A Ruelle Operator for continuous time Markov Chains

0.16 MB·English

by  Alexandre Baraviera

A Ruelle Operator for continuous time Markov Chains
Alexandre Baraviera ∗ Ruy Exel † Artur O. Lopes‡
8
0
March 17, 2008
0
2

r
a
M Abstract
We consider a generalization of the Ruelle theorem for the case of continuous
7
1 time problems. We present a result which we believe is important for future use in
problems in Mathematical Physics related to C∗-Algebras.
]
S We consider a finite state set S and a stationary continuous time Markov Chain
D X , t ≥ 0, taking values on S. We denote by Ω the set of paths w taking values
t
h. on S (the elements w are locally constant with left and right limits and are also
t right continuous on t). We consider an infinitesimal generator L and a stationary
a
m vector p . We denote by P the associated probability on (Ω,B). All functions f we
0
[ consider bellow are in the set L∞(P).
FromtheprobabilityP wedefineaRuelleoperator Lt,t ≥ 0, actingonfunctions
1
v f : Ω → R of L∞(P). Given V : Ω → R, such that is constant in sets of the form
1 {X = c}, we define a modified Ruelle operator L˜t ,t ≥ 0, in the following way:
0 0 V
there exist a certain f such that for each t we consider the operator acting on g
5 V
2 given by
03. L˜tV(g)(w) = [f1V Lt(eR0t(V◦Θs)(.)dsgfV)](w)
8
We are able to show the existence of an eigenfunction uand an eigen-probability
0
v: νV on Ω associated to L˜tV,t ≥ 0.
We also show the following property for the probability ν : for any integrable
i V
X g ∈ L∞(P) and any real and positive t
r
a
e−R0t(V◦Θs)(.)ds [(L˜tV (g)) ◦θt]dνV = gdνV
Z Z
This equation generalize, for the continuous time Markov Chain, a similar one
for discrete time systems (and which is quite important for understandingthe KMS
states of certain C∗-algebras).
∗Instituto de Matema´tica, UFRGS, Porto Alegre 91501-970, Brasil. Beneficiary of grant given by
CAPES - PROCAD.
†DepartamentodeMatema´tica,UFSC,Florianopolis88040-900,Brasil. PartiallysupportedbyCNPq,
Instituto do Milenio, PRONEX - Sistemas Dinaˆmicos.
‡Instituto de Matema´tica, UFRGS, Porto Alegre 91501-970, Brasil. Partially supported by CNPq,
Instituto doMilenio,PRONEX-SistemasDinaˆmicos,BeneficiaryofgrantgivenbyCAPES-PROCAD.
fax number (55)51-33087301
1
1 Introduction
We want to extend the concept of Ruelle operator to continuous time Markov Chains. In
order to do that we need a probability a priori on paths. This fact is not explicit in the
discrete time case (thermodynamic formalism) but it is necessary here.
We consider a continuous time stochastic process: the sample paths are functions of
the positive real line R = {t ∈ R: t ≥ 0} taking values in a finite set S with n elements,
+
that we denote by S = {1,2,...,n}. Now, consider a n by n real matrix L such that:
1) 0 < −L , for all i ∈ S,
ii
2) L ≥ 0, for all i 6= j, i ∈ S,
ij
3) n L = 0 for all fixed j ∈ S.
i=1 ij
WePpoint out that, by convention, we are considering column stochastic matrices and
not line stochastic matrices (see [N] section 2 and 3 for general references).
Wedenoteby Pt = etL thesemigroup generated byL. The left actionofthesemigroup
can be identified with an action over functions from S to R (vectors in Rn) and the right
action can be identified with action on measures on S (also vectors in Rn).
The matrix etL is column stochastic, since from the assumptions on L it follows that
1
(1,...,1)etL = (1,...,1)(I +tL+ t2L2 +···) = (1,...,1).
