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Potential Analysis 21: 47–74, 2004. © 2004 Kluwer Academic Publishers. Printed in the Netherlands. 47 Right Dual Process for Semidynamical Systems MOUNIR BEZZARGA Département de Mathématiques et Physique, Institut Préparatoire aux Etudes d’Ingénieurs de Tunis, 2 rue Jawaher Lel Nehru, 1008 Montfleury-Tunis, Tunisie (e-mail: [email protected]) (Received: 13 December 2001; accepted: 9 July 2003) Abstract. We construct a dual semigroup of kernels associated to a semidynamical system. The above semigroup is in duality with the deterministic semigroup defined by Koopmann. We also prove the existence of right dual process associated to a semidynamical system. Mathematics Subject Classifications (2000): Primary: 47D07; secondary: 60J, 31D05, 58F99. Key words: Koopmann operator, resolvent, right-process, semidynamical system, sub-Markovian semigroup. 0. Introduction The semidynamical systems arise from a dynamical interpretation of functional differential equations with time lag (i.e. Cauchy problem) and evolution type of partial differential equations (i.e. the heat diffusion equation). Starting with a semidynamical system (X, B, , ω) (cf. [3–7]), we construct a metric on X 0 = X \{ω} and we construct a so-called dual semigroup P = (P t ) t R + associated to with respect to the Lebesgue measure given in [4, 5]. It is shown that the dual resolvent V = (V α ) αR + associated to with re- spect to constructed in [4, 5], coincides with the resolvent associated to the semigroup P. We prove that the semigroup P is in duality with the deterministic semigroup H = (H t ) t R + , of Koopmann operators, defined in [19]. The above semigroup H was introduced in [19] in order to transform a nonlinear problem in finite dimension to a linear one in infinite dimension. The potential analysis of the semigroup H in the case of dynamical systems with real parameters is introduced in [18]. We apply the so-called Doob u-transformation to the semi-group P, in order to associate a right process. We also characterize the associated invariant probabilities. 1. Preliminaries In this section, we will introduce some definitions and we establish some proposi- tions which will be useful in the remainder of this paper (for more details see [3–5, 7] and [24]).

Right Dual Process for Semidynamical Systems

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Potential Analysis 21: 47–74, 2004.© 2004 Kluwer Academic Publishers. Printed in the Netherlands.

47

Right Dual Process for Semidynamical Systems

MOUNIR BEZZARGADépartement de Mathématiques et Physique, Institut Préparatoire aux Etudes d’Ingénieurs deTunis, 2 rue Jawaher Lel Nehru, 1008 Montfleury-Tunis, Tunisie(e-mail: [email protected])

(Received: 13 December 2001; accepted: 9 July 2003)

Abstract. We construct a dual semigroup of kernels associated to a semidynamical system. Theabove semigroup is in duality with the deterministic semigroup defined by Koopmann. We also provethe existence of right dual process associated to a semidynamical system.

Mathematics Subject Classifications (2000): Primary: 47D07; secondary: 60J, 31D05, 58F99.

Key words: Koopmann operator, resolvent, right-process, semidynamical system, sub-Markoviansemigroup.

0. Introduction

The semidynamical systems arise from a dynamical interpretation of functionaldifferential equations with time lag (i.e. Cauchy problem) and evolution type ofpartial differential equations (i.e. the heat diffusion equation).

Starting with a semidynamical system (X,B,�,ω) (cf. [3–7]), we construct ametric on X0 = X \ {ω} and we construct a so-called dual semigroup P = (Pt)t∈R+associated to � with respect to the Lebesgue measure given in [4, 5].

It is shown that the dual resolvent V∗ = (V ∗α )α∈R+ associated to � with re-

spect to constructed in [4, 5], coincides with the resolvent associated to thesemigroup P. We prove that the semigroup P is in duality with the deterministicsemigroup H = (Ht)t∈R+ , of Koopmann operators, defined in [19]. The abovesemigroup H was introduced in [19] in order to transform a nonlinear problem infinite dimension to a linear one in infinite dimension. The potential analysis of thesemigroup H in the case of dynamical systems with real parameters is introducedin [18]. We apply the so-called Doob u-transformation to the semi-group P, inorder to associate a right process. We also characterize the associated invariantprobabilities.

1. Preliminaries

In this section, we will introduce some definitions and we establish some proposi-tions which will be useful in the remainder of this paper (for more details see [3–5,7] and [24]).

48 MOUNIR BEZZARGA

In [3, 7], we gave the notion of a semidynamical system as being a separablemeasurable space (X,B) with a distinguished point ω and a measurable map � :R+ ×X → X having the following properties:

(S1) For any x in X, there exists an element ρ(x) in R+ such that �(t, x) �= ω forall t ∈ [0, ρ(x)) and �(t, x) = ω for all t � ρ(x). Moreover ρ(ω) = 0 andρ(x) > 0 for all x ∈ X, x �= ω.

(S2) For any s, t ∈ R+ and any x ∈ X, we have �(s,�(t, x)) = �(s + t, x).(S3) �(0, x) = x, for all x ∈ X.(S4) If �(t, x) = �(t, y) for all t ∈ R

∗+, then x = y.

DEFINITION 1. The collection (X,B,�,ω) with the above properties is calleda semidynamical system having ω as coffin state.

Set X0 = X \ {w}, B0 = {A ∈ B;A ⊂ X0} and for any x ∈ X0, we denote by �xthe trajectory of x, i.e.:

�x = {�(t, x); t ∈ [0, ρ(x))}.So for any x, y ∈ X0, we put x � y if y ∈ �x .

We also define the function �x on [0, ρ(x)) by �x(t) = �(t, x).

DEFINITION 2. We say (cf. [3, 15]) that (X,B,�,ω) is a transient semidynam-ical system if moreover there exists a sequence (An)n ∈ BN such that X0 = ⋃

n An

and for any x ∈ X0, λ(�−1x (An)) < ∞, where λ denotes the Lebesgue measure

on R.

HYPOTHESIS. In what follows, we shall suppose that (X,B,�,ω) is a transientsemidynamical system.

Under this assumption (cf. [3]) �x becomes a measurable isomorphism betweenthe interval [0, ρ(x)) and the trajectory �x endowed with the trace measurablestructures.

Therefore “�” becomes an order on X0.

DEFINITION 3. A maximal trajectory is a totally ordered subset � of X0 withrespect to the above order, such that there is no minorant x0 of �, x0 ∈ X0 \ � andfor any x ∈ �, we have �x ⊂ �.

DEFINITION 4. A point x ∈ X0 is said to be a start point if it is minimalwith respect to “�”. x is said to be a branching point if there exist two maximaltrajectories �1 and �2 such that �1 �= �2 and �1 ∩ �2 = �x .

The set of all start points is denoted by S and the set of all branching points isdenoted by B.

The set S is namely the entrance boundary of the semidynamical system (X,B,

�,ω).

RIGHT DUAL PROCESS FOR SEMIDYNAMICAL SYSTEMS 49

1.1. EXAMPLES

(i) As an example of parabolic evolution type, we consider an analogous of the heatequation problem in the Banach space

�2 ={x = (ζn)n ⊂ R;

∑n

ζ 2n < +∞

},

endowed with the norm ‖x‖2 = (∑

n ζ2n )

1/2. The corresponding semidynamicalsystem � is defined (cf. [7]) by �(t, x) = (ηn)n, where ηn = ζn exp(−(n +12)

2π2t). As limt→+∞ ‖�(t, x)‖2 = 0, the null sequence ! is a coffin state andthe semidynamical system is transient.

