x )uα(x ,x )···uα(x ,z) f (x )dm(x ),
1 1 2 k j j j
Z j=1
Y
and consequently
k ∞
Qz,z e−αζ f (X )dt (2.11)
 j tj j
j=1Z0
Y
  k
= uα(z,x )uα(x ,x )···uα(x ,z) f (x )dm(x ).
1 1 2 k πj j j
πX∈PkZ jY=1
3 The loop measure and its moments
Set
F
µ(F)= Qz,z dm(z). (3.1)
ζ
Z (cid:18) (cid:19)
7
Proof of Theorem 1.1: We use an argument from the proof of [4,
Lemma 2.1], which is due to Symanzik, [11].
It follows from the resolvent equation that the potential densities uβ(x,y)
are continuous and monotone decreasing in β, for x 6= y. Using this together
with the resolvent equation and the monotone convergence theorem we obtain
that for x 6= x
k 1
d
uα(x ,z)uα(z,x )dm(z) = − uα(x ,x ). (3.2)
k 1 k 1
dα
ZS
Hence using (2.10)
Qz,z e−αζMf1,...fk dm(z) (3.3)
∞
Z (cid:16) (cid:17)
k
d
= − uα(x ,x )uα(x ,x )···uα(x ,x ) uα(x ,x ) f (x )dm(x )
1 2 2 3 k−1 k dα k 1 πj j j
πX∈Tk⊙Z jY=1
k
d
= − uα(x ,x )uα(x ,x )···uα(x ,x )uα(x ,x ) f (x )dm(x ).
1 2 2 3 k−1 k k 1 j j j
dα
Z j=1
Y
For the last step we used the product rule for differentiation and the fact that
in the middle line we are summing over all translations mod k.
Since, as mentioned, uα(x,y) is monotone decreasing in α for x 6= y,
v(x,y) = lim uα(x,y) (3.4)
α→∞
exists and
∞
v(x,y)f(y)dm(y) = lim e−αt P (x ,dy)f(y)dt = 0. (3.5)
t k
α→∞
Z Z0 Z
Hence v(x,y) = 0 for m−a.e. y. Integrating (3.3) with respect to α from 0
to ∞ and using Fubini’s theorem we then obtain (1.6). (1.3) then follows by
(1.7).
To show that the right hand side of (1.3) is finite we repeatedly use the
Cauchy-Schwarz inequality and our assumption (1.2). See the proof of [8,
Lemma 3.3].
Proof of Theorem 1.2: The formula (1.10) follows immediately from
(2.11). When k ≥ 2, the right hand side of (1.10) can be shown to be finite
by repeatedly using the Cauchy-Schwarz inequality, our assumption (1.2) and
8
the fact that uα(x,z) is integrable in z for any α > 0. When k = 1, if K is a
compact set containing the support of f and K is a compact neighborhood
1
of K, then
e
uα(z,x)uα(x,z)f (x)dm(x)dm(z)
1
Z Z
= uα(z,x)uα(x,z)f (x)dm(x)dm(z)
1
ZKZ
e
+ uα(z,x)uα(x,z)f (x)dm(x)dm(z).
1
ZKcZ
e
Using (1.2)
uα(z,x)uα(x,z)f (x)dm(x) dm(z) (3.6)
1
ZK(cid:18)Z (cid:19)
e
≤ m(K)sup uα(z,x)uα(x,z)f (x)dm(x) < ∞,
1
z
Z
e
and using (1.9)
uα(z,x)uα(x,z)f (x)dm(x)dm(z) (3.7)
1
ZKcZ
e
≤ C uα(x,z)dm(z) f (x)dm(x) < ∞.
1
Z (cid:18)Z (cid:19)
4 Subordination
The basic idea in our proof that the loop measure is shift invariant is to show
that the loop measure can be obtained as the ‘limit’ of loop measures for
processes with transition densities. These processes will be obtained from the
original process by subordination.
We consider a subordinator T which is a compound Poisson process with
t
Levy measure cψ so that
∞ (ct)j ∞
Ex(f (X ))= e−ct Ex(f (X ))ψ∗j(ds). (4.1)
Tt j! s
j=1 Z0
X
9
If we take ψ to be exponential with parameter θ, then ψ∗j(ds) = sj−1θje−sθds
Γ(j)
so that we have
∞ (ct)j ∞ sj−1θj
Ex(f(X )) = e−ct Ex(f(X )) e−sθds. (4.2)
Tt j! s Γ(j)
j=1 Z0
X
Hence the subordinated transition semigroup
∞ (ct)j ∞ sj−1θj
P (x, dy) = e−ct P (x, dy) e−sθds. (4.3)
t s
j! Γ(j)
j=1 Z0
X
e
Noting that
∞ dj−1 ∞ dj−1
P (x,A)sj−1e−sθds = P (x,A)e−sθds = Uθ(x,A), (4.4)
s dθj−1 s dθj−1
Z0 Z0
we see that P (x, dy) is absolutely continuous with respect to the measure m
t
on S, and we can choose transition densities p (x,y).
t
From noweon we take θ = c= n, and use (n) as a superscript or subscript
(n)
to denote objects with respect to the subordineated process, denoted by X .
t
Lemma 4.1
1
uα (x,y) = uα/(1+α/n)(x,y). (4.5)
(n) (1+α/n)2
In particular,
uα (x,y) ≤ u (x,y) = u(x,y). (4.6)
(n) (n)
10