OSHIMA, YOICHI

Osaka City University Repository

Oshima, Y.
Osaka J. Math.
6 (1969), 29-37
A NECESSARY AND SUFFICIENT CONDITION FOR
A KERNEL TO BE A WEAK POTENTIAL KERNEL
OF A RECURRENT MARKOV CHAIN
YOICHI OSHIMA
(Received December 14, 1968)
1. Introduction
Let P be an irreducible recurrent transition probability on a denumerable
space S with invariant measure a. Let c be an arbitrary (but fixed) state of S.
Then from the work of Kondό [3] and Orey [8], there exist the class of weak
potential kernels A(x, y) defined by the property that, for every null charge/, Af
is bounded and satisfies the equation
(l l) (I-P)Af = f.
Moreover Af is represented by
(1-2) Af='Gf+l(f),
where / is a null charge, 1( ) is an arbitrary linear functional on the space of
null charges and CG is defined as follow;
ΓP(*,
lθ
(1.3) cP(x, y) = , y)
otherwise,
(1.4) 'G(χ,y)=ΓΣ£-*cPm(*,y)
lθ otherwise.
Moreover A satisfies the following maximum principle [4], [5]:
(RSCM)'> If m is a real number and/ is a null charge then the relation that
(1.5) m^Af on the set {/>0}
implies that
(1.6) m—f~:>Af everywhere,
1) This is the abbreviation of "reinforced semi-complete maximum principle"; this
maximum principle corresponds to the semi-complete M.P. as well as the reinforced M.P.
(of Meyer) corresponds to the complete M.P.
30 Y. OSHIMA
where /' = (—/)VO.
In the present paper we are concerned with the following construction
problem. Given a positive measure a and a (not necessarily positive) kernel A
satisfying (RSCM), does there exist an irreducible recurrent transition probability
which has a as its invariant measure, and A as its weak potential kernel ? This
is not true in genera!2), but as Kondo [4] has proved, it is true if a is a finite
measure. In section 2 we shall introduce another necessary condition for the
weak potential kernel A (referred to as condition (*)). Then we shall prove
(theorem 3.1) that, if the pair (A, a) satisfies maximum principle (RSCM) and
condition (*), A is a weak potential kernel of a (unique) recurrent Markov chain
with a as its invariant measure.
I should like to express my hearty gratitude to T. Watanabe for his kind
advices.
2. Some potential theory for a kernel A satisfying (RSCM)
Let a be a strictly positive measure and A, a kernel on S. A function/ on
S is said to be a null charge with respect to a if Σ a(x) \f(x) \ < °° and 2 a(x)f(x)
=0. Let N be the space of null charges vanishing outside a finite subset of S.
We assume that the kernel A satisfies condition (RSCM) for f^N. Fix an
arbitrary state c and define
(2.1) CG(X, y) = A(x, y)-A(c, y)-(A(x, c)-A(c, c))^-.
a(c)
If A is a weak potential kernel then (2.1) is clearly satisfied by taking/ in
equation (1.2) as
(2.2)
a(y)Y' x = c
a(c)
— 1 x = y
0 otherwise,
and calcurating Af(x)—Af(c).
From definition (2.1) cG(c, x)=cG(xy c)=0 for every
Lemma 2.1 For arbitrary elements x, y in S which are different from c
I(X, y)^G(X, y)^G(y, y) .
Proof. By taking/ as (2.2) we have
a(c)
2) Λ counter example was given by Kondo and T, Watanabe,
WEAK POTENTIAL KERNEL OF A RECURRENT MARKOV CHAIN 31
Hence, if we write /+=/vθ, /-=(-/) Vθ, by (RSCM)
A(y, c) , ,
a(c) a(c)
so that
(2.3) A(y, , , ,
a(c) a(c}
^A(c,c)^-A(c,y)-I(X,y),
which proves the lemma.
Corollary. For every x^S there exists a constant C such that
(2.4) cG(x, y) ^  C a(y) for every y e S .