Every x = (ζn)n ∈ �2 such that∑

n n2ζ 2n = +∞, is a start point of�. Moreover,

we remark that any start point is polar in the sence of [14], Chap. XII.(ii) The second example deals with the absence of left uniqueness in finite

dimension. Let X be the subset of R2 of the points xs = (s, 0), s ∈ [0, 1] and

xns = (s, n), s ∈ (−∞, 0), n ∈ Z∗+. The semidynamical system on the phase space

X is defined by parallel uniform movement to the right: �(t, xs) = xs+t and

�(t, xns ) ={xns+t if s + t < 0,

xs+t if not,

where xs+t = (1, 0) if s + t � 1. The point (0, 0) is a branching point for � andwe don’t have any minimal point.

(iii) Next consider an example of a semidynamical system which arises fromthe functional differential equation

x(t) = maxt−1�s�t

|x(s)|, (")

with initial functions space X = C([−1, 0],R) the collection of all continuousfunctions from [−1, 0] to R endowed with ‖·‖∞ the uniform norm on X. As in theexample (ii), there is absence of negative uniqueness. Indeed, let x, y ∈ X suchthat ‖x‖∞ = ‖y‖∞ = 1, x(s) = y(s) for −1/2 � s � 0, and x(0) = y(0) = 1.Then �(t, x) = �(t, y) for t � 1/2.

In [3] we have associated a sub-Markovian resolvent V = (Vα)α∈R+ of kernelson the measurable space (X0,B0), defined by

Vαf (x) =∫ ρ(x)

0e−αtf (�(t, x)) dt, ∀x ∈ X0.

It is shown in [15] that the resolvent V is proper. The family V is also the resolventassociated to the deterministic semigroup H = (Ht)t∈R+ introduced in [18, 19] andgiven on (X0,B0) by

εxHt ={ε�(t,x) if t ∈ [0, ρ(x)),0 if t � ρ(x).

50 MOUNIR BEZZARGA

On the other hand, it is proved in [3] that the map �x is a measurable isomor-phism between the interval [0, ρ(x)) and the trajectory �x endowed with the tracemeasurable structures.

Let (cf. [4]) be the Lebesgue measure associated to the semidynamical system(X,B,�,ω) and given by

(A) = λ(�−1x (A)),

for any x ∈ X0, A ∈ B0 and A ⊂ �x . In the sequel, we shall give some detailsabout the construction of this measure . In fact, for any x ∈ X0, we denote by λxthe measure on the trajectory �x given by

λx(A) = (�−1x (A)), for A ∈ B

and A ⊂ �x .Also, for any countable family (�xn)n of trajectories in X0, we denote by λM

the unique measure on the set M = ⋃n �xn such that λM(A) = λxn(A) whenever

A ∈ B and A ⊂ �xn .So, the Lebesgue measure is given by

(A) = supM

λM(A ∩M),

for any Borel subset A of X0, where M runs the set Bσ0 of all countable union of

trajectories of X0.We recall (cf. [5]) that in the same way can be defined on the σ -algebra

B0() of all subsets A of X0 such that A∩M ∈ B0 for any countable union M oftrajectories of X0. Also remark that the measure is σ -finite on each M in Bσ

0 .One can show that the resolvent family V may be considered on the measurable

space (X0,B0()) and we denote by F (X0,) the set of all positive B0()-measurable functions on X0.

We recall the following definition (cf. Chapter III in [17]).

DEFINITION 5. We call arrival time, the map defined on X0 ×X0 as follows

((x, y) ={t if x � y and y = �(t, x), t ∈ [0, ρ(x)),+∞ if not.

In the example (i) let x = (ζn)n ∈ �2 such that x is not a start point, then x is notnull and (nαζn)n ∈ �2 for any α > 0. Let y = (ηn)n ∈ �2 such that ((y, x) = t ,t > 0, i.e. ηn = ζnexp((n + 1

2 )2π2t), so ζn = 0 if and only if ηn = 0. If (ζnk )k is

the subsequence of all ζn such that ζn �= 0, then ((y, x) = 4π2(2nk+1)2

Log(ηnkζnk).

Note that if x � y, then ([x, y]) = ((x, y), where [x, y] is an interval oforder.

It is shown that the arrival time function ( is measurable if we endow X0 ×X0

with the product measurable structure of the σ -algebra B0() (cf. [4–6]).In fact, we have the following:

RIGHT DUAL PROCESS FOR SEMIDYNAMICAL SYSTEMS 51

THEOREM 6. If (X;B) is a Lusin space in the sence of [11], then the arrivaltime function ( is measurable on X0 × X0 with respect to the product σ -algebraof B0.

Proof. Since (X;B) is a separable measurable space and the map � : R+ ×X → X is measurable, therefore the graph of � is measurable in R+×X×X withrespect to the product σ -algebra.

On the other hand the set

O := {(x, y) ∈ X0 ×X0, x � y}is measurable in X0 ×X0 with respect to the product σ -algebra of B0 (cf. [3]). Sothe set

M := ([0,∞)× O) ∩ {(t, x,�(t, x)); t ∈ [0,∞), x ∈ X}is a measurable subset of the Lusin space R+ ×X0 ×X0.

The projection p defined on M by

p(t, x, y) := (x, y),

is a measurable injection and the set X0 × X0 is a Lusin space. Since for anyα ∈ R

∗+, M ∩ ([0, α) × X0 × X0) is a measurable subset of R+ × X0 × X0, thenby the Lusin’s theorem argument the set

{(x, y) ∈ X0 ×X0;((x, y) < α} = p(M ∩ ([0, α)×X0 ×X0))

is measurable with respect to the product σ -algebra of B0. ✷COROLLARY 7. If (X,B) is a separable measurable space, then the arrivaltime function ( is measurable on X0 × X0 with respect to the product σ -algebraof B0(). ✷

Proof. Since the semi-dynamical system (X,B,�,ω) is transient, then for anyx ∈ X0, the trajectory �x of x is isomorphic to the real interval [0, ρ(x)). There-fore, for any x ∈ X0, �x is a Lusin measurable space. We deduce that, for anyM ∈ Bσ

0 ,( is measurable on M×M with respect to the trace measurable structureof the product σ -algebra of Bσ

0 and so ( is measurable on X0 × X0 with respectto the product σ -algebra of B0(). ✷Using the definition of the Lebesgue measure and of the resolvent V = (Vα)α∈R+associated to (X,B,�,ω), we obtain, on the measurable space (X0,B0()), that

Vαf (x) =∫�x

e−α((x,y)f (y) d(y)

=∫

e−α((x,y)G(x, y)f (y) d(y), ∀x ∈ X0,

52 MOUNIR BEZZARGA

where

G(x, y) ={

1 if x � y,

0 if not,

is the Green function associated to (X,B,�,ω).Next, we shall give an example in finite dimension of Cauchy problem in order

to show how to get the Lebesgue measure associated to a semidynamical system.The semidynamical system � generated by the differential equation d

dt x = −x2

with initial condition in X = (0,+∞) is given on R+ ×X by

�(t, x) = x

xt + 1.

Note that for any x, y ∈ X, we have supt∈R+ |�(t, x) − �(t, y)| � |x − y| andtherefore the family of mappings (�(t, x))t∈R+ is equicontinuous. The associatedarrival time is given on X×X by ((x, y) = 1/y − 1/x and we have x � y if andonly if x − y ∈ R+.

Let f be a positive Borel function on X, then for any x ∈ X we have∫�x

f (y) d(y) =∫ +∞

0f (�(t, x)) dt =

∫ +∞

0f

(x

xt + 1

)dt.

Set y = x/(xt + 1), then we get∫�x

f (y) d(y) =∫ +∞

0f (y)χ(0,x](y)

dy

y2.

Let (xn)n be an increasing sequence of X with respect to the natural order such that

limn→+∞ xn = +∞.

Then X = ⋃n �xn and by monotone convergence theorem we get,∫

X

f (y) d(y) := limn→+∞

∫�xn

f (y) d(y) =∫ ∞

0f (y)

dy

y2.

