TO MARKOV PROCESSES
S. HEIKKIL ¨AReceived 2 December 2004 and in revised form 8 March 2005
New existence and comparison results are proved for fixed points of increasing operators and for common fixed points of operator families in partially ordered sets. These results are then applied to derive existence and comparison results for invariant measures of Markov processes in a partially ordered Polish space.
1. Introduction
A. Tarski proved in his fundamental paper [18] that the set Fix(G) of fixed points of any increasing self-mappingGof a complete lattice is also a complete lattice. Davis completed this work by showing in [3] that a lattice is complete if each of its increasing self-mappings has a fixed point. As a generalization of this result Markowsky proved in [16] that each self-mappingGof a partially ordered set (poset)Xhas the least fixed point if and only if each chain ofX, also the empty chain, has the supremum, and that in such a case each chain of Fix(G) has the supremum in Fix(G) (see also [2]).
In [9,10] it is shown that ifG:X→Xis increasing, if nonempty well-ordered (w.o.) and inversely well-ordered (i.w.o.) subsets ofG[X] have supremums and infimums inX, and if for somec∈Xeither supremums or infimums of{c,x}exist for eachx∈X, then Ghas maximal or minimal fixed points, and least or greatest fixed points in certain order intervals ofX. Applications of these results to operator equations, as well as various types of explicit and implicit differential equations are presented, for example, in [1,8,9,10]. To meet the demands of our applications to Markov processes we will prove inSection 2 similar fixed point results when the existence of supremums or infimums of{c,x},x∈X, is replaced by weaker hypotheses. Results on the structure of the fixed point set are also derived. The proofs are based on a recursion principle introduced in [11].
In [18] existence of common fixed points is also proved for commutative families of increasing self-mappings of a complete latticeX. As for generalizations of these results, see, for example [4,16,19]. InSection 3we derive existence results for common fixed points of a family of mappingsGt:X→X,t∈S, whereSis a nonempty set, in cases when for somet0∈Sresults ofSection 2are applicable toG=Gt0, when (i)GtGt0=Gt0Gtfor
eacht∈S, and when (ii) either Gt0x≤Gtx orGtx≤Gt0x for allt∈Sandx∈X. For
instance, ifXis a closed ball inRm, ordered coordinatewise, a family{G
t}t∈Shas a com-mon fixed point ifGt0is increasing and satisfies (i) and (ii) (cf.Example 3.4). The results
ofSection 2 can also be applied to prove the existence of increasing selectors for fixed points of an increasing family of increasing mappings (cf.Remarks 3.3andExample 3.4). The obtained results are then applied inSection 4to prove existence and comparison results for invariant measures of Markov processes in a partially ordered Polish spaceE. Such results have applications in ergodic theory, in economics and in statistics (see, e.g., [13,17,20]). No compactness hypotheses are imposed onE.
Examples are given to demonstrate the obtained results. For instance, the example of Subsection 4.4is constructed to justify the need of the new fixed point results derived in Section 2.
2. Fixed point results
In this sectionX=(X,≤) is a poset. Recall that a subsetCofXiswell-ordered (respec-tivelyinversely well-ordered) if each nonempty subset ofChas the least (respectively great-est) element.
Whena,b∈X,a≤b, we denote
[a)= {x∈X|a≤x}, (a]= {x∈X|x≤a}, [a,b]= {x∈X|a≤x≤b}. (2.1)
Given a subsetY ofX we say that a mappingG:X→X isincreasingin Y ifGx≤Gy wheneverx,y∈Yandx≤y. We say thatx∈Yis theleast fixed pointofGinYifx=Gx, and ifx≤ywhenevery∈Y andy=Gy. The greatest fixed point ofGinY is defined similarly, by reversing the inequality. A fixed pointxofGis calledmaximalifx=y when-every=Gyandx≤y, andminimalify=Gyandy≤ximplyx=y.
A nonempty subsetY ofX is calledrelatively well-order completeif each nonempty w.o. or i.w.o. subset ofY has supremums and infimums inX. If these supremums and infimums belong toY, we say thatYiswell-order complete. Denote
Y+=y∈X|y=supWfor some w.o. subsetWofY,
Y−=y∈X|y=infWfor some i.w.o. subsetWofY.
(2.2)
If there exists ac∈Xand an increasing mapping fc:Y →Xsuch that fc(y) is an upper bound of{c,y}for ally∈Y, we say that fcis anup-mapofY. If there exists ac∈Xand an increasing mapping fc:Y→Xsuch that fc(y) is a lower bound of{c,y}for ally∈Y, we say that fcis adown-mapofY.