Proof. Exchanging c and x in the second inequality in (2.3), it follows that
°G(X, y) = A(x, y)-A(c, y)-(A(x, c)-A(c, c))2&
a(c)
a(c) a(x) a(c) a(x)
Let CS be the set S— {c}, and CM be the space of all functions on CS vanishing
outside a finite subset of CS. Let CM + be the space of all non-negative functions
in CM.
Theorem 2.1. The kernel CG satisfies the reinforced maximum principle [7] :
(RM) If a is a non-negative constant and if cf and cg are two elements of CM+,
then the relation that
(2.5) a+cGcf—cf^cGcg on the set [cg>fy implies that
(2.6) a+cGcf-cf^cGcg everywhere on CS.
Proof. Let / be the function on S such that f^N and f\ c
s
=
cf. Such /
is obviously unique. The function g^N is defined similarly. Then inequality
(2.5) implies that
f on the set £-
For, since c/and cg are non-negative, the set {g— />0} is contained in the union
of c and {cg>0}. Hence by (RSCM)
32 Y. OSHIMA
<*+A(g-f)(c)-(g-fΓ ^A(g-f) everywhere.
Since the function (g—f)~ is equal to cf on CSΓ\ {£=0}, the above inequality,
combined with (2.5), proves the theorem.
A non-negative function ch on CS is said to be quasi-excessive^ if, for every
cg^cM, the inequality
ch^cGcg on the set {cg>0}
implies that
ch—cg~ ^> °Gcg everywhere.
Moreover Meyer introduced the notion of the pseudo-rtduite cHE
ch for
every quasi-excessive function ch and every subset E of CS. This function
cHE
ch satisfies the following four conditions.
(2.7) cHE°h is quasi-excessive.
(2.8) ΉEch^ch on CS and ΉEch=ch on E.
(2.9) If chl and Ch2 are two quasi-excessive functions such that Ch^ch2 on ZJ,
(2.10) If cf£ΞcM+ vanishes outside of E then ΉEcGcf=cGcf.
For example, the function CGC/, c/ecj/+, and every positive constant are
quasi-excessive ([7] see also [5]).
Now we introduce a condition.
Condition (*): There exists a sequence of finite sets {E
n
}
n
^lf2)... increasing
to S such that c^E
n
 for each n, and a sequence {/*„}„ =ι>2, of function on S
satisfying the following conditions.
(i) O^A^l, A
w
(c)=0, *„=! on F
n
=S-E
n
, and limA^O.
(«) For every f^N and every real number m (^Af(c)) the relation that
m+h
n
^Af on the set {/>0}.
implies that
m+h
n
—f~:>Af everywhere on CS.
In section 3 we shall show that if A is a weak potential kernel of an irreduci-
ble recurrent Markov chain, it satisfies condition (*).
Theorem 2.2 Condition (*) is equivalent to the condition that, there exists
a sequence of finite sets {cE
n
}
n=lt2t... increasing to
 CS such that
(2.11) HmΉc
s
_cBn l = 0.
Proof. Suppose that condition (*) holds and let ch
n
 be the restriction of
h
n
 to CS and cE
n
=
eSr\E
n
. Obviously cS-cE
n
=F
n
, 0^cA
n
^l, and ch
n
= I on
F
n
. It then follows that ch
n
 is a quasi-excessive function for every n. In fact,
3) This definition is slightly different from Meyer's one; this is the discrete version of
Meyer's,
WEAK POTENTIAL KERNEL OF A RECURRENT MARKOV CHAIN 33
let c/be in CM and/, the extention of c/to S such that/<ΞΛΓ. If
ch
n
^
 cGcf on the set {c/>0}
then
h
n
+Af(c)^Af on the set {/>0} ,
since {c/>0} is contained in {/>0} U {c}. Hence from condition (*),
h
n
+Af(c)—f~:>Af everywhere on CS,
that is,
ch
n
-
cf-^°Gcf everywhere on CS.
Since cHp
n
* 1 =ch
n
 by definition,
limc^M l -0.
Conversely, if (2.11) holds, set eEH\J{c}=En, Fn=S—En and
Γ ΉFn l on
 CS
h
» = 1 A'"[ 0 at c.