Consequently, we deduce that d = dy/y2 is the density measure with respect tothe Lebesgue measure on X.

Using the Lebesgue measure and the function (, we associate to the semi-dynamical system a dual resolvent V∗ = (V ∗

α )α∈R+ of kernels on the measurablespace (X0,B0()) with respect to (cf. [5]). This resolvent is given, for f ∈F (X0,), by

V ∗α f (x) =

∫e−α((y,x)G(y, x)f (y) d(y), ∀x ∈ X0.

We consider again the semidynamical system � given on R+ × (0,+∞) by

�(t, x) = x

xt + 1.

RIGHT DUAL PROCESS FOR SEMIDYNAMICAL SYSTEMS 53

Then for any α ∈ R+ and any positive Borel function f on X, we have for eachx ∈ X0,

V ∗α f (x) =

∫ +∞

0e−α(

1x− 1

y )f (y)χ[x,+∞)(y)dy

y2.

DEFINITION 8 (cf. [3]). The fine topology T� associated to the semidynamicalsystem (X,B,�,ω), is the set of all subset D of X0 having the following property

x ∈ D ⇒ [∃ε > 0 : �(t, x) ∈ D,∀t ∈ [0, ε) ∩ [0, ρ(x))].In [3], we characterized the set E , of all V-excessive functions on X0, as beingthe set of all positive B0-measurable functions on X0 which are decreasing withrespect to the associated order “� ” and T�-continuous.

DEFINITION 9 (cf. [4, 9]). Let f be a positive function defined on X0. We saythat f is totally increasing if for any x ∈ X0 and any finite subset A of pairwiseincomparable elements of X0 such that a � x for any a ∈ A, then

∑a∈A f (a) � x.

NOTATION. Let � be a maximal trajectory of X0 and let x be an element of �.We shall denote

�x := {y ∈ �; x � y} = {�(t, x); t ∈ [0, ρ(x))},�−x := {y ∈ �; y � x, y �= x} = � \ �x.

REMARK 3.9 (cf. [4]). If �′, �′′ are two trajectories (or two maximal trajectories)of X0 then we have either �′ ∩�′′ = ∅ or there exists an element x of X0 such that�′x = �′′

x = �x and �′x− ∩ �′′

x− = ∅.

So for any x ∈ X0, we denote by γx the set of all finite family of maximaltrajectories such that x is the smallest common element.

DEFINITION 10 (cf. [4]). Let f be a positive function defined on X0. We say thatf is totally continuous on the left at any x ∈ X0 if for any maximal trajectory �

passing by x, there exists lim y→x

y∈�−x

f (y) with respect to the inherent topology T 0φ

on � and further

f (x) = supH∈γx

{∑�∈H

(limy→x

y∈�−x

f (y)

)}.

f is said totally increasing and totally continuous on the left on X0 if it is totallyincreasing on X0 and totally continuous on the left at every element of X0.

NOTATION. In the sequel, we shall denote by E∗ the set of all excessive functionson (X0,B0()) with respect to the resolvent V∗ = (V ∗

α )α�0.

In the following theorem we shall give a characterization for E∗.

54 MOUNIR BEZZARGA

THEOREM 11 (cf. [4–6]). Let f be an element of F (X0,). Then the followingassertions are equivalent:

(a) f ∈ E∗,(b) f is finite, totally increasing and continuous on the left on X0.

The proof of the following theorem is analogous to the proof of Theorem 2.7 in [4].

THEOREM 12. The following assertions are equivalent:

(i) E is a standard H-cone in the meaning of [8],(ii) E∗ is a standard H-cone,(iii) The H-cone E∗ has a unit in the meaning of [8] (i.e. ∃s∗ ∈ E∗ : s∗(x) > 0,

∀x ∈ X0),(iv) X0 ∈ Bσ

0 ,(v) The Lebesgue measure is σ -finite.

2. On the Inherent Topology Associated to (X,B,�,ω)

Starting with a transient semidynamical system (X,B,�,ω), we have associatedin [3], a suitable topology T 0

� , namely the inherent topology in the meaning of [17]in the dynamical case, as being the set of all subset D of X0 having the followingproperty:

[∀x ∈ X0,∀t0 ∈ [0, ρ(x)) : �(t0, x) ∈ D]⇒ [∃ε > 0 : �(t, x) ∈ D,∀t ∈ (t0 − ε, t0 + ε) ∩ [0, ρ(x))].

We note that the σ -algebra B0() contains the class of Borel sets in X0 withrespect to T 0

� .In the sequel, we shall introduce a metric m on X0 such that T 0

� is finer thanthe associated topology Tm and we give some properties of T 0

� and also give acharacterization of the connected components of X0.

Let x, y ∈ X0, then �x ∩ �y = ∅ or there exists a unique element z of X0 suchthat �x ∩ �y = �z (cf. [4]). Indeed, since the set �x ∩ �y is closed with respect tothe fine topology T� on X0, we deduce that z = �(α, x), where α = inf{((x, u) :u ∈ �x ∩ �y}, is an element of �x ∩ �y and moreover �x ∩ �y = �z. Put

t (x, y) ={((x, z)+ ((y, z) if �x ∩ �y = �z,

+∞ if �x ∩ �y = ∅.

Then we have the following:

PROPOSITION 13. The map m defined on X0 ×X0 by

m(x, y) =

t (x, y)

1 + t (x, y)if �x ∩ �y �= ∅,

1 if �x ∩ �y = ∅,

RIGHT DUAL PROCESS FOR SEMIDYNAMICAL SYSTEMS 55

is a metric on X0. Moreover, the inherent topology T 0� is finer than the suitable

topology Tm.Proof. Let x, y and z be elements of X0, then we have

t (x, y) � t (x, z)+ t (y, z).

Indeed,

(i) If �x ∩ �z = ∅ or �y ∩ �z = ∅ then t (x, z) = +∞ or t (y, z) = +∞.(ii) If �x ∩ �z �= ∅ and �y ∩ �z �= ∅, we also have �x ∩ �y �= ∅.

Consider for example the case where �x ∩ �z = �z1 and �y ∩ �z = �z2 withz1 � z2. Then we obtain

t (x, y) = ((x, z1)+((z1, z2)+((y, z2),

t (x, z) = ((x, z1)+((z, z1),

t (y, z) = ((y, z2)+((z, z1)+((z1, z2).

Then the inequality

m(x, y) � m(x, z)+m(y, z)

holds from the inequality

t (x, y) � t (x, z)+ t (y, z).

One can easily verify that Tm the suitable topology associated with m is includedin T 0

� . ✷PROPOSITION 14. Let x0 ∈ X0. Then x0 has a compact neighborhood if andonly if there is a finite family of maximal trajectories having x0 as first commonpoint.

Proof. Let x0 ∈ X0 be such that there exists a countable or uncountable familyof maximal trajectories (�i)i∈I having x0 as first common point.

A basis of neighborhoods of x0 with respect to T 0� is given by

N ={(⋃

i∈I(xi, x0]

)∪ [x0, y)/x0 < y, xi < x0, xi ∈ �i

},

where (xi, x0] and [x0, y) are intervals of order. Suppose now that K is a compactneighborhood of x0 with respect to T 0

� . Then there exists N ∈ N such that N ⊂ K,where N is the set of adherent points of N with respect to T 0

� . Moreover it is easyto verify that

N =(⋃i∈I

[xi, x0])∪ [x0, y].

For any i ∈ I , we consider x′i , x′′i , x

′′′i ∈ �i such that

x′i < xi < x′′i < x′′′i < x0

56 MOUNIR BEZZARGA

and y′ ∈ �x0 such that y < y′.We put

!0 =(⋃i∈I(x′′i , x0)

)∪ [x0, y

′)

and

!i = (x′i , x′′′i ); i ∈ I,

then we have

N ⊂⋃i

!i.

So N is a compact with respect to T 0� but we can’t extract a finite covering.