2
It is well known that there exist a vector of probability p = (p1,p2,...,pn) ∈ Rn such
0 0 0 0
that etL(p ) = Ptp = p0 for all t > 0. The vector p is a right eigenvector of etL. All
0 0 0
entries pi are strictly positive, as a consequence of hypothesis 1.
0
Now, let’s consider the space Ω˜ = {1,2,...,n}R+ of all functions from R to S. In
+
principlethisseemstobeenoughforourpurposes, buttechnicaldetailsintheconstruction
of probability measures on such a space force us to use a restriction: we consider the space
Ω ⊂ Ω˜ as the set of right-continuous functions from R to S, which also have left limits
+
in every t > 0. These functions are constant in intervals (closed in the left and open in
the right). In this set we consider the sigma algebra B generated by the cylinders of the
form
{w = a ,w = a ,w = a ,...,w = a },
0 0 t1 1 t2 2 tr r
where t ∈ R , r ∈ Z+,a ∈ S and 0 < t < t < ... < t . It is possible to endow Ω
i + i 1 2 r
with a metric, the Skorohod-Stone metric d, which makes Ω complete and separable ([EK]
section 3.5), but the space is not compact.
Now we can introduce a continuous time version of the shift map as follows: we
define for each fixed s ∈ R the B-measurable transformation Θ : Ω → Ω given by
+ s
Θ (w ) = w (we remark that Θ is also a continuous transformation with respect to
s t t+s s
the Skorohod-Stone metric d).
For L and p fixed as above we denote by P the probability on the sigma-algebra B
0
defined for cylinders by
P({w = a ,w = a ,...,w = a }) = Ptr−tr−1...Pt2−t1Pt1 pa0.
0 0 t1 1 tr r arar−1 a2a1 a1a0 0
2
For further details of the construction of this measure we refer the reader to [B].
The probability P on (Ω,B) is stationary in the sense that for any integrable function
f, and, any s ≥ 0
f(w)dP(w) = (f ◦Θ )dP(w).
s
Z Z
From now on the Stationary Process defined by P is denoted by X and all functions
t
f we consider are in the set L∞(P).
There exist a version of P such that for a set of full measure we have that all sample
elements w are locally constant on t, with left and right limits, and w is right continuous
on t. We consider from now on such probability P acting on this space.
From P we are able to define a continuous time Ruelle operator Lt, t > 0, acting
on functions f : Ω → R of L∞(P). It’s also possible to introduce the endomorphism
α : L∞(P) → L∞(P) defined as
t
α (ϕ) = ϕ◦Θ , ∀ϕ ∈ L∞(P).
t t
Werelateinthenext sectiontheconditionalexpectationwithrespect totheσ-algebras
F+ with the operators Lt and α , as follows:
t t
[Lt(f)](Θ ) = E(f|F+).
t t
Given V : Ω → R, such that it is constant insets of theform{X = c} (i.e., V depends
0
only on the value of w(0)), we consider a Ruelle operator family L˜t , for all t > 0, given
V
by
1
L˜tV(g)(w) = [ f Lt(eR0t(V◦Θs)(.)dsgfV)](w),
V
for any given g, where f is a fixed function.
V
We are able to show the existence of an eigen-probability ν on Ω, for the family L˜t ,
V V
for all t > 0, such that satisfies:
Theorem A. For any integrable g ∈ L∞(P) and any positive t
1
Z [ eR0t(V◦Θs)(.)ds fV ] [ Lt( [ eR0t(V◦Θs)(.)dsfV] g ) ◦Θt ] dνV = Z gdνV.
The above functional equation is a natural generalization (for continuous time) of the
similar one presented in Theorem 7.4 in [EL1] and proposition 2.1 in [EL2] .