It is enough to show the property (ii) of condition (*). Suppose that, for some
f£ΞN and some real number m (2^
on
Then one has
m-Af(c)+ΉFn l^cGcf on {c/
where ef is the restriction of/ to CS. The fact that m—Af(c)-\-ΉFn 1 is a quasi-
excessive function implies that
m-Af(c}+ΉFnΛ-cf-^ιcGcf everywhere on CS,
which is nothing but condition (*).
Note. If a is a finite measure, then condition (*) is satisfied.
Let IF be the indicator function of a set JP, then from lemma 2.1 CGIF^1 on
F. Hence from (2.8) and (2.9) ΉF \^CGIF. Hence if Fn decrease to empty
set, inequality
e
ίWl(*)^βσ/,
β
implies that
Where the second inequality follows from the corollary of lemma 2.1.
3. Main result
Let A be a weak potential kernel of an irreducible recurrent transition
34 Y. OSHIMA
probability P with invariant measure a. We shall now prove that A satisfies
condition (*) of section 2.
Define CP and CG as (1.3) and (1.4) respectively. Let ΉFbe the rέduite
defined by CP. Since CHF 1 is the pseudo-rόduite associated with the above
CG (see [5] P. 37, theorem 1.3), it is enough to show that for a sequence of finite
sets {°E
n
}
n=lt2>... increasing to
 CS, lim ΉFn l=Q (Fn=cS—cEn) by theorem 2.2.
One can easily seen that the function ch(x)=\im cHFn l(x) is an invariant function
for CP (i.e. cPch=ch) and bounded by 1. On the other hand,
1
 =
 cG(l-cP \)(x)+\imcPn \(x)
and
lim CPΛ \(x) = lim P
x
[σ{c]>n] = 0 ,
implies that 1 is a potential of non-gengative function (where σ{
c
} is the
hitting time of the Markov chain with transition probability P). Hence ch is
also a potential. The fact that ch is an invariant function and also a potential
shows that ch=0.
The main result of the present paper is this.
Theorem 3.1. Given a positive measure a and a kernel A satisfying maximum
principle (RSCM) and condition (*), there exists a unique irreducible recurrent
transition probability P which has a as its invariant measure, and A as its weak
potential kernel.
Uniqueness was proved by Kondό [4]. We shalll divide the proof of
existence into several lemmas. In the following we shall use the notation of
section 2 with no further reference.
Lemma 3.1. There exists a sub-Markov transition probability €P(xy y) on
CS such that
cpn(χ> y^ f°r every *> y ίn °s
Proof. See Meyer [7] P. 238 lemma 10.
Lemma 3.2. For every y e cSy Σ*ΦC oί(x)€P(x, y) ^  a(y).
Proof. To the contrary, suppose that there exists some state y^cS such
that
Then there exists a finite subset F of CS containing y and satisfying
Σ*
e
F a(x)cP(x, y)-a(y) = a>0 .
Define a function f^N by
WEAK POTENTIAL KERNEL OF A RECURRENT MARKOV CHAIN 35
CP(X, y)-I(X, y) x^F
a
7 \ ^—C
a(c)
0 otherwise.
Since Af+f~ attains its maximum on the set {/>0} and since /(c)<0, there
exists a state x0^F such that,
Af(x0)^Af+f" everywhere on S.
In particular,
α(c)
Hence,
°>-^^ Af(c)-Af(Xΰ) = cG(~J)(Xo) ,
where cf is the restriction of / to CS. On the other hand,
CG(-7)(*0) = °G(x<n jO-Σ,^ CG(*0, z)cP(z, y)
^
 cG(Xo, y)-(cG(Xo, y)-I(Xΰ, y)) = /(*„, j)^0 .
This lead us to a contradiction.
Lemma 3.3. CG(1-CP \}=\ on CS .
Proof. For any positive integer n, we have
1 - ΣA cP*(l-βP l)(*)+cP"+1 l(*) .