Now if there exists a family (zj )j ⊂ B of branching points in !i for some i, wereplace !i by

!′i = !i ∪

(⋃j

[((· , zj ) < εj ])

for some convenient positive real numbers εj .Conversely, suppose that there exists a finite family (�i)1�i�n of maximal tra-

jectories having x0 as first common point. We consider a neighborhood K of x0

having the following form

K =⋃

1�i�n

[xi, y], xi � x0 � y and xi ∈ �i.

Let (!j )j be an arbitrary family of open subsets with respect to T 0� such that

K ⊂ ⋃j !j . So for any i, 1 � i � n and any j, �i ∩!j is open in �i with respect

to the trace topology of T 0� and [xi, y] is a compact with respect this topology.

Then we can extract a finite covering of K.Next, if x0 ∈ S is a start point, then [x0,�(ε, x0)] is a compact neighborhood

of x0 with respect to T 0� for any ε ∈ (0, ρ(x0)). ✷

COROLLARY 15. The space X0 is locally compact with respect to T 0� if and only

if for any x0 ∈ X0, there exists a finite family of maximal trajectories having x0 asfirst common point.

COROLLARY 16. If the inherent topology T 0� coincides with Tm, then (X0,T

0�)

is locally compact.Proof. If (X0,T

0�) is not locally compact, then by the above corollary there

exists x0 ∈ X0 such that there is a non-finite family (�i)i of maximal trajectorieshaving x0 as first common point. Let y be a fixed point in �x0 \ {x0} and for each i,

RIGHT DUAL PROCESS FOR SEMIDYNAMICAL SYSTEMS 57

we consider an element yi ∈ �i, yi < x0 such that we can extract a sequence(yin)n which converges to x0 with respect to T 0

� . It follows that⋃

i(yi, y) is aneighborhood of x0 with respect to T 0

� which doesn’t contain any ball with respectto Tm, having x0 as center. ✷REMARK 17. If (X0,T

0�) is locally compact and the set B of all branching points

doesn’t have any accumulation point, then T 0� coincides with Tm.

PROPOSITION 18. The connected component of X0 with respect to T 0� contain-

ing a point x0 ∈ X0 coincides with the union of all maximal trajectories comingacross �x0 .

Proof. Let M be a connected component of X0 with respect to T 0� . Then, for

any maximal trajectory � such that � ∩M �= ∅, we have � ⊂ M.Let x0 be an element of M, so M is the union of all maximal trajectories coming

across �x0 . ✷COROLLARY 19. Let f be a positive function on X0. If f is totally increas-ing then, on each connected component, f vanishes outside a countable union ofmaximal trajectories.

Proof. Let M be a connected component of X0 w.r. to T 0� and let x ∈ M. Also

consider a sequence of positive real numbers (tn)n with limit ρ(x) and for each nwe put xn = �(tn, x). As M coincides with the union of all maximal trajectoriescoming across �x we deduce that M = ⋃

n[y � xn] and therefore

[f > 0] ∩M =⋃n

⋃m

([f (y) >

1

m

]∩ [y � xn]

).

Since f is finite and totally increasing on X0, we conclude that there exists at most[mf (xn)] maximal trajectories covering [f (y) > 1

m] ∩ [y � xn]. So that, on M,

f vanishes outside a countable union of maximal trajectories. ✷In the following, we shall suppose:

(C) For any point x ∈ X0, there exists at most a countable family of maximaltrajectories passing by x (cf. [5]).

REMARK 20. If (X0,T0�) is locally compact, it isn’t necessary to suppose that (C)

is satisfied. Indeed, it is suffices to consider the uniform movement on some deter-ministic fractal structures.

Also, by the above proposition if (C) is satisfied, it isn’t necessary that (X0,T0�)

is locally compact. In fact in the example of the uniform movement given in the pre-liminaries, the point (0, 0) doesn’t have any compact neighborhood although (C)is satisfied.

58 MOUNIR BEZZARGA

PROPOSITION 21. For any connected component M of X0 with respect to T 0�

there exists at most a countable family (�n)n of maximal trajectories such thatM = ⋃

n �n.Proof. Let x0 be an element of M, so M is the union of all maximal trajectories

coming across �x0 ; on the other hand there exists at most a countable family (�n)nof maximal trajectories passing through x0 and therefore M = ⋃

n �n. ✷COROLLARY 22. Let M be a connected component with respect to T 0

� . Then Mendowed with the trace measurable structure of B0 is a Lusin space.

Proof. The semidynamical system (X,B,�,ω) being transient, then for anyx ∈ X0 the trajectory �x is isomorphic to the real interval [0, ρ(x)) and therefore�x is a Lusin measurable space.

Using now the last proposition, we deduce thatM is a Lusin measurable space. ✷COROLLARY 23. (X0,T

0�) is a metrisable space.

Proof. Let M be a connected component with respect to T 0� . Then M is open

with respect to T 0� (see [3]) and there exists a sequence (�n)n of maximal trajec-

tories such that M = ⋃n �n. So M endowed with the trace topology of T 0

� is aseparable space with countable basis and therefore is metrisable.

Set X0 = ⋃i Ci where (Ci)i is the family of all connected components and let

di be a metric on Ci endowed with the trace topology of T 0� . Then, the map d given

on X0 ×X0 by

d(x, y) =

di(x, y)

1 + di(x, y), if x, y ∈ Ci for some i,

1, if �x ∩ �y = ∅,defines a metric on X0 such that the associated topology coincides with T 0

� . ✷THEOREM 24. The map t → �(t, x) is continuous on the right with respect tothe inherent topology T 0

φ .Proof. Let f : X0 → R

+ be a bounded B0-measurable function. For anyx ∈ X0, we have

(V0f )(�x(t)) =∫ +∞

0f ◦�(s,�(t, x)) ds

=∫ +∞

t

f (�(s, x)) ds.

Therefore, t → (V0f ) ◦ �x(t) is continuous on the right, i.e. t → �(t, x) iscontinuous on the right with respect to the inherent topology T 0

φ . ✷COROLLARY 25. The deterministic semigroup H is continuous on the right.

RIGHT DUAL PROCESS FOR SEMIDYNAMICAL SYSTEMS 59

3. Dual Semigroup Associated to (X,B,�,ω)

In this section, we shall construct a dual semigroup P of kernels associated to asemidynamical system (X,B,�,ω).

In Theorem 11 it is given the following characterization of coexcessive func-tions: f ∈ E∗ if and only if f is finite, totally increasing and continuous on theleft on X0. By Corollary 19, f ∈ E∗ implies that, on each connected componentwith respect to the inherent topology T 0

φ , f vanishes outside a countable union ofmaximal trajectories. As coexcessive functions are the only useful functions of thedual resolvent V∗ = (V ∗

α )α∈R+ associated to � with respect to the Lebesgue mea-sure , so except explicit mention all results are obtained under the supplementaryhypothesis (C).

Such assumption is, by Proposition 21, equivalent with the fact that any con-nected component with respect to the inherent topology T 0

φ is a countable unionof maximal trajectories. Obviously, all results under this condition (C) are alsovalid in general case for any function f which, on each connected component withrespect to the inherent topology T 0

φ , vanishes outside a countable union of maximaltrajectories.

In the following we will give further comments in live with the above mentionedarguments about condition (C).

3.1. COMMENTS

Let

� : R+ ×9 → 9

be an unstable continuous global semidynamical system with state space an opensubset 9 of R

n, which arises from ordinary differential equation (i.e. �(t, x) isassumed to be the solution with initial condition x). Note that if � deals with theabsence of the assumption (C), we can take back the study of such semidynamicalsystem to another one satisfying the countability condition (C). An example isgiven by the autonomous differential equation in R:

x = −x1/3,

where the right-hand side is only continuous and there is an uncountable family ofpositive integral curves still passing through each point of the t-axis.

