Functional Analysis in Markov Processes Proceedings of the Internati
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for u-ra e z , and v
(2) Let U
v
E
VeE)
(1) Let U
E
V (1:) and V
for ]1-a.e.z, D(u(
fB
u Ldz )
t-
E
V (B*).
,z]v»
t.. 1 ( u ( [ · , z ] v »
=
PVDU(
Then
r- ,z]v)
E1 ( u ) .
"
and {V be a canonical sequence. n}n:1
fB u Cdz ) t. 1 ( u - u ( [ · , Z] v n »
.... 0
as n ....
(X)
Then
1/2
8
3.
Standard Ornstein-Uhlenbeck process. Definition 3.1. w(t);
0
oS
We say that a B-valued stochastic process
< co } is a standard Wiener process associated
t
with (lJ,H,B) if
o
w(·) : [O,co) ->B is continuous and w(O)
(1)
with probability
one, w(t
(2)
W(t ) - W(t
l),
2
l),
w(t ) - w(t ) , ••• , W(t 2
3
n)
- w(t
n_ l)
are
independent B-valued random variables for any integer nand
o
-
co ]
= 1
by
Proposition 3.2, we see that for any quasi-continuous map f:B->-M, P{WEW; f(w(·)): [0,00) ->-M is continuous} = 1. )J Definition 3.4.
We say that a subset K of B is quasi-closed
if there exist some topological space M, a quasi-continuous map f : B ->- M and a closed subset E of M satisfying K = f
-1
(E).
We say that a subset G of B is quasi-open if B \ G is quasiclosed. Remark 3.2. P)J{ w E W;
By Remark 3.1, we see that {tE [0,00); w(t) E K
is closed in [O,oo)}
1
for any quasi-closed subset K of B, and P)J{ w E W;
{t E [0,00); w(t) E G
is open in [O,oo)}
1
for any quasi-open subset G of B. The following is obvious. (1) Let M n = 1,2, ... , be topological spaces n, n = 1,2, ..• , be quasi -continuous maps. Then
Proposition 3.3. and f
n
: B ->- M n,
the product map
00
IT f : B ->- IT M n=l n n=l n
is quasi-continuous.
(2) Let K n = 1,2, .•. , be quasi-closed subsets of B. n, K u K is quasi-closed and n K 2 l n=l n
is quasi-closed.
(3) A subset of capacity zero in B is quasi-closed.
Then
11
The following lemma is useful. Lemma 3.1.
(1) Let {Kn}n:l be a decreasing sequence of quasi-
closed Borel subsets of B.
Then Cap (K
n
) .. Cap ( ';:; K ) as n + n=l n
(2) Let G be a quasiopen Borel subset of B.
Then Cap (G)
co.
0
provided that W(G) = O. Proof.
For any T > 0, it is obvious that
(1)
n {t; 0'; t ,; T, w (t) E K } n=l n
{ t ; 0,; t ,; T, w (t) E
n K l. n=l n
{t; O,;t ';T, W(t)E K l , n= 1,2, ... , are compact for
However,
n
Pwa.e.w by Remark 3.2.
Thus we obtain
P {Wi oK (w) ,;T} .. P {Wi 0nK (w),;T }, n+ WnW n n
This implies
e
K n
(z) ..
e
nK n n
(z),
n +
00,
for u-ia v e v z .
By virtue
of Lemma 3.1.1 in Fukushimall], we get
t 1 (e Kn Hence {e product
K
n
Cl
e
Km
) = Cap (K
n
) Cap (K) m
}n:l is convergent in V(E) .
On the other hand, e
with respect to the inner + e nK
K
n
, n
+
2
00, in L (B;dW).
n n 00
= Cap ( n K ) , n=l n
This proves that n
for any n > m ,
>
(2) Suppose that W(G) = O.
By Remark 3.2, we get
PW{w; 0G(w) = inf{r>O; r is a rational number, w(r)EG}}=l.
o
However, Pw{w; w(t) EG} = W(G) accordingly
P { w ; 0G (w) < w
00
}
follows from Proposition 3.2.
O.
for any t '" 0, and Therefore ODr assertion
12
Proposition 3.4. continuous.
Assume that UEV(E:) and u:B+IR is qu
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