Self-similar branching Markov chains

0.29 MB·English

by  Nathalie Krell

Self-similar bran
hing Markov
hains
8
0
Nathalie Krell
0
2

n February 1, 2008
a
J

4
2 Laboratoire de Probabilités et Modèles Aléatoires,

Université Pierre et Marie Curie,
]
R 175 rue du Chevaleret, 75013 Paris, Fran
e.
P
.
h
t
a
m
Abstra
t
[
The main purpose of this work is to study self-similar bran
hing Markov
hains.
2
First we will
onstru
t su
h a pro
ess. Then we will establish
ertain Limit Theorems
v
8 using the theory of self-similar Markov pro
esses.
3
1
0
Key Words. Bran
hing pro
ess, Self-similar Markov pro
ess, Tree of generations, Limit
.
8
Theorems.
0
7
0 A.M.S. Classi(cid:28)
ation. 60J80, 60G18, 60F25, 60J27.
:
v
i e-mail. krell

r.jussieu.fr
X
r
a
1 Introdu
tion.
This work is a
ontribution to the study of a spe
ial type of bran
hing Markov
hains. We
X
will
onstru
t a
ontinuous time bran
hing
hain whi
h hasRa∗ self-similar property and
whi
h takes its values in the spa
e of (cid:28)nite point measures of +. This type of pro
ess is
a generalization of a self-similar fragmentation (see [4℄), whi
h may apply to
ases where
the size models non additive quantities as e.g. surfa
e energy in aerosols. We will fo
us on
α
the
ase where the index of self-similarity is non-negative, whi
h means that the bigger
individuals will reprodu
e faster thanxth→e sxm−a1ller ones. TherRe∗is no loss of generality by
+

onsidering this model, as the map on atoms in transforms a self-similar
α −α
pro
ess with index into another one with index (and preserves the Markov property).
1
In this arti
le we
hoose to
onstru
t the pro
ess by bare hand. We extend the method
used in [4℄ to deal with more general pro
esses where we allow an individual to have a mass
bigger than that of its parent. We will explain in the sequel, whi
h di(cid:30)
ulties this new
set-up entails. There exists
losely related arti
les about bran
hing pro
esses, like among
others [18℄, [19℄ from Kyprianou and [12℄, [13℄ from Chauvin. However noti
e that the time
of splitting of the pro
ess depends on the size of the atoms of the pro
ess.
More pre
isely we will (cid:28)rst introdu
e a bran
hing Markov
hains as a marked tree and
we will obtain a pro
ess, indexed by generations (it is simply a random mark on the tree of
generation, see Se
tion 2). Thanks to a martingale whi
h is asso
iated to the latter and the
theory of random stopping lines on a tree of generation, we will de(cid:28)ne the pro
ess indexed
by time. After having
onstru
ted the pro
ess, we will study the evolution of the randomly

hosen bran
h of the
hain, from whi
h we shall dedu
e some Limit Theorems, relying on
the theory of self-similar Markov pro
esses. In an appendix we will
onsider the intrinsi
pro
ess and give some properties in the spirit of the arti
le of Jagers [15℄. By the way we
will show properties about the earlier martingale.
2 The marked tree.
In this part we will introdu
e a bran
hing Markov
hain as a marked tree, whi
h gives a
genealogi
des
ription of the pro
ess that we will
onstru
t. This terminology
omes from
Neveu in [21℄ even if here the marked tree we
onsider is slightlydi(cid:27)erent. First we introdu
e
some notations and de(cid:28)nitionsR. ∗ s = n δ
A (cid:28)nite point measure on + is a (cid:28)nite sum of Dira
point masses i=1 si, where
s s n ≥ 0
i
♯thse= na=re s
(aRll∗ed) the atoms of and iss an arbMitra(Rry∗)integer. We shPall often write
R+∗ for the number offat:oRm∗s →of R, and p + for the spas
e∈oMf (cid:28)n(Rite∗)point
measures on +. We also de(cid:28)ne for + measurable fun
tion and p +
♯s
hf,si := f(s ),
i
i=1
X
s
by taking the sum over the atoms of repeated a

ording to their multipli
ity and we will
sometimes use the slight abuse of notation
♯s
hf(x),si := f(s )
i
i=1
X
f x M (R∗)
when is de(cid:28)ned as a fun
tion depending on the variable . We endow the spa
e p +
s s
n
with the topology of weak
onvergen
e, whi
h means that
onverge to if and only if
hf,s i hf,si f
n