Consider D a dense countable subset in 9 and put S(D) = ⋃x∈D �x , the

right saturated of D. Then S(D) is a �-invariant subset of 9 and for any x ∈ 9,there exists a sequence (xn)n in D such that the sequence of functions (�(· , xn))nconverges uniformly on compact subset of R+ toward �(· , x) the solution withinitial condition x.

Now suppose that

� : R+ ×X → X,

60 MOUNIR BEZZARGA

arises from a dynamical interpretation of evolution type hyperbolic partial differen-tial equation with time lag (i.e. the wave equation), where the state space (X, ‖ · ‖)is a separable Banach space. Therefore there exists a such dense countable subset Din (X, ‖ · ‖). So if the family of mappings (�(t, · ))t∈R+ is equicontinuous, then forany x ∈ X there exists a sequence (xn)n in D such that the sequence of functions(�(· , xn))n converges uniformly on R+ toward �(· , x) the solution with initialcondition x, i.e. limn supt∈R+ ‖�(t, xn)−�(t, x)‖ = 0.

So for a dynamical interpretation we study �0 the restriction of � to R+×S(D)in order to get the countability condition (C). Note that all start points of �0 are inD and so the entrance boundary of �0 is included in D.

A concrete example with an equicontinuity condition is given, in Chapter VIII[24], by a type of wave equation with weak damping of the form

∂2

∂t2u = ∂2

∂x2u− a(t, x)

∂tu+ f (t, x), x ∈ [0, 1], (∗)

with boundary conditions u(t, 0) = u(t, 1) = 0, ∀t > 0 and with initial displace-ment u(0, x) = u0(x) and initial velocity ∂

∂tu(0, x) = v0, ∀x ∈ [0, 1].

Set L2 = L2([0, 1],R), endowed with the quadratic norm and assume thata(t, x) > 0 on R+ × [0, 1] with both a, f ∈ L1([0,+∞);L2) and that there existsa" ∈ L2 with a − a" ∈ L1([0,+∞);L2). We choose X = H 1

0 × L2 as state spacefor the associated semidynamical system, where H 1

0 = H 10 ([0, 1];R) is the set of

the functions u ∈ L2 which are absolutely continuous with u′ := ∂∂xu ∈ L2 and

satisfying the boundary condition.We endow X with the norm

‖(u, v)‖ = √‖u′‖2 + ‖v‖2.

If u := ∂∂tu(· , x) denotes the time derivative, then the equation (∗) becomes in L2

u(t)+ a(t)u(t)−=u(t) = f (t), t � 0,

with initial values u(0) = u0, u(0) = u1.We will assume that the equation (∗) on [0,+∞) deals with right uniqueness

of solution through the initial value (u0, u1) ∈ X. Then we define the associatedsemidynamical system by

�(t, (u0, u1)) = (u(t), u(t)),

where u is the solution with initial value (u0, u1).Under the above assumptions the family of mappings (�(t, · ))t∈R+ is equicon-

tinuous. Moreover for any initial condition (u0, u1) ∈ X, we have limt→+∞ ‖u(t, · )‖= 0. So that (X,�) is transient with coffin state null.

NOTATION. H(X0) denotes the set of all positive functions on X0.

RIGHT DUAL PROCESS FOR SEMIDYNAMICAL SYSTEMS 61

DEFINITION 26. For any function f ∈ H(X0) and any t ∈ R+, we define thefunction Ptf given by

Ptf (x) =∑

((y,x)=tf (y), ∀x ∈ X0.

Note that, if there exists a start point y0 ∈ X0 such that ((y0, x) < t , we put in thissum, f (y) = f (y0) in place of y � y0,((y, x) = t and that a sum with emptycarrier is null.

PROPOSITION 27. For any t ∈ R+ the map Pt : H(X0) � f → Ptf ∈ H(X0)

is linear and σ -additive. Moreover, for any s, t ∈ R+, we have Ps(Ptf ) = Ps+t (f ).Proof. Let f ∈ H(X0) and let s, t ∈ R+. Then

Ps(Ptf )(x) =∑

((y,x)=sPtf (y) =

∑((y,x)=s

∑((z,y)=t

f (z),

i.e.

Ps(Ptf )(x) =∑z∈γ

f (z),

where

γ = {z : ((z, y)+((y, x) = s + t,∀y ∈ [((z, · ) = t] ∩ [((· , x) = s]}.Then

Ps(Ptf )(x) =∑

((z,x)=s+tf (z) = Ps+t (f )(x). ✷

NOTATION. Pσ denotes the set of all connected components of X0 with respectto the inherent topology T 0

φ given in [3], [17], Chap. III.

REMARK 28. (1) For any M ∈ Pσ , there exists a countable family (�n)n∈N∗ ofmaximal trajectories such that M = ⋃

n∈N∗ �n.(2) Let f ∈ H(X0), then f ∈ F (X0,) if and only if for any M ∈ Pσ , the

function f is measurable on M with respect to the σ -algebra B0.

THEOREM 29. For any f ∈ F (X0,) the map R+ × X0 � (t, x) → Ptf (x) ∈R+ is measurable if we endow X0 with the σ -algebra B0().

In particular, f → Ptf is a kernel on F (X0,) and the family P = (Pt)t∈R+is a measurable semigroup of kernels on (X0,B0()).

Proof. Let x0, y0 be two elements ofX0, such that x0 � y0 and let φ : [x0, y0] →R+ be a measurable function, where [x0, y0] is an interval of order. We put

Tφ := {(t, x) ∈ R+ × [x0, y0], t ∈ [0, φ(x)]}

62 MOUNIR BEZZARGA

and◦T φ := {(t, x) ∈ R+ × [x0, y0], t ∈ [0, φ(x))}.

Obviously, Tφ and◦T φ are measurable if we endow [x0, y0] with the trace measur-

able structure.On the other hand, we can reduce the proof for f = χ[x0,y0]. Indeed, one can

easily verify that

Ptf (x) = χ[T((x0,·)∩(R+×[x0,y0])]∪[(T((x0,·)\◦T((y0,·))∩(R+×�y0 )]

(t, x),

where ( is the arrival time and

T((x0, · ) := {(t, x) ∈ R+ × �x0; t ∈ [0,((x0, x)]},◦T ((y0, · ) := {(t, x) ∈ R+ × �y0; t ∈ [0,((y0, x))}.

It follows that the function (t, x) → Ptf (x) is measurable with respect to theproduct measurable structure. The proof then holds for any f ∈ F (X0,) by amonotone class argument. ✷REMARK 30. Note that, in the case where (C) isn’t satisfied, the above theorem isstill valid for all positive B0()-measurable function f which, on each connectedcomponent with respect to the inherent topology T 0

φ , vanishes outside a countableunion of trajectories.

THEOREM 31. The resolvent (Rα)α∈R+, associated with a semigroup P, coin-cides with the dual resolvent V∗ associated with (X,B,�,ω).

Proof. The proof can be reduced to f = χ[x0,y0], x0, y0 ∈ X0 such that x0 � y0.So we have

Rαf (x) =

1

α(1 − e−α((x0,x)) if x ∈ [x0, y0],

1

α(e−α((y0,x) − e−α((x0,x)) if x ∈ �y0 ,

0 if not.

Since the same expressions are obtained for V ∗α f , we deduce that Rαf = V ∗

α f . ✷REMARK 32. In [6], we have constructed the dual resolvent V∗, associated witha semidynamical system without the supplementary hypothesis (C), by setting oncemore

V ∗α f (x) =

∫e−α((y,x)G(y, x)f (y)d(y), ∀x ∈ X0,

for any α ∈ R+ and any f ∈ F (X0,).

RIGHT DUAL PROCESS FOR SEMIDYNAMICAL SYSTEMS 63

In the following corollary we don’t suppose that (C) is satisfied but we show thatthe dual resolvent V∗ is completely determined by the dual semigroup P.