For instance, if sup{c,x}exists for eachx∈Y, the mapping fc(y)=sup{c,y}is an up-map ofY. Similarly, fc(y)=inf{c,y}is a down-map ofY if inf{c,x}exists for each x∈Y. Ifcis a lower bound ofY inX, then fc(y)≡yis an up-map ofY, and ifcis an upper bound ofY inX, thenfc(y)≡yis a down-map ofY.
Lemma 2.1. Given a setᏰof subsets of X containing the empty set∅, and a mapping Ᏺ:Ᏸ→X, there is a unique well-ordered subsetCofXwith property
(A)x∈Cif and only ifx=Ᏺ(C<x), whereC<x= {y∈C|y < x}. IfᏲ(C)is defined, it is not a strict upper bound ofC.
As an application ofLemma 2.1we prove the following auxiliary result. Lemma2.2. Assume thatG:X→Xis an increasing mapping.
(a)If all nonempty w.o. subsets ofG[X]have supremums, and ifG[X]+has an up-map fc, then equationx=fc(Gx)has the least solutionx, and
x=minx∈X|fc(Gx)≤x. (2.3)
(b)If all nonempty i.w.o. subsets ofG[X]have infimums, and ifG[X]−has a down-map
fc, then equationx=fc(Gx)has the greatest solutionx, and
x=maxx∈X|x≤fc(Gx)
. (2.4)
Moreover, bothxandxare increasing with respect toG.
Proof. (a) Letfc:G[X]+→Xbe an up-map ofG[X]+, and letᏰdenote the family of all well-ordered subsets ofX. LetW∈Ᏸ\ ∅be given. To prove thatG[W] is well-ordered, letBbe a nonempty subset ofG[W]. ThenA= {x∈W|Gx∈B}is a nonempty subset ofW. BecauseWis well-ordered, thenx0=minAexists. Ify∈B, theny=Gxfor some x∈A. Thusx0≤x, whenceGx0≤Gx=y becauseG is increasing. Consequently,Gx0 is the least element ofB. This implies, by definition, thatG[W] is well-ordered. Thus supG[W] exists, by a hypothesis.
The above result guarantees that we can define a mappingᏲ:Ᏸ→Xby
Ᏺ(∅)=c, Ᏺ(W)=fcsupG[W], W∈Ᏸ\ ∅. (2.5)
According to this definition condition (A) ofLemma 2.1can be rewritten in the following form:
c=minC, c < x∈C iffx= fcsupGC<x. (2.6)
ByLemma 2.1there exists only one well-ordered setCinXsatisfying (2.6). It is nonempty becausec∈C. Since G is increasing, thenG[C] is a nonempty well-ordered subset of G[X]. Thus supG[C] exists and belongs toG[X]+, whenceᏲ(C)= fc(supG[C]) is de-fined. Sincefcis increasing, it follows from (2.6) thatᏲ(C) is an upper bound ofC. This result and the last conclusion ofLemma 2.1 imply thatx:=Ᏺ(C)=maxC. SinceG is increasing, thenGx=maxG[C]=supG[C], whencex=Ᏺ(C)= fc(Gx), that is,x=x is a solution of equationx=fc(Gx).
whencex= fc(supG[C<x])≤ fc(Gy)≤y. Consequently, by transfinite induction,x≤y for eachx∈C, that is,x=maxC≤y. This result implies that (2.3) holds, and thatx=x is the least solution of equationx=fc(Gx).
The proof of (b) is dual to the above one, and the last conclusion of Lemma is a
con-sequence of (2.3) and (2.4).
ApplyingLemma 2.2we obtain existence and comparison results for fixed points of an increasing mappingG:X→Xif one of the following hypotheses holds.
(Ga)G[X] has a lower bound, and nonempty w.o. subsets ofG[X] have supremums inX.
(Gb)G[X] has an upper bound, and nonempty i.w.o. subsets ofG[X] have infimums inX.
Corollary2.3. LetG:X→Xbe increasing.
(a)If (Ga) holds, thenGhas the least fixed point which is increasing with respect toG. (b)If (Gb) holds, thenGhas the greatest fixed point which is increasing with respect toG. Proof. Assume that (Ga) holds, and letc∈Xbe a lower bound ofG[X]. Then fc(y)≡y is an up-map ofG[X]+, and equationx= fc(Gx) is reduced to the fixed point equation x=Gx. The first conclusion of (a) follows then fromLemma 2.2(a). Similarly, the first conclusion of (b) follows from Lemma 2.2(b), and the last conclusions of (a) and (b)
follow from the last conclusion ofLemma 2.2.