αon≥ve0rge to for all
ontinuous bouνnded fun
tions . M (R∗)
Let beanindexofself-similarityand besomeprobability measureon p + .
X = (( ♯X(t)δ ) )
The aim of this work is to
onstru
t a bran
hing Markov
hain i=1 Xi(t) t≥0 with
P
2
M (R∗) α ν
p +
values in , whi
h is self-similar with index and has reprodu
tion law . The index
of self-similarity will play a part in the rate at whi
h an individual will reprodu
e and the
ν
reprodu
tion law will spe
ify the distribution of the o(cid:27)spring. We stress that our setting
in
ludes the
ase when
ν(∃i : s > 1) > 0,
i
(1)
whi
h means that with a positive probability the size of a daughter
an ex
eed that of her
mother.
To do that, exa
tly as des
ribed in Chapter 1 se
tion 1.2.1 of [4℄, we will
onstru
t a
marked tree.
We
onsider the Ulam Harris labelling system
U := ∪∞ Nn,
n=0
N = {1,2,...} N0 = {∅} U
with the notation and . In the sequel the elements of are
alled
∅ u =
nodes (or sometimes also individuals) and the distinguished node the root. For ea
h
(u ,...,u ) ∈ U n u |u| = n
1 n
|∅| = 0 ,nw≥e
0allu =th(euge,n..e.,ruati)on∈oNfn andiw∈ritNe , wiuthi =the(uob,v..i.o,uus ,
io)nv∈enNtnio+n1
1 n 1 n
. When , and , we write
i u u = (u ,...,u ) n ≥ 2
1 n
for the -th
hild of . We also de(cid:28)ne for with ,
mu = (u ,...,u )
1 n−1
u mu = ∅ u ∈ N v = mnu n ≥ 0 v (cid:22) u
the mother of , if . If for some we write and say that
u v M U M (cid:22) v u (cid:22) v u ∈ M
stems from . Additionally for a set of , means that for some .
M (cid:22) L x ∈ L M
Generally we write if all stem from .
s
Here it will be
onvenient to identify the point measure with the in(cid:28)nite sequen
e
(s ,...,s ,0,...)
1 n
obtained by aggregation of in(cid:28)nitely many 0's to the (cid:28)nite sequen
e of the
s
atoms of . (ξ ,i ∈ N) ν
i
In parti
ular wne say that aξra=nd0om⇔ini(cid:28)>ninte sequen
e has the lawn ,δif there is
a (random) index su
h that i and the (cid:28)nite point measure i=1 ξi has the
ν
law . P
De(cid:28)nition 1. Let two independent families of i.i.d. variables be indexed by the nodes of the
(ξu,u ∈ U) (eu,u ∈ U) u ∈ U ξu = (ξui)i∈N
tree, and , where for ea
h is distributed a