COROLLARY 33. For any α ∈ R+ and any f ∈ F (X0,), we have for eachx ∈ X0

V ∗α f (x) = sup

M∈Bσ0

∫M

e−αtPt (f χM)(x) dt.

Proof. Using the definition of the Lebesgue measure , we get∫f d =

supM∈Bσ0

∫Mf d, and therefore V ∗

α f (x) = supM∈Bσ0V ∗α (f χM)(x), ∀x ∈ X0. By

the above theorem, we get

V ∗α f (x) = sup

M∈Bσ0

∫M

e−αtPt (f χM)(x) dt, ∀x ∈ X0. ✷

THEOREM 34. The semigroup P on (X0,B0()) is in duality with the deter-ministic semigroup H on (X0,B0()) associated to the semidynamical system(X,B,�,ω)with respect to the Lebesgue measure, i.e. for any f, g ∈ F (X0,)

we have∫g.Ptf d =

∫f ·Htg d, ∀t ∈ R+.

The semigroup P is called the dual semigroup associated with the semidynamicalsystem (X,B,�,ω).

Proof. Arguing as in the proof of Theorem 3.2, we consider f and g as follows:f = χ[x0,y0] and g = χ[u0,v0]; x0, y0, u0, v0,∈ X0 satisfying x0 � y0 and u0 � v0.Then we have

g(x)Ptf (x) = χ[u0,v0](x)χ[T((x0,·)∩(R+×[x0,y0])]∪[(T((x0,·)\◦T ((y0,·))∩(R+×�y0 )]

(t, x),

where

T((x0, · ) := {(t, x) ∈ R+ × �x0; t ∈ [0,((x0, x)]}and

◦T ((y0, · ) := {(t, x) ∈ R+ × �y0; t ∈ [0,((y0, x))}.

So, we distinguish the following cases:Case 1. If v0 /∈ �x0 , then gPtf = (Htg)f = 0 and therefore∫

gPtf d =∫(Htg)f d = 0.

Case 2. If v0 ∈ �x0 \�y0 , we put �x1 = �x0 ∩�u0 for the convenient element x1

of �x0 . Then we have

gPtf = χ[x1,v0]χ[((x0,·)�t ].

64 MOUNIR BEZZARGA

In particular, gPtf = 0 if t � ((x0, v0). So∫gPtf d =

{((x0, v0)− max(t,((x0, x1)) if t � ((x0, v0),

0 if not.

On the other hand, we have

(Htg)f = χ[x0,v0]χ[((x0,x1)�t+((x0,·)�((x0,v0)].

In particular (Htg)f = 0 if t � ((x0, v0). So∫(Htg)f d =

{((x0, v0)− t − max(0,((x0, x1)− t) if t � ((x0, v0),

0 if not,

i.e. ∫(Htg)f d =

{((x0, v0)− max(t,((x0, x1)) if t � ((x0, v0),

0 if not.

In consequence, we get∫(Htg)f d =

∫gPtf d.

Case 3. If v0 ∈ �y0 and x1 /∈ �y0 , then we have

gPtf = χ[x1,v0]χ[max(t,((x0,x1))�((x0,·)�((x0,y0)]∪[max(t,((x0,y0))�((x0,·)�min(t+((x0,y0),((x0,v0))].

Which gives∫gPtf d

=

min(t +((x0, y0),((x0, v0))−−max(t,((x0, x1)) if t � ((x0, y0),

min(((x0, v0), t +((x0, y0))− t if ((x0, y0) � t � ((x0, v0),0 if ((x0, v0) � t.

On the other hand, we have

(Htg)f = χ[x0,y0]χ[((x0,x1)�t+((x0,·)�((x0,v0)],

and therefore∫(Htg)f d

=

min(((x0, v0)− t, ((x0, y0))−−max(0,((x0, x1)− t) if t � ((x0, y0),

min(((x0, v0)− t, ((x0, y0)) if ((x0, y0) � t � ((x0, v0),

0 if ((x0, v0) � t,

RIGHT DUAL PROCESS FOR SEMIDYNAMICAL SYSTEMS 65

i.e. ∫(Htg)f d

=

min(((x0, v0), t +((x0, y0))−−max(t,((x0, x1)) if t � ((x0, y0),

min(((x0, v0), t +((x0, y0))− t if ((x0, y0) � t � ((x0, v0),

0 if ((x0, v0) � t.

So ∫(Htg)f d =

∫gPtf d.

Case 4. If v0 ∈ �y0 and x1 ∈ �y0 , then we have

gPtf = χ[x1,v0]χ[t�((x0,·)�t+((x0,y0)],

and ∫gPtf d = min(((x0, v0), t +((x0, y0))− max(t,((x0, x1)).

On the other hand, we have

(Htg)f = χ[x0,y0]χ[((x0,x1)�t+((x0,·)�((x0,v0)]

and therefore∫(Htg)f d = min(((x0, v0)− t, ((x0, y0))− max(0,((x0, x1)− t)

=

min(((x0, v0), t +((x0, y0))−−max(t,((x0, x1)) if t � ((x0, v0),

0 if ((x0, v0) � t.

So we conclude that∫g.Ptf d =

∫f ·Htg d. ✷

REMARK 35. As in the last theorem in the case where the assymption (C)isn’t satisfied, the above theorem is yet valid for any positive B0()-measurablefunction f which, on each connected component with respect to the inherent topol-ogy T 0

φ , vanishes outside a countable union of maximal trajectories, i.e. for anyg ∈ F (X0,) we have∫

gPtf d =∫fHtg d, (∀)t ∈ R+.

66 MOUNIR BEZZARGA

NOTE. Using the above theorem, we can obtain again the duality between theresolvents V∗ and V, as it is shown in [5, 6] without hypothesis (C), i.e. for anyf, g ∈ F (X0,) and for any α ∈ R+,∫

gV ∗α f d =

∫fVαg d.

PROPOSITION 36. The dual resolvent V∗ (respectively the dual semigroup P)associated with the semidynamical system (X,B,�,ω) is sub-Markovian if andonly if the maximal trajectories are pairwise disjoint.

We consider again the example (ii) in the preliminaries of a semidynamical systemwith the absence of left uniqueness.

The dual semigroup P = (Pt )t>0 associated to � has the following form:

Ptf (xs) =

f (xs−t ) if s − t � 0,+∞∑n=1

f (xns−t ) if not

and Ptf (xns ) = f (xns−t ), for any positive Borel function f on X0 = X \ {(1, 0)}.

4. Doob u-Transformation of the Dual Semigroup P

We shall apply the so-called Doob u-transformation to the dual semigroup P, inorder to get the sub-Markovian property.

REMARK 37. As in Proposition 21, for any M ∈ Pσ , there exists a countablefamily (�n)n∈N∗ of maximal trajectories such that M = ⋃

n∈N∗ �n.We know also that the function u = ∑+∞

n=112n χ�n is supermedian with respect to

the resolvent V∗ and it is positive on M and satisfies u � 1 on X0 (cf. [5]).

THEOREM 38 (cf. [21]). Let M be an element of Pσ and u defined as above, thenthe family P

u := (P ut )t�0 given by

Put f (x) =

Pt(uf )(x)

u(x)if x ∈ M,

0 if x ∈ X0 \M,

for any f ∈ F (X0,), defines a sub-Markovian semigroup on (X0,B0()). Theassociated resolvent (V u

α )α�0 is given by

V uα f (x) =

V ∗α (uf )(x)

u(x)if x ∈ M,

0 if x ∈ X0 \M.

The above semigroup Pu (respectively resolvent (V u

α )α�0) is in duality with the de-terministic semigroup H (respectively the resolvent V) on (X0,B0()) associatedto the semidynamical system (X,B,�,ω) with respect to the measure u.