In [EL1] and [EL2] the important probability in the Bernoulli space is an eigen-
probability ν for the Ruelle operator associated to a certain potential V = logH :
{1,2,..,d}N → R. This probability ν satisfies: for any m ∈ N, g ∈ C(X),
λ E (λ−1g)dν = gdν,
m m m
Z Z
where
Hβ[m] Hβ(x)Hβ(σ(x))...Hβ(σn−1(x))
λ (x) = (x) = =
m
Λ[m] Λ[m](x)
3
elog(Hβ(x))+log(Hβ(σ(x)))+...+log(Hβ)(σn−1(x))
,
Λ[m](x)
and σ is the shift onthe Bernoulli space {1,2,..,d}N. Here E (f) = E(f|σ−m(B)) denotes
m
the expected projection (with respect to a initial probability P on the Bernoulii space)
on the sigma-algebra σ−m(B), where B is the Borel sigma-algebra and Λ[m] is associated
to the Jones index.
We refer the reader to [CL] for a Thermodynamic view of C∗-Algebras which include
concepts like pressure, entropy, etc..
We believe it will be important in the analysis of certain C∗ algebras associated to
continuous time dynamical systems a characterization of KMS states by means of the
above theorem. We point out, however, that we are able to show this theorem for a
certain ρ just for a quite simple function V as above. In a forthcoming paper we will
V
consider more general potentials V.
One could consider a continuous time version of the C∗-algebra considered in [EL1].
We just give an idea of what we are talking about. Given the above defined P for each
t > 0, denote by s : L2(P) → L2(P), the Koopman operator, where for η ∈ L2(P) we
t
define (s η)(x) = η(θ (x)).
t t
Another important class of linear operators is M : L2(P) → L2(P), for a given fixed
f
f ∈ C(Ω), and defined by M (η)(x) = f(x)η(x), for any η in L2(P). We assume that f
f
is such defines an operator on L2(P) (remember that Ω is not compact).
In this way we can consider a C∗-algebra generated by the above defined operators
(for all different values of t > 0), then the concept of state, and finally given V and β
we can ask about KMS states. There are several technical difficulties in the definition of
the above C∗-algebra, etc... Anyway, at least formally, there is a need for finding ν which
is a solution of an equation of the kind we describe here. We need this in order to be
able to obtain a characterization of KMS states by means of an eigen-probability for the
continuous time Ruelle operator. This setting will be the subject of a future work. This
was the motivation for our result.
With the operators α and L we can rewrite the theorem above as
ρ (G−1E (G ϕ)) = ρ (ϕ),
V T T T V
for all ϕ ∈ L∞ and all T > 0, where, as usual, ρ (ϕ) = ϕdρ , E = α LT is in fact a
V V T T
projection on a subalgebra of B, and G : Ω → R is given by
T
R
T
G (x) = exp( V(x(s))ds).
T
Z0
For the map V : Ω → R, which is constant in cylinders of the form {w = i},
0
i ∈ {1,2,...,n}, we denote by V the corresponding value. We also denote by V the
i
diagonal matrix with the i-diagonal element equal to V .
i
Now, consider Pt = et(L+V). The classical Perron-Frobenius Theorem for such semi-
V
group will be one of the main ingredients of our main proof.
4
As usual, we denote by F the sigma-algebra generated by X . We also denote by F+
s s s
the sigma-algebra generated σ({X ,s ≤ u}). Note that a F+-measurable function f(w)
u s
on Ω does depend of the value w .
s
We also denote by I the indicator function of a measurable set A in Ω.
A
2 A continuous time Ruelle Operator
We consider the disintegration of P given by the family of measures, indexed by the
elements of Ω and t > 0 defined as follows: first, consider a sequence 0 = t < t < ··· <
0 1
t < t ≤ t < ··· < t . Then for w ∈ Ω and t > 0 we have on cylinders:
j−1 j r
µw([X = a ,...,X = a ]) =
t 0 0 tr r
1 Pt−tj−1 ···Pt2−t1Pt pa0 if a = w(t ),...,a = w(t )
pw(t) w(t)aj−1 a2a1 a1a0 0 j j r r
0
( 0 otherwise.