The next result is a consequence ofLemma 2.2andCorollary 2.3.
Theorem2.4. Assume thatG:X→Xis increasing, thatG[X]is relatively well-order com-plete, and thatG[X]+has an up-mapfc. Then
(a)Ghas the greatest fixed pointx∗in(x], wherex=min{x∈X|x= fc(Gx)}; (b)xandx∗are increasing with respect toG.
Proof. The given hypotheses ensure byLemma 2.2that the least solutionxof equation x= fc(Gx) exists, and thatxis increasing with respect toG. SinceG is increasing and Gx≤ fc(Gx)=x, thenxis an upper bound ofG[(x]], whenceCorollary 2.3, withX= (x], implies thatGhas the greatest fixed pointx∗in (x], and thatx∗is increasing with
respect toG.
The following result is a dual toTheorem 2.4.
Proposition2.5. Assume thatG:X→Xis increasing, thatG[X]is relatively well-order complete, and thatG[X]−has a down-map fc. Then
(a)Ghas the least fixed pointx∗in[x), wherex=max{x∈X|x=fc(Gx)};
(b)x∗andxare increasing with respect toG.
The next result gives some information on the structure of the fixed point set Fix(G). Proposition2.6. If the hypotheses ofTheorem 2.4orProposition 2.5hold, thenFix(G)is well-order complete. In particular,Ghas minimal and maximal fixed points.
subset ofG[X], which is relatively well-order complete, thenx0=supWexists inX. Be-causex≤x0for eachx∈WandGis increasing, thenx=Gx≤Gx0for eachx∈W. Thus Gx0is an upper bound ofW, and sincex0=supW, thenx0≤Gx0. ThusG, restricted to [x0), satisfies the hypothesis (Ga), whenceGhas byCorollary 2.3(a) the least fixed point x1in [x0). Obviously,x1is the least upper bound ofWin Fix(G).
The fact that each nonempty i.w.o. subset of Fix(G) has the greatest lower bound in Fix(G) follows similarly fromCorollary 2.3(b). Thus Fix(G) is well-order complete.
Since each nonempty well-ordered subset of Fix(G) has an upper bound in Fix(G), it follows from the version of Zorn’s Lemma due to Bourbaki (cf., e.g., [11, page 6]) that Fix(G) has a maximal element, which is a maximal fixed point ofG. The dual reasoning
shows thatGhas a minimal fixed point.
As a consequence ofTheorem 2.4and Propositions2.5and2.6we obtain the following fixed point results, which generalize those of [10, Theorem 2.1 and Proposition 2.3]. Corollary2.7. LetG:X→X be an increasing mapping whose range is relatively well-order complete, and assume thatc∈X.
(a)Ifsup{c,x}exist for allx∈G[X]+, thenFix(G)is well-order complete,Ghas minimal and maximal fixed points, and the greatest fixed pointx∗in(x], wherexis the least solution of equationx=sup{c,Gx}.
(b)Ifinf{c,x}exist for allx∈G[X]−, thenFix(G)is well-order complete,Ghas minimal
and maximal fixed points, and the least fixed pointx∗in[x), wherexis the greatest
solution of equationx=inf{c,Gx}.
(c)x∗,x∗,xandxare increasing with respect toG.
Proof. The hypotheses ofTheorem 2.4andProposition 2.5hold whenfc(y)=sup{c,y}, y∈G[X]+in (a) and fc(y)=inf{c,y},y∈G[X]−in (b).
Example 2.8. Letl1be the space of all absolutely summable real sequences, ordered by (xn)∞n=0≤(yn)∞n=0if and only ifxn≤ynfor eachn∈N, and
Y=
(xn)∞n=0∈l1|
∞
n=0
xn ≤1. (2.7)
Show that the results ofCorollary 2.7hold for each increasing mappingG:X→X for whichG[X]⊆Y⊆X⊆l1.
Solution. Assume thatG:X→X is increasing, and thatG[X]⊆Y ⊆X⊆l1. It is well-known (cf., e.g., [11, Corollary 5.8.6]) that all nonempty w.o. and i.w.o. subsets ofYhave supremums and infimums inX. ThusG[X] is relatively well-order complete. Moreover, choosingc=(0)∞n=0, it is easy to see that both sup{c,x}and inf{c,x}exist for allx∈Y.