RIGHT DUAL PROCESS FOR SEMIDYNAMICAL SYSTEMS 67

REMARK 39. Let M be an element of Pσ and (�n)n∈N∗ be the countable familyof maximal trajectories such that M = ⋃

n∈N∗ �n.Let σ be a permutation of N

∗ and put u = ∑+∞n=1

12n χ�n and v = ∑+∞

n=112n χ�σ(n) .

Then the function v/u extended by zero on X0 \M, is supermedian with respect toPu. Moreover the Doob v

u-transformation of the semigroup P

u coincides with theDoob v-transformation of the semigroup P, i.e. for any t ∈ R+,

P vt = (P u

t )v/u.

THEOREM 40. Let M be an element of Pσ and u defined as above, then thesemigroup P

u = (Pt )t�0 is right continuous.Proof. The minimal points of X0 with respect to the order associated to the

semidynamical system (X,B,�,ω) are polar, then we can assume that X0 iswithout minimal points (cf. [14], Chap. XII). Let f : X0 → R+ be a boundedcontinuous function and x0 be an element of X0. For any maximal trajectory �0,the map t → Pt(χ�0f )(x0) is right continuous and Pt(χ�0f ) � ‖f ‖∞, where‖f ‖∞ = supx∈X0

|f (x)|.In the other hand, let u = ∑+∞

n=11

2n χ�n , so the series of functions∑+∞n=1

12n Pt (χ�nf )(x) normally converges on R+ ×X0 to Pt(uf )(x). So we obtain

limt→t+0

Pt(uf )(x0) = Pt0(uf )(x0).

That is

limt→t+0

Put f = Pu

t0f. ✷

Now, for any M ∈ Pσ , we put uM = ∑+∞n=1

12n χ�n , for some sequence (�n)n∈N∗

of maximal trajectories such that M = ⋃n∈N∗ �n. Denote by ′ the measure on

(X0,B0()) defined by

′(A) =∫A

uM d, ∀M ∈ Pσ , A ∈ B0 and A ⊂ M.

Then, we have the following result:

PROPOSITION 41. The family P′ = (P ′

t )t∈R+ of measurable kernels given on(X0,B0()) by

P ′t f = Pu

t f on M,

∀M ∈ Pσ , ∀f ∈ F (X0,), defines a sub-Markovian semigroup on (X0,B0()),which is in duality with the deterministic semigroup H with respect to ′.

The semigroup P′ is called the direct some of the family of semigroups (PuM )M∈Pσ

and we write

P′ =

⊕M∈Pσ

PuM .

68 MOUNIR BEZZARGA

THEOREM 42. We assume that X0 is a connected space with respect to T 0� and let

(�n)n∈N∗ be a countable family of maximal trajectories such that X0 = ⋃n∈N∗ �n.

We put also u = ∑+∞n=1

12n χ�n . Then X0 is semisaturated with respect to H

(respectively Pu).

In particular, there exists a right process on (X0,B0) having Pu as transition

semigroup.Proof. Let I = (x0, y0) be an interval of order in X0 endowed with the H -

cone S (in the sense of [8]) of all positive increasing functions which are lowersemicontinuous functions on I .

Let now θ be a H -integral on S, then there exists a measure m on (x0, y0] suchthat for any s ∈ S, we have

θ(s) =∫s dm,

where

s(x) = s(x) if x ∈ I,

s(y0) = limx→y0

s(x).

Let µ be a finite measure on I such that θ � µ on S. We consider an increasingsequence (xn)n of elements of I such that limn xn = y0 and we put sn = χ(xn,y0).So we have sn ∈ S and

θ(sn) � µ(sn), ∀n ∈ N.

But (µ(sn))n decreases to 0 and sn(y0) = 1. Therefore m({y0}) = 0, that is m notcharging the point y0 and we can consider that θ is a measure on I . The proof thenholds for X0 by a Riesz decomposition. ✷PROPOSITION 43 (cf. [6]). A function s is excessive (resp. supermedian) withrespect to P

u, if and only if the function us is an excessive (resp. supermedian)function with respect to P.

Next, we shall construct the associated right dual process.First, let B be the set of all branching points of the semidynamical system

(X,B,�,w). Then B is a countable set of semipolar points. So the set X1 =X0 \B is also semisaturated with respect to P

u and the cone Eu of all Pu-excessive

functions, is minstable. For any x ∈ X1, we put ρ∗(x) the element of R+ such that

[((· , x) � t] ∩ B = ∅, ∀t ∈ [0, ρ∗(x))

and

[((· , x) � t] ∩ B �= ∅, ∀t � ρ∗(x).

RIGHT DUAL PROCESS FOR SEMIDYNAMICAL SYSTEMS 69

ρ∗ is called the colife time and the set wx := [((· , x) < ρ∗(x)] ∩X1 is called thecotrajectory of x. We denote by 9 the set of all cotrajectories wx , where x runs thephase space X1.

Now, consider the map Y defined on R+×9 by Y (t, wx) = wx(t) where wx(t)

is the point of X1 such that

((wx(t), x) = t, ∀t ∈ [0, ρ∗(x))

and

((wx(t), x) = w∗, ∀t � ρ∗(x),

where w∗ is a distinguished point. For any t � 0, we put also Yt(wx) = Y (t, wx)

and denote by F0 (resp. F 0t ) the σ -algebra generated by the family (Ys, s � 0)

(resp. (Ys, s � t)). For any x ∈ X1, we define on (9, F 0) the probability P x bysetting

P x(Yt ∈ A) = Put (χA)(x), ∀t � 0,∀A ∈ B0, A ⊂ X1.

We also have

Ex(f ◦ Yt) = Put f (x),

for any positive measurable function f and for t < t ′

Ex(f ◦ Yt ′/Ft ) = (P ut ′−t f ) ◦ Yt .

We also define for any probability law µ, the measure Pµ by setting

Pµ(Yt ∈ A) =∫P x(Yt ∈ A) dµ(x).

Then we endow (9, F 0) with the measure Pµ for which the process (Yt)t�0 isMarkovian, having µ as initial law and P

u as transition semigroup. (Yt)t�0 isnamely the dual u-process associated with (X,B,�,w).

Finally, we shall prove that (Yt)t�0 is a right process.Let x0 ∈ X1 and let wx0 its cotrajectory. If y0 ∈ wx0 then x0 = �(t0, y0) for

some t0 ∈ [0, ρ∗(x0)). Consider a sequence (yn)n in X1 which is increasing to y0

and (tn)n a sequence of real numbers which is decreasing to t0 with x0 = �(tn, yn),i.e. Ytn(wx0) = yn. Let s∗ be an excessive function with respect to P

u. Then us∗is increasing and continuous on the left with respect to T 0

� , so we have (us∗(yn))nis increasing to us∗(y0). Since y0 /∈ B, then (u(yn))n is increasing to u(y0) and(s∗(yn))n converges to s∗(y0).

5. Invariant Probabilities with Respect to Pu

In the sequel we assume that X0 is a connected space with respect to T 0� and set

X0 = ⋃n∈N∗ �n for a convenient sequence (�n)n of maximal trajectories. In this

70 MOUNIR BEZZARGA

case B0() coincides with the σ -algebra B0. We also set u = ∑∞n=1

12n χ�n , which

is supermedian with respect to the dual resolvent V∗ and verify 0 < u � 1 onX0. As in the above section, P

u denotes the u-Doob transformation of the dualsemigroup P.

The main result of this section is provides by the description of the associatedinvariant measures.

The following definitions and propositions are satisfied for Q = H (resp. Q =P). Set Q = (Qt)t>0. We recall (see [2, 12, 14, 16]) that a σ -finite measure µ on(X0,B0) is called excessive with respect to Q provided µQt � µ for all t > 0 andµ = supt>0 µQt .