Proposition 2.1. µw is the disintegration of P along the fibers Θ−1(.).
t t
Proof: It is enough to show that for any integrable f
fdP = f(x)dµw(x)dP(w).
t
ZΩ ZΩZΘ−t1(w)
For doing that we can assume that f is in fact the indicator of the cylinder [X =
0
a ,...,X = a ]; then the right hand side becomes
0 tr r
1
fdµw(x)dP(w) = I Pt−tj−1 ···Pt1 pa0dP(w) =
t [w(tj)=aj,...,w(tr)=ar]pw(t) w(t)aj−1 a1a0 0
Z Z Z 0
n
1
I Pt−tj−1 ···Pt1 pa0dP(w) =
[w(t)=a,w(tj)=aj,...,w(tr)=ar]pw(t) w(t)aj−1 a1a0 0
a=1Z 0
X
n
1
Ptr−tr−1...Ptj−tpa Pt−tj−1 ···Pt1 pa0 = Ptr−tr−1 ···Pt1 pa0 =
arar−1 aja 0pa aaj−1 a1a0 0 arar−1 a1a0 0
a=1 0
X
P([X = a ,...,X = a ]) = fdP.
0 0 tr r
Z
In the second inequality we use the fact that P is stationary.
The proof for a general f follows from standard arguments.
Definition 2.2. For t fixed we define the operator Lt : L∞(Ω,P) → L∞(Ω,P) as follows:
Lt(ϕ)(x) = ϕ(y¯)dµx(y¯).
t
Zy¯∈Θ−t1(x)
5
Remark 2.3. The definition above can be rewritten as
Lt(ϕ)(x) = ϕ(yx)dµx(yx),
t
Zy∈D[0,t)
where the symbol yx means the concatenation of the path y with the translation of x:
y(s) if s ∈ [0,t)
xy(s) =
x(s−t) if s ≥ t,
(cid:26)
and, D[0,t) is the set of right-continuous functions from [0,t) to S. This follows simply
from the fact that, in this notation, Θ−1(x) = {yx: y ∈ D[0,t)}.
t
Note that the value lim y(s) do not have to be necessarily equal to x(0).
s→t
In order to understand better the definitions above we apply the operator to some
simple functions. For example, we can see the effect of Lt on some indicator function of a
given cylinder: consider the sequence 0 = t < t < .. < t < t ≤ t < ... < t and then
0 1 j−1 j r
takef = I . Then, forapathz ∈ Ωsuchthatz = a ,...,z = a
{X0=a0,Xt1=a1,...,Xtr=ar} tj−t j tr−t r
(the future condition) we have
1
Lt(f)(z) = Pt−tj−1...Pt2−t1Pt1 pa0,
pz0 z0aj−1 a2a1 a1a0 0
0
otherwise (i.e., if the path z does not satisfy the condition above) we get Lt(f)(z) = 0.
Note that if t < t, then Lt(f)(z) depends only on z . For example, if f = I
r 0 {X0=i0}
then
1
Lt(f)(z) = I (yx)dµz(yx) = µz([X = i ]) = Pt pi0.
{X0=i0} t t 0 0 pz0 z0,i0 0
Zy∈D[0,t) 0
In the case f = I , then Lt(f)(z) = Pt pi00, if z = j , and Lt(f)(z) = 0
{X0=i0,Xt=j0} z0,i0pz0 0 0
0
otherwise.
We describe bellow some properties of Lt.
Proposition 2.4. Lt(1) = 1, where 1 is the function that maps every point in Ω to 1.
Proof: Indeed,
Lt(1)(x) = 1(yx)dµx(yx) = dµx(yx) = µx([X = x(0)]) =
t t t t
Zy∈D[0,t) Z
n n
1
µx([X = a,X = x(0)]) = Pt pa = 1.
t 0 t px(0) x(0)a 0
a=1 0 a=1
X X
We can also define the dual of Lt, denoted by (Lt)∗, acting on the measures. Then we
get:
6
Proposition 2.5. For any positive t we have that (Lt)∗(P) = (P).