SinceG[X]−∪G[X]+⊆Y, thenGsatisfies the hypotheses ofCorollary 2.7.
Solution. The constant sequences (−1)∞n=0and (1)∞n=0are least and greatest elements ofY, whence they are lower and upper bounds ofG[X]. Moreover, as inExample 2.8, one can show thatG[X] is relatively well-order complete. Thus the hypotheses ofCorollary 2.3
hold.
Remarks 2.10. Formula (2.6) is reduced to the generalized iteration method (I) intro-duced in [11, Theorem 1.1.1] when fc(y)≡y. In particular,Corollary 2.3(a) is a spe-cial case of [11, Theorem 1.2.1], andCorollary 2.3(b) is a special cases of [11, Proposi-tion 1.2.1].
It can be shown that the first elements of the well-ordered subsetCwhich satisfies (2.6) are the following iterations:x0=c,xn+1= fc(Gxn),n=0, 1,..., as long asxn+1 ex-ists andxn< xn+1. Ifxn+1=xn, thenx=xn. If (xn)∞n=0 is strictly increasing, thenxω=
fc(sup{Gxn}
n∈N) is the next element ofC. Choosingx0=xωabove we obtain the next possible elements ofC, and so on.
Similarly, it is easy to show that ifG[X] inTheorem 2.4and inProposition 2.5is finite, the fixed pointsx∗andx∗ofGare the last elements of the finite sequences determined
by the following algorithms:
(i)x0=c. Fornfrom 0 whilexn=Gxn,xn+1=GxnifGxn< xnelsexn+1=fc(Gxn). (ii)x0=c−. Fornfrom 0 whilexn=Gxn,xn+1=GxnifGxn> xnelsexn+1=fc(Gxn). Compared to results of [6], no sequencibility hypotheses are needed in the proof of Proposition 2.6. Propositions 2.1 and 2.2 of [10] could also have been used to prove the existence of maximal and minimal fixed points ofGinProposition 2.6.
3. On common fixed points of mapping families
In this section we apply results ofSection 2to derive existence results for common fixed points of a family of mappingsGt:X→X,t∈S, whereXis a poset andSis a nonempty set. By acommon fixed pointof{Gt}t∈Swe mean a pointx∈Xfor whichGtx=xfor each t∈S. Least, greatest, minimal and maximal common fixed points are defined as in the case of a single operator. We assume that for a fixedt0∈S
(G0)GtGt0=Gt0Gtfor allt∈S,
and that one of the following hypotheses is valid. (G1)Gtx≤Gt0xfor allt∈Sandx∈X.
(G2)Gt0x≤Gtxfor allt∈Sandx∈X.
As an application ofProposition 2.6we prove the following existence result for the existence of common fixed points of the operator family{Gt}t∈S.
Theorem3.1. Assume that (G0) holds, thatGt0is increasing, thatGt0[X]is relatively
well-order complete, and thatGt0[X]+has an up-map orGt0[X]−has a down-map.
(a)If (G1) holds, the family{Gt}t∈Shas a minimal common fixed point. (b)If (G2) holds, the family{Gt}t∈Shas a maximal common fixed point.
Proof. (a) The hypotheses assumed forGt0 imply byProposition 2.6thatGt0has a
min-imal fixed pointx−. Applying this result and the hypothesis (G0) we see thatGt0Gtx−=
GtGt0x−=Gtx−. Thus eachGtx−is also a fixed point ofGt0. Hence, if (G1) holds, then
whencex−is a common fixed point of{Gt}t∈S. To prove thatx−is minimal, assume that
xis a common fixed point of{Gt}t∈S, and thatx≤x−. Thenxis a fixed point ofGt0and
x−is its minimal fixed point, so thatx=x−. Thusx−is a minimal common fixed point
of{Gt}t∈S.
(b) The result of (b) is similar consequence ofProposition 2.6. The next result is a consequence ofCorollary 2.3.
Theorem3.2. (a)If (G0) and (G1) hold, ifGt0is increasing, and if (Ga) holds forG=Gt0,
then{Gt}t∈Shas the least common fixed point, and it is increasing with respect toGt0.
(b)If (G0) and (G2) hold, ifGt0is increasing, and if (Gb) holds forG=Gt0, then{Gt}t∈S
has the greatest common fixed point, and it is increasing with respect toGt0.