In [14], Chap. XII it is shown that, in our situation, µ is an excessive measurewith respect to Q if and only if µQt � µ for all t > 0. We write Exc(Q) for theclass of excessive measures with respect to Q.

Now, a σ -finite measure µ on (X0,B0) is termed invariant with respect to Q iffor each t > 0, µQt = µ. It is clear that any invariant measure is also excessive.Moreover, if µ is a probability on (X0,B0), we say that µ is an invariant probabilitywith respect to Q. For details about Exc(H) see [4].

THEOREM 44. There is no invariant probability on (X0,B0) with respect to H.Proof. Since (X,B,�,w) is a transient semidynamical system, so the associ-

ated resolvent V is proper, i.e. there exists a strictly positive measurable function ϕon X0 such that V0ϕ is bounded (see [15]). Set s = V0ϕ and remark that s isa decreasing excessive function with respect to H. Suppose that there exists aninvariant probability µ on (X0,B0) with respect to H and fix t > 0. Then

µHt(s) = µ(s).

As the measurable function s −Hts is nonnegative, so s −Hts = 0 µ-a.s. That iss(x) = s(�(t, x)) µ-a.s., which is impossible. ✷THEOREM 45. (i) For any s ∈ E , the density measure (su) is excessive withrespect to P

u. Moreover, for any s1, s2 ∈ E , we have

s1 � s2 ⇐⇒ (s1u) � (s2u).

(ii) For any positive measurable function f on X0, the density measure f isexcessive with respect to P

u if and only if there exists s ∈ E such that f = su;-a.s. on X0.

(iii) The set uE:= {(su), s ∈ E}, is naturally solid convex subcone of Exc(Pu)

which is closed in order from below.Proof. (i) Let s ∈ E and let f be a positive measurable function on X0. We have

for each t > 0,∫(sP u

t f )u d =∫sPt (uf ) d

RIGHT DUAL PROCESS FOR SEMIDYNAMICAL SYSTEMS 71

=∫(Hts)f u d

�∫sf u d.

Therefore the σ -finite measure (su) is excessive with respect to Pu. Obviously,

if s1 � s2 then (s1u) � (s2u).Conversely, if (s1u) � (s2u), then s1 � s2 -a.s. on X0. Since s1 and s2 are

lower semicontinous with respect to T 0� , then s1 � s2 on X0.

(ii) Let f be a positive measurable function on X0 such that f is an excessivemeasure with respect to P

u. Since the associated resolvent is basic with respect tothe σ -finite measure u and f is absolutely continuous with respect to , thenby [14], Chap. XII there exists s ∈ E such that f = (su), i.e. f = su -a.s.

(iii) Let (si)i be an increasing family of elements of E such that the measure µgiven on (X0,B0) by:

µ(A) = supi

((siu))(A),

is excessive with respect to Pu. Since E is a standard H -cone (cf. [4]) in the

meaning of [8], so there exists a sequence (in)n such that supi si = supn sin andtherefore

µ =((

supn

sin

)u),

i.e. the set uE is a convex subcone of Exc(Pu) closed in order from below.Now, let µ ∈ Exc(Pu) and s ∈ E be such that µ � (su). In particular µ

is absolutely continuous with respect to the σ -finite measure u. Using Radon–Nikodym theorem, there exists a positive measurable function f on X0 such thatµ = (f u).

By (ii), we get µ = (tu) for some t ∈ E . Hence uE is a naturally solid convexsubcone of Exc(Pu). ✷NOTATION. In the following ρ ′ denotes the function on X0 given by

ρ ′(x) = supy�x

((y, x).

PROPOSITION 46. ρ ′ is measurable on X0 with respect to the σ -algebra B0.Proof. Since X0 = ⋃

n∈N∗ �n is a countable union of maximal trajectories, thenthere exists a countable subset A of X0 which is dense in X0 with respect to T 0

� .Then, for any real number α we have

[ρ ′ � α] =⋂a∈A

[((a, · ) � α]

which is in B0 (see Theorem 6). ✷Now, we give a characterization of the invariant probabilities with respect to P

u.

72 MOUNIR BEZZARGA

THEOREM 47. A probability µ on (X0,B0) is invariant with respect to Pu if and

only if there exists a sequence (αn)n of positive real numbers with∑

n αn = 1 anda sequence (xn)n of start points of X0 such that µ = ∑

n αnεxn .Proof. Let µ be an invariant probability with respect to P

u. For any supermedianfunction with respect to P, we have s/u is supermedian with respect to P

u andtherefore

∫Put (

su) dµ = ∫

su

dµ.This implies that

Pts = s, µ-a.s.

Now, let x0 be a non-start point of X0 (i.e. ρ ′(x0) > 0) and let α ∈ ]0, min(ρ(x0),

ρ ′(x0))[. We fix t ∈ ]α, ρ ′(x0)[ and we consider an increasing sequence (xn)n∈N ofelements of �x0 such that �x0 = ⋃

n[xn, xn+1] with ((xn, xn+1) � α, ∀n ∈ N.For each n, the function sxn := χ�xn is supermedian with respect to P and we havePtsxn = 0 �= sxn on the ordered interval [xn, xn+1]. Then µ([xn, xn+1]) = 0, for anyn ∈ N and therefore µ(�x0) = 0. As any maximal trajectory is a countable unionof trajectories and X0 = ⋃

n �n, we deduce that µ(X0\S) = 0, where S is the setof all start points which is at most countable subset of X0.

Consequently, there exists a sequence (αn)n of positive real numbers with∑

n αn= 1 and a sequence (xn)n of start points such that µ = ∑n αnεxn .

Conversely, if x0 is a start point of X0 (i.e. ρ ′(x0) = 0), then Pts(x0) = s(x0)

for each t > 0 and any positive measurable function s on X0. So the Dirac measureεx0 is invariant with respect to P

u. ✷

5.1. ON THE JUMP’S SEMIDYNAMICAL SYSTEMS

Finally, we will introduce an important class of semidynamical systems which willbe needed as source of examples with discrete state space. The semidynamicalsystem to be constructed, has interesting physical application, in the jump’s con-duction phenomena through semiconductor materials. In order to better understandthe nature of this hopping phenomena see references ([1], Chap. VI, 9.3 in [23] andChap. 9, 9.8 in [13]). Let E = {xn, n ∈ N × {∞}} be a countable set and let ζbe a function on E satisfying 0 < ζ(x) < +∞ for all x ∈ E. A particle startingfrom a point xn0 ∈ E remains there for an exponentially holding time τn1 withparameter ζ(xn0) at which time it jumps to a new position xn1 . It then remains atxn1 for a length of time τn2 which is exponentially distributed with parameter ζ(xn1)

but which, given xn1 , is independent of τn1 . Then it jumps to xn2 . . . . However if itjumps to x∞, then it remains there for all time to come. A rigorous construction ofa such general semidynamical system

� : R+ × E → E,

imposes the following property:

∀n ∈ N ∪ {∞},∀t ∈ R+, ∃τ, τ ′ ∈ R+, τ < τ ′

RIGHT DUAL PROCESS FOR SEMIDYNAMICAL SYSTEMS 73

such that ∀x ∈ E, �(s, x) = xn, ∀s ∈ [τ, τ ′)whenever �(t, x) = xn.

We say that � is jump’s semidynamical system.On the time phase space E := R+ × E, we consider the associated transient

semidynamical � given on R+ × E by

�(t, (s, x)) ={(t + s,�(t + s, x)) if �(t + s, x) �= x∞,ω if �(t + s, x) = x∞,

where ω be a point adjoint to E in the usual manner.The entrance boundary S of � is a countable subset of {0} × E and therefore

the assumption (C) is satisfied. Moreover all invariant probabilities are convexcombinations of Dirac masses carried in S.

Acknowledgments

I would like to thank the referee for the suggestions. They helped me considerablyto refine my paper.

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