Passing to the limit we obtain
where r(x)=lim cpn+1. l(^). It remains to show that r(x)=0. From condition
(*) for arbitrary £>0 there exists a number M such that for any integer mΞ>M,
Hence
Σ,ΦC
 cP"+l(x, y) = cP"+ίIFm(X)+cP"+ΊEm(x)^Hfm. l(x)+cP"^IEm(X) ,
where IF is the indicator function of F. Tending n to infinity we obtain
Lemma 3.4. Σ*ΦC ^  (*)( 1 ~ cp 1 ) W ^
Proof. Let JP be an arbitrary finite subset of C*S, and define
36 Y. OSHIMA
1-CP-1(*) x<=F
0 otherwise.
As noted in the proof of lemma 3.2, there exists a state xQ^F such that
Af(x0)^Af+f- on S.
In particular,
cGef(x0) = Af(x0)-Af(c)^f-(c) = Σ^F a(y)(l-cP l ) ) / a ( c ) ,
and by lemma 3.3, the left side of the above inequality is bounded by 1.
Now we can define the desired transition probability P.
ί
 cP(χ,y)
(3.1) P(*,y)= (αOO-
α
«P(j,))/
α
(
e
)
From lemmas 3.2 and 3.4, P is a transition probability on S.
Lemma 3.5. aP=a and (I—P)Af=ffor any
Proof. If #Φ£, then
and
(I-P)Af(x) = (I-P)(Af(c)+°Gf)(X) =f(x) .
By the same argument for ΛJ— c, lemma follows.
Lemma 3.6. The transition probability P is recurrent and irreducible.
Proof. Let σ[χ} be the hitting time for x of the Markov chain with
transition probability P. Then for every #Φ£,
P,['u<<*>] = ΣjΦc'Gfo y)P(y, c) = CG(1-CP 1) (*)=!,
by lemma 3.3. Hence,
where σ>} is the positive hitting time for state c. Thus c is a recurrent state
for P and hence also for P, where P is defined by,
(3.2) P(X,y)= ^\a(x)
WEAK POTENTIAL KERNEL OF A RECURRENT MARKOV CHAIN 37
Moreover,
•P*[<r(c}<°°]=l>
α (C)
shows that
Pc[σ{χ}<°°]>0 for a11 *eΞcS.
Hence x is a recurrent state for P and hence for P. Since # is recurrent and
P*[σ {
c
}<oo]=l, it follows that P
c
[σ{χ}<°°]=l for all Λ:e5. Irreducibility
follows from the fact that, -P
r
[σ {
c
}<00]— 1 and P
c
[cτl:v}<oo]=l for every xyy
in 5.
OSAKA CITY UNIVERSITY
Bibliography
[1] G.A. Hunt: Markov processes and potentials II, Illinois J. Math. 1 (1957), 316-395.
[2] J.G. Kemeny and J.L. Snell: Potentials for denumerable Markov chains, J. Math.
Anal. Appl. 3 (1961), 196-260.
[3] R. Kondό: On weak potential operators for recurrent Markov chains with con-
tinuous parameters, Osaka J. Math. 4 (1967), 327-344.
[4] R. Kondδ: On a construction of recurrent Markov chains, Osaka J. Math. 6 (1969),
13-28.
[5] R. Kondδ, Y. Oshima and T. Watanabe: Topics in Markov Chain, Seminar on
Probability, vol. 28, 1968 (Japanese).
[6] P.A. Meyer: Probability and Potentials, Blaisdell Publishing Company, 1966,
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[7] P.A. Meyer: Caractέήsation des noyaux potentiels des semi-groupes discrets,
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[8] S. Orey: Potential kernels for recurrent Markov chains, J. Math. Anal. Appl. 8
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A NECESSARY AND SUFFICIENT CONDITION FOR A KERNEL TO BE A WEAK POTENTIAL KERNEL OF A RECURRENT MARKOV CHAIN

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A NECESSARY AND SUFFICIENT CONDITION FOR A KERNEL TO BE A WEAK POTENTIAL KERNEL OF A RECURRENT MARKOV CHAIN

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