Proof. (a) The hypotheses (Ga) given forG=Gt0imply byCorollary 2.3thatGt0has the
least fixed pointx∗, and it is increasing with respect toGt0. This result and the hypotheses
(G0) and (G1) imply thatGtx∗is a fixed point ofGt0 andGtx∗≤Gt0x∗=x∗ for each
t∈S. Sincex∗is the least fixed point ofGt0, thenGtx∗=x∗, whencex∗is a common
fixed point of{Gt}t∈S. Ifxis a common fixed point of{Gt}t∈S, thenxis a fixed point ofGt0, andx∗is its least fixed point, whencex∗≤x. Thusx∗is the least common fixed
point of{Gt}t∈S.
The proof of (b) is similar to the above one.
Remarks 3.3. The phrase: “increasing with respect toG” of a least (greatest) fixed point x∗(x∗) ofG:X→X in some order interval means that if ˜G:X→X satisfies the same
hypotheses asG, and ifGx≤Gx˜ for allx∈X, thenx≤x˜∗(x∗≤x˜∗), where ˜x∗ (˜x∗)
denotes the corresponding least (greatest) fixed point of ˜G. In particular, the results of Section 2can be applied to find increasing selectors for fixed points of increasing families of increasing mappings, as demonstrated in the last part of the next example. The first part of it proves a result stated in the Introduction.
Example 3.4. Whenc=(c1,...,cm)∈RmandR∈(0,∞), the space
X=
x=x1,...,xm
∈Rm|m i=1
xi−ci 2
≤R2
, (3.1)
ordered coordinatewise, is well-order complete. The centerc=(c1,...,cm) ofXis an order center ofX, that is, sup{c,x}and inf{c,x}belong toXfor eachx∈C. Hence, if{Gt}t∈S satisfies (G0) and (G1), (respectively (G0) and (G2)), and ifGt0 is increasing, it follows
fromTheorem 3.1that {Gt}t∈S has a minimal (respectively a maximal) common fixed point. Moreover, ifSis a poset ifG:X×S→Xis increasing with respect to the product ordering of X×S, it follows fromCorollary 2.7 thatGt=G(·,t) has the greatest fixed pointxt∗in (xt], wherext=min{x∈X|x=sup{c,Gtx}}, and that the mappingt→xt∗ is increasing.
4. Applications to Markov processes
in the sense that ifxn→x,yn→yandxn≤ynfor eachn, thenx≤y. For instance, closed subsets of separable ordered Banach spaces are p.o. Polish spaces. LetᏮbe a family of all Borel subsets ofE, and letᏹdenote the space of probability measures onE, that is, the space of all countably additive functionsp:Ꮾ→[0, 1] for whichp(E)=1.
We say that a sequence (pn) ofᏹconverges weaklytop∈ᏹif limn→∞Ef(x)pn(dx)=
Ef(x)p(dx) for each bounded and continuous functionf :E→R.
In view of [5, Theorem 11.3.3] the weak convergence is equivalent to the convergence in theProhorov metricρonᏹdefined by
ρ(p,q) :=inf>0|p(A)≤qx∈E|d(x,A)<+∀A∈Ꮾ, p,q∈ᏹ. (4.1)
This result and [14, Theorem 2] imply that relation:pqif and only if p(A)≤q(A) for eachA∈Ꮾwhich is increasing, that is,y∈Awheneverx∈Aandx≤y, defines a closed partial ordering on (ᏹ,ρ).
Lemma4.1. LetBbe a nonempty closed set inE, andᏼ:= {p∈ᏹ|p(B)=1}.
(a)If increasing sequences ofBconverge, then each nonempty w.o. subsetWofᏼcontains an increasing sequence which converges tosupW.
(b)If decreasing sequences ofBconverge, then each nonempty i.w.o. subsetWofᏼ con-tains a decreasing sequence which converges toinfW.
(c)If monotone sequences ofBconverge, thenᏼis relatively well-order complete inᏹ. (d)ᏼis a closed subset of(ᏹ,ρ).
Proof. (a) Assume that each increasing sequence ofBconverges, and let (pn)∞n=0be an in-creasing sequence inᏼ. By [15, Proposition 4] there exists an a.s. increasing sequence {Yn|n∈N} of B-valued random variables, defined on a common probability space (Ω,µ) such thatµ(Y−1
n (A))=pn(A) for alln∈NandA∈Ꮾ. Since increasing sequences ofBconverge, then (Yn)∞n=0converges a.s. inΩ, and the limit is by [5, Theorem 4.2.2] a random variable on (Ω,µ). Thus, by [15, Theorem 6] the sequence (pn)∞n=0 converges weakly, and hence in (ᏹ,ρ). The assertion of (a) follows then from [11, Proposition 1.1.5].
