605
Iernat. J. lath. & lath. Sci.
Vol. 5 No. 3 (1982)605-607
SQUARE VARIATION OF BROWNIAN PATHS
IN BANACH SPACES
MOU-HSIUNG CHANG
Department of Mathematics
University of Alabama in Huntsville
Huntsville, Alabama 35899
U.S.A.
.
It is known that if {W(t), 0 <_ t
2n
IR then lip
IW(i/2 n) W((i- i)/2 n)
n-o
ABSTRACT.
12
i} is a standard Brownian motion in
<_
1 almost surely.
We generalize this
celebrated theorem of Levy to Brownian motion in real separable Banach spaces.
KEY WORDS AND PPSES. Gausian measures, abstra Wiener space, Brownian motion
in Banach spaces, square variation of Browian paths.
AMS (MOS) SUBJECT CLASSIFICATIONS (1970).
Seconddry 2 8A40.
1.
Primary 60J65, 60G15, 60G17;
INTRODUCTION.
This note is addressed to those familiar with
Levy’s
theorem [i, p. 510].
It
should be readable for those who have an elementary understanding of Banach space
and knowledge on Gaussian measure, Borel-Cantelli lemma, etc.
Let B be a real separable Bausch space with norm
l’II
and let B* be the topo-
logical dual of B, i.e. the space of bounded linear functionals on B.
mean zero Gaussian
meas,re on
bert space H
norm.,
sense of Gross
I’II.
B*
H*
with
.,II’ll
If D is a
B then it is known [2, p. 35] that B contains a Hilsuch that
[2, p. 34], [3, p. 127].
I’;I
is a measurable norm on H
As a consequence,
If’If
in the
is weaker than
Thus through an injection map it was shown in [2] that the relation
H
B holds.
distribution on H
to
Furthermore, D is the extension of a canonical normal
B, and we shall say that
is generated by H
606
M. HANG
Let
when t
{Ut
t >
O, and
60
subset of B and
80(A)
O} be the family of Gaussian measures on B given by ut(A)=o(A)
I/2
(A/t
U{t -I/2 x: x E A} when t > O, where A is any Borel
is the unit mass concentrated at the origin of B, i.e.
1 if 0 g A,
A.
0 if 0
measures under convolution, i.e.
(t
*
IB t(A-
s )(A)
x)d
t
U s(x),
continuous functions a defined on
{t
t >
t+s’
where
Note that
*
Us
O} is a semigroup of Gausslan
for any Borel set A.
B be the space of
Let
O, and let F be the
[0,1] Into B such that a(O)
B generated by the functions 00(t). It is clear that B is a real
separable Banach space under the II
I0 sup ll(t) ll and F coincides with the
Borel o-fleld of
B" A stochastic process {W(t) 0 t < 1}, W(t)() --(t), on
B is called -Brownian motion (restricted to the unit time interval) on B if wheno-fleld of
O<t_<l
<
ever 0
t
o
pendent and
< t
<
I
<
I, then
tn
(tj) -(tj_l)
has distribution
(j
(tj_ I)
(tj)
tj
t j-1
1,2,...,n) are inde-
on B.
Let
PW denote
the
B induced by {W(t) 0 t I}. PW usually called the
measure on
B" Here we shall denote the expectation operator
<
mean zero measure on
abstract Wiener
with respect to the measure
<
is
PW"
The purpose of this note is to prove an analogue of a celebrated theorem of
Lvy [Theorem
2.
5, p. 510] for -Brownlan motion in a general separable Banach space.
RESULT.
THEOREM i.
Let
,2
_,
PROOF.
{W(t)
0 _< t -< i} be -Brownian motion in B.
Iw(/2n)
W((’-])/2n)112
{W(t)
n
I I1112
B
a.(,o
,
Our proof is elementary and straight-forward, and it goes as follows.
2n
Set S
Then
n
lw(i/2n)
w((t-l) 12 n)
12
n
0,1,2,
0 < t < I} has independent increments and that
Then, from the fact that
r- i/2 W (rt)
the same distribution for each r > 0 and t > 0 [3], we have
and W(t) have
607
BROWNIAN PATHS IN BANACH SPACES
2
n
W((i-l)/2n)ll 2
EWI IW(i/2 n)
Ew(S n)
i=l
n
2
EWI IW(2-n) ll 2
i=l
n
2
2-n
mw iW(1) 112
i=l
Furthermore,
2
n
VaT (Sn)
i=l
2
n
2-2nvar(l]W(1) ll2)
i=l
2
Note that
Var(IIW(1)
inequality, we have
7. n 2
n=l
2-n
12
<
-n
War(
by a theorem of Fernique [4].
Therefore, by Chebyshev’s
PW
< oo, and the theorem follows from the Borel-Cantelli lemma [5, p. 76].
REMARK.
When B
Theorem I reduces to that of Levy for the standard Brown-
ian motion, since
BI Ixl 12 d(x)
1
x
2
exp{-x2/2}
dx
i.
REFERENCES
i.
LVY,
2.
GROSS, L.
3.
GROSS, L. Potential theory on Hilbert space, J. Functional Analysis 1 (1967),
123-181.
4.
FERNIQUE, X. Integrabilit des Vecteurs Gaussiens, C.R. Acad. Sci., Paris,
SeT. A 270 (1970), 1698-1699.
5.
CHUNG, K.L. A Course in Probability Theor..y, 2nd edition, Academic Press,
New York, 1974.
P.
Le Mouvement brownien plan, Amer. J. Math. 62 (1940), 487-550.
Abstract Wiener space, Proceeding of 5th Berkeley Symposium on
Math. Star. and Prob. 2 (1965), 31-42.