Actes, Congrès intern. Math., 1970. Tome 2, p. 507 à 512.
ENTRANCE AND EXIT SPACES
FOR A MARKOV PROCESS
by E . B .
DYNKIN
Entrance and exit spaces (closely related to Martin entrance and exit boundaries) play an important rôle in the theory of Markov processes (see [1] — [6]).
We shall outline a new method of constructing these spaces based on the consideration of conditional processes. The method is applicable to Markov processes
in the most wide sence. The theory becomes not only more general but also much
simpler. Particularly it becomes completely invariant with respect to the time
reversion.
We consider inhomogeneous processes. The relation of the stated theory to
the usual one dealing with the homogeneous processes will be treated in another
place.
1. Denumerable time set.
1.1. - We shall consider processes having the state space (E,(R) where E is
a locally compact Hausdorff space with countable basis and 03 is the a-field of
its Borel sets.
Let T be a denumerable subset of the interval (a, + °°) where — °° < a < 4- °°
and let a be a limit point of T. Denote by £2 the set of all functions co(f) defined
on T with values in E. For each T' Ç T denote by 8H(T') the a-field generated by
the sets {co : w ( 0 e r } (teT ;TGtB). Set
&lt = EK(Tn (a , t])t&Ü = 01(7*0 17,00)) N =SfC(T) .
Denote by 5ft the set of all measures P on &t satisfying condition P(fì) < 1.
The set of all P G ft for which P(ft) = 1 will be denoted by \ . Let S be the collection of all functions £ (co) of the form
/iMf,)].../„[«(*,)]
where n=\,2,...,tl
t„GT
and
fx
/„
are non-negative continuous functions with compact supports. Introduce into ft
a topology setting Pfc -> P if Mk £ -> M £(*) for all £ E S. The topological space 31
is a compactum. We shall study stochastic processes (xt, P) where xt(u) = œ(t)
(tET) and PG ft0. Set Pe 3Ä if the process (xt, P) has the Markov property :
for each f G 7 \ £ G 0 l , , q e SC*9
(1) M means the integral with respect to the measure P.
508
KB. DYNKIN
(1.1)
Mito\xt)
D 5
(a.s.PX1)
= M(Ì\xt)Miv\xt)
This property is equivalent to the following one : for each tGT, 17 G 91*
(1.2)
M(rì\Snt) = M(rì\xt)
(a.s.P)
Let P° G 1 . For each (s, x) G r x E, a measure P
on SI may be constructed
in such a way that, for every £ G $ I , M JJC £ is a Borei function of x and
(1.3)
M°{£U,} = M ^ £
(a.s.P°)
Let us fixe a family of measures {PJ|JC} and let us denote by ® the class of all
P G ft0 for which
(1.4)
M{<q\9lt}=MtiXt'n
(a.s.P)
for each
teT^G&C*.
It is clear that ®C3Jl. The formulae (1.2) and (1.3) imply that P°G®.
Starting from the family i^xh we shall construct a measurable space (V, CI)
and we shall associate with any v G V a measure Pv G ® in such a way that, for
any P G ® ,
(1.5)
lim ?sx = P p
s^a
s
(a.s.P)
where p is a measurable function on £2 with values in V. Moreover the formulae
(1.6)
W(n
(1.7)
= F{pGr}
P G 4 ) = jT PVC4) M(</v)
establish a one-to-one correspondence between P G ® and probability measures u
on (V, &). The collection (K, 6L , Pv) is called the entrance space for the process
(xt, P) (and for the class ® too).
1.2. — Set P Gfta+ if there exists a sequence (sn , xn) E T x £ such that sn ^ a
and ?s x - » P . The set fta+ is compact. Consider now an arbitrary compactum S
homeomorphic to fta+ and denote by Pv a measure corresponding to v G g under
a fixed homeomorphism from S onto ftfl+.
Set0ia+=
a+
ngit.
teT
x
Let P G Ä . By virtue of (1.4)
(1.8)
Urn M s x n = limM {1? \Sds} = M{n | 0lfl+} .
The right side is an integral of 17 with respect to a measure depending on co
(conditional probability distribution). Evidently this measure belongs to fta+ .
Denote by p(co) the corresponding element of &. Then
(1.9)
M0rì = M(rì\9la+}
(a.s.P)
(1) For any a-field ÏÏ* the notation £Gg» means that £ is a non-negative &i-measurable
function. The notation (a.s.P) means "almost surely with respect to measure P".
SPACES FOR A MARKOV PROCESS
509
The formulae (1.8) and (1.9) imply (1.5).
The equality (1.9) may be extended to all n G ^t. Particularly we have Mpn = n
(a.s.P) for 7}ESda+. Therefore the a-field &Ca+ is generated by the sets
{o> : p(œ)er)(reoL)
and by the sets of measure 0.