The proof in the case (b) is dual to the above one, and (c) follows from (a) and (b). (d) If (pn)∞n=0is a sequence inᏼwhich converges topin (ᏹ,ρ), thenpn→pweakly by [5, Theorem 11.3.3]. Thus, by [5, Theorem 11.1.1],p(B)≥lim supn→∞pn(B)=1, whence p∈ᏼ. This result implies thatᏼis a closed subset of (ᏹ,ρ).
4.2. Existence results for invariant measures of a Markov process. LetS=(S, +) be a commutative groupoid. For instance,Scan be a set of all nonnegative or positive real numbers, rational numbers or integers. Following the terminology adopted in [20] we say that a mappingP:S×E×Ꮾ→[0, 1], called atransition function, defines aMarkov processP(t,x,A) on thephase space(E,Ꮾ) if the following conditions hold:
(i)P(t,·,A) is aᏮ-measurable function onEfor all fixedt∈SandA∈Ꮾ. (ii)P(t,x,·)∈ᏹfor all fixedt∈Sandx∈E.
We say thatp∈ᏹis aninvariant measureofP(t,x,A) if
p(A)=
EP(t,x,A)p(dx) ∀t∈S,A∈Ꮾ. (4.2) Conditions (i) and (ii) ensure that the equation
Gtp(A)=
EP(t,x,A)p(dx), A∈Ꮾ, (4.3)
defines for eacht∈S a mappingGt:ᏹ→ᏹ. Thus p∈ᏹis an invariant measure of P(t,x,A) if and only if pis a common fixed point of the operatorsGt,t∈S. As an easy consequence of theChapman-Kolmogorov equation(iii) and the definition (4.3) ofGtwe obtain the following result.
Lemma4.2. The operatorsGt,t∈Scommute, that is,GtGs=GsGtfor allt,s∈S.
Results ofSection 3andSubsection 4.1will now be applied to derive existence results for extremal invariant measures of a Markov processP(t,x,A).
Theorem4.3. Assume there exists at0∈Ssuch that (P0)x≤yinEimpliesP(t0,x,·)P(t0,y,·)inᏹ; (P1)P(t,x,·)P(t0,x,·)for allt∈Sandx∈E;
(P2)there exists a closed subset B of Ewhose monotone sequences converge such that P(t0,x,B)=1for eachx∈E.
IfBin (P2) has a lower bound inE, thenP(t,x,A)has the least invariant measure, and it is increasing with respect toP.
Proof. It suffices to show that the family of the operatorsGt:ᏹ→ᏹ,t∈S, defined by (4.3) satisfies the hypotheses ofTheorem 3.2. The hypothesis (G0) holds byLemma 4.2, (P1) implies that the hypothesis (G1) is valid, andGt0 is increasing by (P0) (cf. [13]).
The hypothesis (P2) implies thatGt0p(B)=1 for eachp∈ᏹ, whenceGt0[ᏹ] is relatively
well-order complete byLemma 4.1. IfBhas a lower bounda∈E, then relation
p(A)=
1 ifa∈A,
0 otherwise ,A∈Ꮾ, (4.4)
defines a lower bound ofGt0[ᏹ]. Thus also the hypothesis (Ga) holds, whence the
fam-ily{Gt}t∈Shas byTheorem 3.2the least common fixed point p∗. This result, (4.2) and
(4.3) imply that p∗is the least invariant measure ofP(t,x,A). Moreover, it follows from
Theorem 3.2thatp∗is increasing with respect toGt0, and hence also toPby (4.3).
The next result is the dual to that ofTheorem 4.3, and it is a consequence ofTheorem 3.2(b).
and ifBin (P2) has an upper bound inE, thenP(t,x,A)has the greatest invariant measure, and it is increasing with respect toP.
The next result is an application ofTheorem 3.1andLemma 4.1,
Theorem4.5. Assume there exists at0∈Ssuch that the hypotheses (P0) and (P2) hold, and that the setᏼ= {p∈ᏹ|p(B)=1}, whereB is given by (P2), has an up-map or a down-map. ThenP(t,x,A)has
(a)a minimal invariant measure if (P1) holds; (b)a maximal invariant measure if (P3) holds.