Proof: For a fixed t we have that (Lt)∗(P) = (P), because for any f of the form
f = I , 0 = t < t < .. < t < t ≤ t < ... < t , we get
{X0=a0,Xt1=a1,...,Xtr=ar} 0 1 j−1 j r
n
Lt(f)(z)dP(z) = Lt(f)(z)dP(z) =
Z b=1 Z{X0=b}
X
n
1
I (z)dP(z) Pt−tj−1...Pt2−t1Pt1 pa0 =
{X0=b,Xtj−t=aj,...,Xtr−t=ar} pb baj−1 a2a1 a1a0 0
b=1 Z 0
X
n
1
P({X = b,X = a ,X = a ,...,X = a }) Pt−tj−1...Pt2−t1Pt1 pa0 =
0 tj−t j tj+1−t j+1 tr−t r pb baj−1 a2a1 a1a0 0
b=1 0
X
n
1
Ptr−tr−1...Ptj+1−tjPtj−tpb Pt−tj−1...Pt2−t1Pt1 pa0 =
arar−1 aj+1aj ajb 0 pb baj−1 a2a1 a1a0 0
b=1 0
X
f(w)dP(w).
Z
Proposition 2.6. Given t ∈ R , and the functions ϕ,ψ ∈ L∞(P), then
+
Lt(ϕ×(ψ ◦Θ ))(z) = ψ(z)×Lt(ϕ)(z).
t
Proof:
Lt(ϕ(ψ ◦Θ ))(x) = ϕ(ix)(ψ ◦Θ )(ix)dµx(i) =
t t t
Zi∈D[0,t)
ψ(x) ϕ(ix)dµx(i) = (ψLt(ϕ))(x) = ψ(x)Lt(ϕ)(x),
t
Zi∈D[0,t)
since ψ ◦Θ (ix) = ψ(x), independently of i.
t
We just recall that the last proposition can be restated as
Lt(ϕα (ψ)) = ψLt(ϕ).
t
Then we get:
Proposition 2.7. α is the dual of Lt on L2(P).
t
7
Proof: From last two propositions
Lt(f)gdP = Lt(f ×(g ◦Θ ))dP = f ×(g ◦Θ )dP = fα (g)dP,
t t t
Z Z Z Z
as claimed.
We want to obtain conditional expectations in a more explicit form. For a given
f, recall that the function Z(w) = E(f|F+) is the Z (almost everywhere defined) F+-
t t
measurable function such that for any F+-measurable set B we have Z(w)dP(w) =
t B
f(w)dP(w).
B R
RProposition 2.8. The conditional expectation is given by
E(f|F+)(x) = fdµx.
t t
Z
Proof: For t fixed, consider a F+-measurable set B. Then we have
t
fdµwdP(w) = (I (w) fdµw)dP(w) =
t B t
ZBZ Z Z
(fI )dµwdP(w) = f(w)I (w)dP(w) = fdP.
B t B
Z Z Z ZB
Now we can relate the conditional expectation with respect to the σ-algebras F+ with
t
the operators Lt and α as follows:
t
Proposition 2.9. [Lt(f)](Θ ) = E(f|F+).
t t
Proof: ThisfollowsfromthefactthatforanyB = {X = b ,X = b ,...,X = b },
s1 1 s2 2 su u
with t < s < ... < s , we have I = I ◦Θ for some measurable A and
1 u B A t
Lt(f)(Θ (w))dP(w) = I (w)Lt(f)(Θ (w))dP(w) =
t B t
ZB Z
(I ◦Θ )(w)Lt(f)(Θ (w))dP(w) = I (w)Lt(f)(w)dP(w) =
A t t A
Z Z
I (Θ (w))f(w)dP(w)= f(w)dP(w).
A t
Z ZB
8
3 The modified operator Ruelle Operator associated
to V
We are interested in the perturbation by V (defined above) of the Lt operator.