ording
ν (eui)i∈N
to the law , and is a sequen
e of i.i.d. exponential variables with parameter 1. We
x > 0 e
de(cid:28)ne re
ursively for some (cid:28)xed
ξ := x, a := 0, ζ := x−αe ,
∅ ∅ ∅ ∅
u ∈ U i ∈ N
and for and :
ξ := ξ ξ , a := a +ζ , ζ := ξ−αe .
ui ui u ui u u ui ui ui
e
3
u U (ξ ,a ,ζ ) ξ a
u u u u u
To ea
h node of the tree , we asso
iate the mark where is the size, the
ζ u
u
birth-time and the lifetime of the individual with label . We
all
T = ((ξ ,a ,ζ ) )
x u u u u∈U
x P Ω¯
x
a marked tree with root of size , and the law asso
iated is denoted by . Let be the set
of all the possible marked trees.
(ξ ,u ∈ U)
u
The size of the individuals de(cid:28)nes a multipli
ative
as
ade (see the referen
es
X
in Se
tion 3 of [5℄). However the latter is not su(cid:30)
ient to
onstru
t the pro
ess , in fa
t
((a ,ζ ),u ∈ U)
u u
we also need the information given by .
L ⊂ U u,v ∈ L
Another useful
on
ept is that of line. A subset is a line if for every ,
u (cid:22) v ⇒ u = v
. The pre-L-sigma algebra is
H := σ(ξ ,e ;∃l ∈ L : u (cid:22) l).
L u u
e
A random set of individuals J : Ω¯ → P(U)
{J (cid:22) L} ∈ H L ⊂ U P(U) U
L
is optional if for all line , where is the power set of . An
J
optional line is a random line whi
h is optional. For any optional set we de(cid:28)ne the
J
pre- -algebra by:
A ∈ H ⇔ ∀L ⊂ U : A∩{J (cid:22) L} ∈ H .
J L
line
The (cid:28)rst result is:
Lemma 1. ThJe markedtree
onstru
teϕd in: DΩ¯e→(cid:28)ni[t0i,o1n],1usa∈tisU(cid:28)es the strong Markov bran
hing
u
property: for an optional line and , measurable fun
tions, we get
that,
E ϕ ◦Tξu H = E (ϕ ),
1 u J ξu u
(cid:12) !
uY∈J (cid:12) uY∈J
(cid:12)
Tξu (cid:12)T (ξ ,a ,ζ )
where is the marked tree extra
ted from(cid:12) 1 at the node u u u . More pre
isely
Tξu = ((ξ ,a −a ,ζ ) ).
uv uv u uv v∈U
(ξ ,u ∈ U) (e ,u ∈ U)
u u
Proof. Thanks to the i.i.d properties of the random variables and ,
the Markov property for lines is of
ourse easily
he
ked. In order to get the result for a more
e
general optional line, we use Theorem 4.14 of [15℄. Indeed, the tree we have
onstru
ted is
ρ
u
a spe
ial
ase of the tree
onstru
ted by Jagers in [15℄. In our
ase the Jagers's notation ,
τ σ ρ u ∈ U u ξ σ a
u u u u u u
and are su
h that the type of , is the mass of : , the birth time is
τ ζ
u mu
and is here equal to (be
ause the mother dies when she gives birth to her daughters).
u ∈ U
Wein∈otNi
e that all the sτisters have the sameζbirth time, whi
h means that for all and
ui u
all , we have that is here equal to .
4
3 Malthusian hypotheses and the intrinsi
martingale.
We introdu
e some notations to formulate the fundamental assumptions of this work:
p := inf p ∈ R : hxp,siν(ds) < ∞ ,
( ZMp(R∗+) )
and
p := inf p > p : hxp,siν(ds) = ∞
∞
( ZMp(R∗+) )
inf∅ = ∞ p ∈ (p,p )
∞
(with the
onvention ) and then for every :
κ(p) := (1−hxp,si)ν(ds).
ZMp(R∗+)
κ
Note that is a
ontinuous and
on
ave fun
tion (but not ne
essarily a stri
tly in
reasing
(p,p ) p → hxp,siν(ds)
fun
tion) on ∞ , as Mp(R∗+) is a
onvex appli
ation. By
on
avity, the
κ(p) = 0 (p,p )
R ∞
equation has at most two solutions on . When a solution exists, we denote
p := inf{p ∈ (p,p ) : κ(p) = 0} p