Proof. (a) The hypothesis (P0) implies thatGt0 is increasing. Since Bin (P2) is closed,
and sinceGt0[ᏹ]⊆ᏼby (P2), it follows fromLemma 4.1thatGt0[ᏹ] is relatively
well-order complete, and that ᏼcontains bothGt0[ᏹ]+ andGt0[ᏹ]−. Thus an up-map of
ᏼis also an up-map of Gt0[ᏹ]+, and a down-map ofᏼis a down-map of Gt0[ᏹ]−.
The hypothesis (P1) implies that (G1) is valid forG=Gt0, which thus satisfies all the
hypotheses ofTheorem 3.1(a). Consequently, the family{Gt}t∈Shas a minimal common fixed pointp−. In view of (4.2) and (4.3)p−is a minimal invariant measure ofP(t,x,A).
(b) The given hypotheses and (P3) imply that the hypotheses ofTheorem 3.1(b) hold,
which implies the assertion of (b).
4.3. Special cases. In this subsection we consider the special case when the transition functionP:E×Ꮾ→[0, 1] which defines a Markov processP(x,A) on (E,Ꮾ) is indepen-dent on the parametert, and satisfies the following hypotheses.
(a)P(·,A) is aᏮ-measurable function onEfor all fixedA∈Ꮾ. (b)P(x,·)∈ᏹfor all fixedx∈E.
In this case (4.2) is reduced to the form
p(A)=
EP(x,A)p(dx), A∈Ꮾ. (4.5)
Thusp∈ᏹis a invariant measure ofP(x,A) if and only ifpis a fixed point ofG:ᏹ→ᏹ, defined by
Gp(A)=
EP(x,A)p(dx), A∈Ꮾ. (4.6)
The hypotheses (P1) and (P2) are reduced to the following form. (Pa)x≤yinEimpliesP(x,·)P(y,·) inᏹ.
(Pb) There exists a closed setBinEwhose monotone sequences converge inEsuch thatP(x,B)=1 for eachx∈E.
As a special case ofTheorem 4.3andProposition 4.4we obtain the following. Proposition4.6. Let the hypotheses (Pa) and (Pb) hold.
(b)IfBin (Pb) has an upper bound inE, thenP(x,A)has the greatest invariant measure, and it is increasing with respect toP.
As a consequence ofProposition 4.6we obtain the following result.
Corollary4.7. LetHbe a p.o. Polish space whose order-bounded and monotone sequences converge, and assume that a closed subsetEofHcontains an order interval[a,b]ofH. If the transition functionP of a Markov processP(x,A)on(E,Ꮾ)satisfies the hypotheses (a), (b) and (Pa), and ifP(x, [a,b])=1for eachx∈E, thenP(x,A)has least and greatest invariant measures, and they are increasing with respect toP.
Proof. [a,b] is a closed subset ofEwhose monotone sequences converge by assumption, and [a,b] has least and greatest elements, so that the assertions follow fromProposition
4.6whenB=[a,b].
The next result is a consequence ofTheorem 2.4and Propositions2.5and2.6. Proposition4.8. Let the hypotheses (Pa) and (Pb) hold, letBis as in (Pb), and denote ᏼ= {p∈ᏹ|p(B)=1}.
(a)Ifᏼhas an up-map, thenP(x,A)has minimal and maximal invariant measures, and the greatest invariant measurep∗in(p], wherep=min{p|p=fc(
EP(x,·)p(dx))}. (b)If ᏼhas a down-map, thenP(x,A)has minimal and maximal invariant measures,
and the least invariant measurep∗in[p), wherep=max{p|p=fc(
EP(x,·)p(dx))}. (c)p∗,p∗,pandpare increasing with respect toP.
Proof. (a) As in the proof ofTheorem 4.5it can be shown thatG, defined by (4.6), satis-fies the hypotheses ofTheorem 2.4, whence the assertions follow fromTheorem 2.4and Proposition 2.6.