Definition 3.1. We define G : Ω → R as
t
t
G (x) = exp( V(x(s))ds)
t
Z0
Definition 3.2. We define the G-weigthed transfer operator Lt : L∞(Ω,P) → L∞(Ω,P)
V
acting on measurable functions f (of the form f = I ) by
{X0=a0,Xt1=a1,...,Xtr=ar}
Lt (f)(w) := Lt(G f) =
V t
n
= Lt(eR0t(V◦Θs)(.)dsf ) = Lt(eR0t(V◦Θs)(.)dsI{Xt=b}f )(w).
b=1
X
Note that eR0t(V◦Θs)(.)dsI{Xt=b} does not depend on information for time larger than t.
In the case f is such that t ≤ t (in the above notation), then Lt (f)(w) depends only on
r V
w(0).
The integration on s above is consider over the open interval (0,t).
We will show next the existence of an eigenfunction and an eigen-measure for such
operator Lt . First we need the following:
V
Theorem (Perron-Frobenius for continuous time). ([S] page 111) Given L, p and
0
V as above, there exists
a) a unique positive function u : Ω → R, constant equal to the value ui in each
V V
cylinder X = i, i ∈ {1,2,..,n}, (sometimes we will consider u as a vector in Rn).
0 V
b) a unique probability vector µ in Rn(a probability over over the set {1,2,..,n} such
V
that µ ({i}) > 0, ∀i), that is,
V
n
(u ) (µ ) = 1,
V i V i
i=1
X
c) a real positive value λ(V), such that for any positive s
e−sλ(V)u es(L+V) = u .
V V
d) Moreover, for any v = (v ,.,v ) ∈ Rn
1 n
n
lim e−tλ(V)vet(L+V) = ( v (µ ) )u ,
i V i V
t→∞
i=1
X
9
e) for any positive s
e−sλ(V)es(L+V)µ = µ .
V V
From property e) it follows that
(L+V)µ = λ(V)µ ,
V V
or
(L+V −λ(V)I)µ = 0.
V
From c) it follows that
u (L+V) = λ(V)u ,
V V
or
u (L+V −λ(V)I) = 0.
v
We point out that e) means that for any positive t we have (Pt)∗µ = eλ(V)tµ .
V V V
Note that when V = 0, then λ(V) = 0, µ = p0 and u is constant equal to 1.
V V
Now we return to our setting: for each i and t fixed one can consider the probability
0
µt defined over the sigma-algebra F− = σ({X |s ≤ t}) with support on {X = i } such
i0 t s 0 0
that for cylinder sets with 0 < t < ... < t ≤ t
1 r
µt ({X = i ,X = a ,...,X = a ,X = j }) = Pt−tr...Pt2−t1Pt1 .
i0 0 0 t1 1 tr−1 r−1 t 0 j0ar a2a1 a1i0
The probability µt is not stationary.
i0
We denote by Q(j,i) the i,j entry of the matrix et(L+V), that is (et(L+V)) .
t j,i
It is known ([K] page 52 or [S] Lemma 5.15) that
Q(j0,i0)t = E{X0=i0}{eR0t(V◦Θs)(w)ds; X(t) = j0} =
I{Xt=j0} eR0t(V◦Θs)(w)dsdµti0(w).
Z
For example,
I{Xt=j0}eR0t(V◦Θs)(w)dsdP = Q(j0,i)tp0i.
Z i=1,2,..,n
X
In the particular case where V is constant equal 0, then p0 = µ and λ(V) = 0.
V
We denote by f = f , where f(w) = f(w(0)), the density of probability µ in S with
V V
respect to the probability p0 in S.
Therefore, fdp0 = 1.
Proposition R3.3. f (w) = µV(w) = (µV)w(0), f : Ω → R, is an eigenfunction for Lt
V p0(w) (p0)w(0) V V
with eigenvalue etλ(V).
10