Using (1.9), we have, for any nG3£ and any Borel function if > 0 on S
(1.10)
M^(p)n = M ( ^ ( p ) M p n ) = J& <p(v) Mv n p(dv) '
where the measure p is defined by (1.6). Setting here r? = \j/(p) we have
Jsty(v) ip(v)p(dv) = J Mv \ls(p) <p(v)p(dv) .
This implies that, for almost all v, Mv\jj(p) = \J/(v) and hence
(1.11)
Pv(p = v ) = 1 .
Relying on (1.9) and (1.4) it is easy to deduce that Mp%t\ = Mp(%Mtx r})
(a.s.P) for all t G T, £ G 0 l „ rj E ETC*. It follows from here that Pv G ß for u-almost
all v.
Denote by V the set of all v G ê for which P V GS and Pv(p = v) = 1. Let CX
be the totality of all Borel sets of ê contained in V. It has been proved that
p(&\V) = 0. Therefore the formula (1.10) may be rewritten in the form
(1.12)
M^(p)r?= /
ip(v)Mvrip(dv)
v
By setting <p = 1, n = %A we obtain (1.7).
It is easy to show that, for any probability measure p , a measure P defined
by (1.7) belongs to ®. On the other hand if u and P are connected by (1.7) then
by virtue of (1.11), for any TEA
P{p G T} = /
Pv{p G T} p(dv) = J xT(V) p(dv) = p(D .
V
V
Thus formulae (1.6) and (1.7) determine a one-to-one correspondence between
P e S and probability measures u( 1 ).
1.3. — Now let — °° < b < + °° and let T be a denumerable subset of the
interval (— °°, b) and b be a limit point of T. The exit space (F* , (St* , Pv) may
be constructed in the same way as the entrance space (V, OL, Pv). Instead of the
family P ^ , we consider measures P*'x on &ls satisfying the condition
(1.13)
M°il>\xs}=Ms'x*i;
(a.s.P°) (£G0l J )
Instead of the class ® we define the class ®* of measures P G ft0 such that
(1) This correspondence implies that the set of all extremal points of convex set B
coincides with the set of measures PM(i>G V),
510
E3. DYNKIN
D 5
M(£|3tf) = M''*'£ (a.s.P) for all teT, £G0l,. Note that the time reversion
transforms the exit and the entrance spaces into each other.
1.4. — The family PJX may be constructed starting from an arbitrary transition
function p(s, x ; t, T) (s < t G T, x G E, T G03). The associated class ft may be
described as a set of all measures P G ft0 such that
T>(xter\8fls) = p(s,xs;t,r)
(a.s.P)
for all s < t G T, r G (B. Analogously, the family P** may be constructed starting
from any "co-transition function" p*(s, x ; t, Y) (s > t G T, x G E , V Gtö). The
class ft* consists of all measures P G ft0 for which
?(xter\9V)
= p*(s,xs;t,T)
(a.sJP)
forall
s>teT,Fe(&.
2. Harmonic functions.
2.1. — A non-negative function h on the space T x E is called P-harmonic
if (h(t, xt), Sftt, P) is a martingale. Functions hx and h2 are called equivalent if
hx(t, xt) = h2(t, xt) (a.sJP) for all f G T. Our aim is to describe, up to equivalence, all P-harmonic functions subject to condition M h(t, xt) = 1.
To each harmonic function h, there corresponds one and only one measure PA
on the ff-field Sit such that
M„£ = M£/z(f,x,)
for every
teT, £G3l, .
s x
If {V ' } is a family of measures connected with P by relation (1.13) then
M„(£ \SfCf) = Mf'*'£ (a.s.P„) for all £GS^, hence PA G È*.
Denote by ft* the set of all P' Gft* such that the measure p'(t, T) = ?'(xt G T)
is absolutely continuous relative to p(t, T) = P(xter) for each teT. It is clear
that ph(t,dy) = h(t,y) p(t, dy) and therefore P„Gft*. On the other hand,
if P' G ft* and p'(t, dy) = h(t, y) p(t, dy), then
M£h(t,xt) = MM(£|2tf) h(t,xt) = M(M*'*'£) h(t, xt) = M'M*'*'£ =
= M'M'(£|0l') = M'£,
for
s<teT,£e$ls.
It is obvious from here that (h(t,xt), dlt ,P) is a martingal. Thus we have a
one-to-one correspondence between P'Gft£ and classes of equivalent P-harmonic
functions.
2.2. — To proceed further, we need the following assumption about the
measures {P**} :
(P) For any t<ueT,
there exists a measure m such that ?u'y(xteT) = 0
for all T of m-measure 0 and all y G E.