0 ∞ 0
by the smallest, and
all the Malthusian exponent.
We now make the fundamental:
p
0
Malthusian Hypotheses. We suppose that the Malthusian exponent exists, that
p > 0
0
, and that
κ(p) > 0 p > p .
0
for some (2)
Furthermore we suppose that the integral
(hxp0,si)pν(ds)
ZMp(R∗+) (3)
p > 1
is (cid:28)nite for some .
Throughout the rest of this arti
le, these hypotheses will always be taken for
granted.
ν(s ≤ 1 i) = 1
i
Note that (2) always holds when for all (fragmentation
ase). We stress
κ p
that may not be stri
tly in
reasing, and may not be negative when is su(cid:30)
iently large
(see Subse
tion 6.1 for a
onsequen
e of this fa
t.)
Wne≥w2ill(gυiv,e..o.,nυe)exnample based on the Diri
hleυt =pro
enss υ(see the book Kingman [16∆℄).
Fix , 1 n positive real numbers and i=1 i. We de(cid:28)ne the simplex n
by P
n
∆ := (p ,p ,...,p ) ∈ Rn, p = 1 .
n 1 2 n + i
( )
j=1
X
5
(υ ,...,υ ) ∆
1 n n
The Diri
hlet distributionof parameter over the simplex has the density (with
(n−1) ∆
n
respe
t to the -dimensional Lebesgue measure on ):
Γ(υ)
f(p ,...,p ) = pυ1−1...pυn−1.
1 n Γ(υ )...Γ(υ ) 1 n
1 n
a := υ(υ + 1)/( n υ (υ + 1)) a
Let i=1 i i . Note that is stri
tly larger than 1. Let the
(aX ,...,aX ) (X ,...,X )
1 n 1 n
reprodu
tion measure bePthe law of , where is a random ve
tor
(υ ,...,υ )
1 n
with Diri
hlet distribution of parameter . Therefore
n
Γ(υ) Γ(p+υ )
κ(p) = ap i ,
Γ(υ +p) Γ(υ )
i
i=1
X
p = −υ p = 1
0
, and the Malthusian hypotheses are veri(cid:28)ed. n ∈ N u
In this arti
le we will
all extin
tion the event that for some , all nodes at the
n
-th generation have zero size, and non-extin
tion the
omplementary event. We see that
ν(s = 0) > 0
1
the probability of extin
tion is always stri
tly positive whenever , and equals
ν(s = 0) = 0
1
zero if and only if (as we have suppose (3); see p.28 [4℄).
(ξ ,u ∈ U)
u
After these de(cid:28)nitions, we introdu
e a fundamental martingale asso
iated to .
Theorem 1. The pro
ess
M := ξp0, n ∈ N
n u
|u|=n
X
(H ) L n
is a mLar:t=in{gaule∈iUn :th|eu|(cid:28)=ltrnat}ion Ln , with n the line asso
Lipa(tPed) to the -tph>ge1neration
n
(i.e. ). This martingale is bounded in for some , and in
parti
ular is uniformly integrable.
M
∞
Moreover,
onditionally on non-extin
tion the terminal value is stri
tly positive a.s.
κ κ(p) = 0 p := inf{p >
+
Remark 1. As is
on
ave the equation may have a se
ond root
p , κ(p) = 0}
0
). This se
ond root is less interesting: even though
M+ := ξp+, n ∈ N,
n u
|u|=n
X
p > 1 p M+
n
is alEso(a m∞ar|tMingale−, iMt is|pe)as=y∞to
he
k that for all the -variation of is in(cid:28)nite,
i.e. n=0 n+1 n ).
WeP
an noti
e that for all p ∈ (p0,p+) (Mn(p))n∈N := ( |u|=nξup)n∈N is a supermartingale.
E(Mp) < ∞
The assumption (3) means a
tually that 1 P.
•
Proof. We will use the fa
t that the empiri
al measure of the logarithm of the sizes of
fragments
Z(n) := δ
logξu (4)
|u|=n
X
6
an be viewed as a bran
hing random walk (see the arti
le of Biggins [8℄) and use Theorem
θ > p
1 of [8℄. In order to do that we (cid:28)rst introdu
e some notation: for , we de(cid:28)ne
m(θ) := E eθxZ(1)(dx) = E ξθ = 1−κ(θ)
 u
(cid:18)Z (cid:19) |u|=1
X
 