The assertions of (b) are similar consequences of Propositions2.5. and2.6, and the assertions of (c) follow fromTheorem 2.4(b) and fromProposition 2.5(b). Remarks 4.9. In [13] the existence of an invariant measure of a Markov processP(x,A) with property (Pa) is proved in the case whenEis compact and has the least element. This result follows, for example, fromProposition 4.6(a) whenB=E, since in a compact par-tially ordered Polish space all monotone sequences converge. This convergence property holds also whenE(orB) is a closed and order-bounded subset of an ordered separable Banach spaceHwhose order coneKis regular.E(orB) may be also norm-bounded ifK is fully regular. This holds, for instance, ifHis weakly sequentially complete andKis nor-mal (cf. [7, Theorem 2.4.5]). In particular, ifHis infinite-dimensional, and if its subsetE contains an open set, thenEis not compact, not even locally compact, as assumed in [20] in the proof of the existence of an invariant measure. As for examples of such ordered Banach spacesHsee, for example, [7, Section 2], and [11, Section 5.8].
In [12] a number of existence results for invariant measures are derived for a Markov process whose transition function has properties (a) and (b).
Proposition 4.8to overcome this difficulty. This situation is demonstrated in the follow-ing example.
4.4. An example. Assume that an Euclidean m-space Rm is ordered coordinatewise. Givenβ,R∈(0,∞) and (a1,...,am)∈Rm, denote
B=x1,...,xm
∈Rm| x1−a1 β+···+ x
m−am β≤Rβ
. (4.7)
Letᏹbe the set of all probability measures on a closed subsetEofRm which contains B. Assume thatP(x,A) is a Markov process on (E,Ꮾ) which satisfies the hypothesis (Pa), and for whichP(x,B)=1 for eachx∈E. The setBis closed and each monotone sequence ofBconverges, whence the hypothesis (Pb) holds. Defining
c(A)=
1 if (a1,...,am)∈A,
0 otherwise, A∈Ꮾ, (4.8)
we obtain an elementcofᏹ. For eachp∈ᏹ, denote byFpthe (cumulative) distribution function ofp, and let fc(p) and f
c(p) be the probability measures onBwhose distribu-tion funcdistribu-tions are
Ffc(p)x1,...,xm=
Fp
x1,...,xm ifxi≥ai,i=1,...,m,
0 otherwise,
Ffc(p)
x1,...,xm
=
1 ifxi≥ai,i=1,...,m, Fpx1,...,xm otherwise.
(4.9)
Routine calculations show that fcis an up-map and f
cis a down-map ofᏼ= {p∈ᏹ| p(B)=1}. Thus the hypotheses ofProposition 4.8hold.
According to the conclusions ofProposition 4.8the Markov processP(x,A) has min-imal and maxmin-imal invariant measures pm and pm, the greatest invariant measurep∗in the order interval (p] ofᏹ, wherep=min{p∈ᏹ|p(·)= fc(
EP(x,·)p(dx))}, and the least invariant measure in the order interval [p) ofᏹ, where p=max{p∈ᏹ|p(·)=
fc(EP(x,·)p(dx))}.
Consider next the case when the mappingp→EP(x,·)p(dx) has a finite number of values. In view ofRemarks 2.10the definition (4.6) ofGand the above choices of fcand
fcwe can determinep,p,p∗,p∗,pmandpmin the following manner. (1) pis the last element of the finite sequence of iterations
p0=c, pn+1(·)= fc
EP(x,·)pn(dx)
, as long aspn≺pn+1. (4.10)
(2) pis the last element of the finite sequence of iterations
p0=c, pn+1(·)=fc
EP(x,·)pn(dx)
(3) p∗is the last element of the finite sequence of iterations
q0=p, qn+1(·)=
EP(x,·)qn(dx), as long asqn+1≺qn. (4.12) (4) p∗is the last element of the finite sequence of iterations
q0=p, qn+1(·)=
EP(x,·)qn(dx), as long asqn≺qn+1. (4.13) (5) pmis the last element of the finite sequence formed byq0=p, andqn+1is a strict
lower bound ofEP(x,·)qn(dx) as long as such a lower bound exists. (6) pmis the last element of the finite sequence formed byq0=p, andq
n+1is a strict upper bound ofEP(x,·)qn(dx) as long as such an upper bound exists.
Remarks 4.10. Notice thatB, defined by (4.7), is not convex whenβ∈(0, 1). In fact,B can be any closed subset ofRm which has an order centera=(a1,...,am)∈B, that is, sup{a,x}and inf{a,x}exist and belong toBfor eachx∈B.
The sequences (qn) in (5) and (6), as well their last elements, minimal and maximal invariant measures ofP(x,A), may not be uniquely determined.
The above example can easily be extended to the case when the transition function depends also on the parametert∈S.
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S. Heikkil¨a: Department of Mathematical Sciences, University of Oulu, P.O. Box 3000, Oulu 90014, Finland