Let P'Gft*. Then
p'(t, T) = P'(x,GD = M'?u'x«(xter)
,
SPACES FOR A MARKOV PROCESS
511
so that p'(t, T) = 0 if m(T) = 0. The densities of measures p'(t, —), p(t, —)
relative to m will be denoted by p'(t, y), p(t, y), It is easy to see that the measure
p'(t, —) is absolutely continuous relative to the measure p(t, —) if and only if
p'(t, At) = 0, where At = (y : p(t, y) = 0). Thus the class ft* may be described
as the set of all measures P' G ft* satisfying condition :
(2.2)
P'(x,G,4,) = 0
forall
teT.
2.3. - Let (V* , Ct* ,PV) be the exit space for Sì*. According to the section 1,
each measure P' may be uniquely represented in the form
P' = f
•V*
?vp(dv).
It is clear that condition (2.2) is fulfilled for P' if and only if u(F*\K P *) = 0
where
Fp*={v : vev*
,?vixteAt
= 0} for all
v
v
The measure P belongs to ftp ; therefore p (t,
kv is P-harmonic. The formula
P'= /
teT}.
v
dy) = k (t, y) p(t, dy) where
PV(tfv)
determines a one-to-one correspondence between probability measures p on Vp
and P' G ftp, and the formula
(2.3)
h(t,y)=
f
kv(t,y)p(dv)
determines a one-to-one correspondence between the same measures u and the
classes of equivalent P-harmonic functions.
3. Continuous time parameter.
3.1. — A family of probability measures ut(T) (teT,
entrance law for the transition function p(s, x ; t, T) if
/ vs(dx) p(s,x,t,r)
= i>t(r)
Te(ß)
is called an
(s<teT,xeE,refà).
E
Formulae
p+(r)=p{xtery,
?{xtiedyt
,...
,xtnedyn}=
vtl(dyl)p(t1,
y, ;
t2,dy2)...
P(tn-i,yn-iltn,
dyn)
establish the one-to-one correspondence between the class #C defined by the
condition (1.4) and the set & of all entrance laws.
Now let T be an interval (a, b) and let 71 bea denumerable everywhere dence
subset of the set T. The restriction iÊ"of the entrance law v G& to the set T belongs
to â . On the other hand, to every Ve S> there corresponds an entrance law
512
E.B. DYNKIN
D 5
vt(r) = / vs(dx)p(ß.x;t.D
(.se(a,t)nf)
E
(it is clear that the value vt(T) is independent of s).
We have a chain of one-to-one mappings ft -> & -> & -> ft and therefore an
one-to-one mapping ft onto ft .
Let (V, d , Pv) be the entrance space related to the^lassjft. Set V = V, <%=&
and denote by Pv the elements of ft corresponding to P„Gft. Then the collection
(V, d , Pv) determines the entrance space for the class ft.
3.2. — For every function h on T x E, denote by h its restriction to T x E.
If h is P-harmonic, then h is P-harmonic and
M{h(u,xu)\xt}=M{h(u,xu)\dlt}=
h(t, xt)
(a.s.P)
forall
t<ueT.
On the other hand, if q is P-harmonic, then M{q(u , xu) \xt} is independent of
u e T H [t, b). Therefore there exists a function h on T x E such that
(3.1)
M{q(u,xu)\xt}=
h(t,xt)
(a.s.P)
forali
re(a,ô).
It is easy to show that h is P-harmonic and h is equivalent to q.
Assume the condition (P) and consider the space Kp and the P-harmonic
functions kv constructed in section _2. Set V* = Vf and denote by kv the Pharmonic function corresponding to kv. Then the formula (2.3) establishes a oneto-one correspondence between probability measures on Fp and the classes of
equivalent P-harmonic functions.
REFERENCES
[1] DOOB J.L. — Discrete potential theory and boundaries, J. Math. Mech., 8, 3, 1959,
p. 433-458.
[2] HUNT G.A. — Markoff chains and Martin boundaries, Illinois I. Math., 4, 1960,
p. 313-340.
[3] KUNITA H. and WATANABE T. — Markov processes and Martin boundaries I,
Illinois I. Math., 9, 3, 1965, p. 485-526.
[4] MEYER P.-A. — Processus de Markov : la frontière de Martin, Springer-Verlag,
Berlin-Heidelberg-New York, 1968.
[5] J^hiuKHH. E.B. — IIpocTpaHCTBO BbixoAOB MapKOBCKoro npoiiecca, YcnexH
MaTeM. HayK, 24, 4 (148), 1969, p. 89-152.
[6] AhlHKHH E. E. — SKCIjeCCHBHLie <j)YHKITHH H IipOCTpaHCTBO BMXOAOB MapXOBCKoro npoueçca, Teopna BeposrraocTeH H eë npHMeH.. 15, 1, 1970,
p. 38-55.
Added in proof. See also :
[7] DYNKIN E. B. — Classes of similar Markov processes and corresponding exit and
entrance spaces, /. Faculty of Sci., Univ. Tokyo, Sec. 1, 17, 1-2, 1970, p. 87100.
Academy of Sciences
of USSR
CEMI
Moscow
V71 (USSR)