and
W(n)(θ) := m(θ)−n eθxZ(n)(dx) = (1−κ(θ))−n ξθ.
u
Z |u|=n
X
M = W(n)(p )
n 0
We noti
e that . Therefore in order to apply Theorem 1 of [8℄ and to get the
p p > 1

onvergen
e almost surely and in th mean for some , it is enough to show that
E(W(1)(p )γ) < ∞
0
γ ∈ (1,2]
for some and
m(pp )/|m(p )|p < 1
0 0
p ∈ (1,γ]
for some . The (cid:28)rst
ondition is a
onsequen
e of the Malthusian assumption.
Moreover the se
ond follows from the identities
m(pp )/|m(p )|p = (1−κ(pp ))/|1−κ(p )|p = 1−κ(pp )
0 0 0 0 0
p p > 1
0
whi
h, by the de(cid:28)nition of , is smaller than 1 for well
hosen.
• M > 0 a.s.
∞
q = PF(Minall=y,0le)t us now
he
Ek(Mthat) = 1
oqnd<it1ionally on non-extin
tion. De(cid:28)ne
∞ ∞
, therefore as we get that . Moreover, an appli
ation of the
bran
hing property yields E(qZn) = q,
Z n
n
where is the number of individuals with positive size at the -th generation. Noti
e that
Z = hZ(n),1i ν
n
(Z ,n ∈ N) . By the
onstru
tion of the marked tree and as is aqprobability measure:
n
is of
ourse a Galton-Watson pro
ess and it follows that is its probability of
M = 0
∞
extin
tion. Sin
e
onditionally on the extin
tion, the two events
oin
ide a.s.
4 Evolution of the pro
ess in
ontinuous time.
AMfterhLapv(iPng) de(cid:28)nedthepro
essindexedbygenerationandhavingshown thatthemartingale
n
is bounded, we are now able to de(cid:28)ne properly the main objet of this paper. In
u ξ > 0 I :=
u u
order to do this, when an individual labelled by has a positive size, , let
[a ,a + ζ )
u u u
be the interval of times during whi
h this individual is alive. Otherwise, i.e.
ξ = 0 I = ∅
u u
when , we de
ide that . With this de(cid:28)nition, we set:
7
X = (X(t),t ≥ 0)
De(cid:28)nition 2. We de(cid:28)ne the pro
ess by
X(t) = 1l δ ,t ≥ 0.
{t∈Iu} ξu (5)
u∈U
X
f : R → R
+
In parti
ular we have for measurable fun
tion
hf,X(t)i = f(ξ )1l .
u {t∈Iu}
u∈U
X
x > 0 P X
x
Fxor every , let be the law of thPe pro
Pess starti(nFg )from a single individual with
1 t t≥0
size . And for simpli(cid:28)
ation, we denote for , and let be the natural (cid:28)ltration
(X(t),t ≥ 0) (X (t),...,X (t))
1 ♯X(t)
of the pro
ess . We use the notation for the sequen
e of
X(t)
atoms of . In the following we will show that this sequen
e is almost surely (cid:28)nite. Of
(X (t),...,X (t)) ((ξ );t ∈ I )
1 ♯X(t) u u

ourse the set is the same as the set ; but sometimes it will
(X (t))
i
be
learer to use tuhe∈nRotation .
+
We de(cid:28)ne for :
F(u) := u♯sν(ds).
ZMp(R∗+)
F(u) (Z ,n ≥ 0) =
n
We noti
e that is the generating fun
tion of the Galton-Watson pro
ess
(♯{u ∈ U : ξ > 0 |u| = n},n ≥ 0)
u
and .
ǫ > 0
From now on, we will suppose that for every
1 du
= ∞.
F(u)−u (6)
Z1−ǫ
F′(1) = E(Z ) < ∞
1
Of
ourse if this last assumption is ful(cid:28)lled. Therefore we get the (cid:28)rst
theorem about the
ontinous time pro
ess:
X M (R∗)
Theorem 2. The pro
ess takes its values in the set p + . It is a bran
hing Markov
X(t+r) X(r) = s

hain, more pre
isely tXhe(i)
(ot)nditional distribution ofi X(i)(t) given that X(ti)s the samPe
as that of the sum , where for ea
h index , is distributed as under si
X(i)(t)
and the variables P are independent.
X c > 0
The pro
ess al(scoXh(acαstt)h,et ≥s
a0l)ing propPerty, Pnamely for every , the distribution of
1 c
the res
aled pro
ess under is .
In the fragmentation
ase, the fa
t that the size of the fragments de
reases with time
ǫ
entails that the pro
ess of the fragments of size larger than or equal to is Markovian, and
whi
h leads easily to Theorem 2. This property is lost in the present
ase.
• t ≥ 0 X(t)
Proof. First we will
he
k that for all , is a (random) (cid:28)nite point measure. By
Lp p > 1
Theorem 1 and the Doob's -inequality we get that for some :
supM = sup ξp0 ∈ Lp(P).
n u
n∈N n∈N
|u|=n
X
8
As a
onsequen
e: supξp0 ∈ Lp(P)
u
u∈U
X X (t),...
1
and then by the de(cid:28)nition of the pro
ess , writing for the (possibly in(cid:28)nite)
X(t)
sequen
e of atoms of sup sup X (t)p0 ∈ Lp(P).
i
i t∈R+
p > 0 m > 0
0
Re
all that by assumption. We (cid:28)x some arbitrarily large . We now work
m

onditionally on the event that the size of all individuals is bounded by , and we will show
t t ≥ 0
that the number of the individuals alive at time is almost surely (cid:28)nite for all .
{sup ξ ≤ m}
u∈U u
As we are
onditioning on the event , by the
onstru
tion of the marked
tree, wegetthatthelifetimeofanindividual
anbesto
hasti
allyboundedfrombelowbyan
mα
exponential variable of parameter . Therefore we
an bound the number of individuals
t
present at time by the number of individuals of a
ontinuous time bran
hing pro
ess
GW
denoted by in whi
h ea
h individual lives for a random time whose law is exponential
mα ♯s∨1
of parameter and the probability distribution of the o(cid:27)spring is the law of under
ν
(we have taken the supremum with 1 to ensure the absen
e of death). For the Markov
GW
bran
hing pro
ess , we are in the temporally homogeneous
ase and, we noti
e that
u(ns)∨1ν(ds) = (f(u)−u)ν(n 6= 0)+u,
s
ZMp(R∗+)
therefore as we have supposed (6), we
an use Theorem 1 p.105 of the book of Athreya and
Ney [3℄ (proved in Theorem 9 p.107 of the book of Harris [14℄) and get that we are in the
GW GW
non-explosive
ase for the . As the number of the individuals is bounded by that of
t
we get that the number of individuals at time is a.s. (cid:28)nite.
{sup ξ ≤ m} t ≥ 0
u∈U u
Therefore
onditioning on the event , we have that for all , the
t X(t)
num•ber of individuals at time is a.s. (cid:28)nite, i.e. ris∈aR(cid:28)nite poiτnt measure. {u ∈ U :
+ r
Se
ond we will show the Markov property. Fix . Let be equal to
r ∈ I } τ L ⊂ U
u r
. We noti
e that is an optional line. In fa
t for all lines we have that
{τ (cid:22) L} = {r < a +ζ ∀u ∈ L} ∈ H .
r u u L
By de(cid:28)nition, we have the identity
♯X(t+r)
1l δ = 1l δ .
{Xj(t+r)>0} Xj(t+r) {t+r∈Iu} ξu
i=1 u∈U
X X
X(r) = n δ ∈ M (R∗) n = ♯X(r) (v ,...,v ) U
Let i=1 ξvn p + with and 1 n the nodes of . De(cid:28)ne for
i ≤ n
all , P
T˜(i) := ((ξ ,a −a ,ζ −1l (r −a )) ) = ((ξ˜(i),a˜(i),ζ˜(i)) ),
viu viu vi viu {u=∅} vi u∈U u u u u∈U
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I˜(i) := [a˜(i),a˜(i) +ζ˜(i)[
u u u u
and
X(i)(t) = 1l δ .
{t∈I˜u(i)} ξ˜u(i)
u∈U
X
Then n
X(t+r) = X(i)(t).
i=1
X
u ∈ U s ∈ I
u
By the la
k of memory oT˜f(it)he exponential variable, we have that for , given
the law of the marked tree is the same as that of
Tξvi := ((ξviu,aviu −avi,ζviu)u∈U) := ((ξui,aiu,ζui)u∈U).
Thus we have the equality in law:
(d)
1l δ = 1l δ ,
{t∈I˜u(i)} ξ˜u(i) {t∈Iui} ξui
u∈U u∈U
X X
Ii := [ai,ai +ζi[
with u u u u .
τi := {v u ∈ U : r ∈ Ii} L ∈ U
Let r i u . Moreover for all lines we have that
{τi (cid:22) L} = {r < a +ζ ∀v u ∈ L} ∈ H .
r viu viu i L
τi τi
r s
Therefore is an optional line and by applying Lemma 1 for the optional line , we have
that the
ondition distribution of the point measure
1l δ
{t+r∈Ii} ξi
u u
u∈U
X
H X(t) P H = σ(ξ˜ ,e : a ≤ s)
given τr is the laFw =of σ(X(su′)nd:ers′ ≤xi.s)We noti
e th(aXt(1),τXs (2),...,Xu (nu)) u is the
s
same (cid:28)ltration as . ThieXre(fio)(rte) X(ti)s a sequPen
e of
independent random pro
esses, where for ea
h is distributed as under xi. We
then have proven the Markovian property.
• T
x
The s
aling property is an easy
onsequen
e of the de(cid:28)nition of the tree .
g : R∗ → R∗
+ +
Remark 2. For every measurable fun
tion , de(cid:28)ne a multipli
ative fun
tional
s = ♯s δ ∈ M (R∗)
su
h that for every i=1 si p + :
P ♯s
φ (s) := exp(−hg,si) = exp(− g(s )).
g i
i=1
X
G X(t) y = ♯y δ ∈ M (R∗)
Then the generator of the Markov pro
ess ful(cid:28)lls for every i=1 yi p + :
P
Gφg(y) = yiαe−Pj6=ig(yj)ZMp(R∗+)(e−hg(xyi),si −e−g(yi))ν(ds). (7)
X
10