I ntern. J. h. & Nath. Sci.
Vol. 2 No. 4 (1979) 669-676
669
A NOTE ON LOCAL ASYMPTOTIC BEHAVIOR FOR
BROWNIAN MOTION IN BANACH SPACES
MOU-HSlUNG CHANG
Department of Mathematlcs
The University of Alabama in Huntsville
Huntsville, Alabama 35807
U.S.A.
(Received July 26, 1978)
ABSTRACT.
In this paper we obtain an integral characterization of a
two-slded upper function for Brownian motion in a real separable Banach
space.
where B
This characterization generalizes that of Jaln and Taylor
n.
[2]
The integral test obtained involves the index of a mean
zero Gaussian measure on the Banach space, which is due to Kuelbs
[3]
The special case that when B is itself a real separable Hilbert space
is also illustrated.
KEY WORDS AND PHRASES. Gaussian measur on B-space, abstract Wienzr
spaces, covariane operators, Bwnian motion in B-space, upper and lower
functions, integral test.
AMS(MOS) SUBJECT CLASSIFICATION (1970) CODES.
Primary 60J65, 60G17.
M. CHANG
670
INTRODUCTION
i.
Let B be a real separable Banach space with norm
be the topological dual of B.
I’II
such that
that B contains a Hilbert space H
II" II
is a measurable norm on H
II II
As a consequence, the B norm
of [i]
is weaker than
Furthermore, it is also shown in [I] that
let F
lxll.
sup
xeK
[I
I11
Thus
B.
is the extension of the
to B and we shall say that
canonical normal distribution on H
H.
in the sense
H* -H
through a restriction map we have the relation that B*
generated by
and let B*
is a mean zero Gaussian measure on
If
B then it is well known from [i
with norm
II’II
If K denotes the unit ball of
H in the norm I’II,
,
The definition of index of
is
n I, is due to
Kuelbs [3], where
n
sup {k:
I
fl
llfj II
and
It is known from [3] that n
Hilbert space then n
that (t)+
.
lower function for
denoted by
L,
if
e
A function
P(I IW(t+v)
u+v < 6)
Let
tl/2(t)+0 as
II’II,
with respect to the norm
with 0 <
H;
F (i < j < k)}.
on B equals the multiplicity of the maximal eigenvalue
I
as t+0 and
such that
orthogonal in
{W(t):
0
_<
t <
oo} denote -Brownian
denote the class of functions from (0,) to [0, =) such
Let
DEFINITION i.
fk
fl’
is finite and if B itself is a
I exists and
of the covariance operator for
motion in B.
B*;
fkE
W(t-u)
I.
is called an upper function for
if given t > 0, there exists a
<
21/2(u+v)I/2(u+v)
In this case, we say
{W(t):
U.
ll
t+0.
t >
{W(t):
> 0
F for all u, v > 0
EU.
0} with respect to the norm
is called a
t >
0}
671
ASYMPTOTIC BEHAVIOR FOR MTION IN BANACII SPACES
d,
In the case that B
e
{W(t): < < }
[(t)]d+2--e-2(t)/2
is an upper function for d-dimension standard Brownian
[2] have shown that
0
motion
if
O+
a d-dimensional Euclidean space, Jain and Taylor
.
with respect to the Euclidean norm
t
dt <
11"112
if and only
This integral test for two-sided growth in
is the same as that for one-slded growth in
d+2.
In the case that B is an
infinite dimensional real separable Banach space, Kuelbs
(t), a nonnegative, non-decreasing
t
l’II I
respect to
if P
dt <
>
<_
t <
on B if and only if
tl/m(t) F
for only a bounded set of t’s)
Based on the results of Jain-Taylor and Kuelbs, it is very natural to
g
is in U (Definition i) for
equivalent norm
[I-II I
}
is called one-sided upper function with
where the
(I IW(t) II I
-Brownian motion {W(t): 0
l’I, ll
with respect to some equivalent norm
[(t)]nl e-2(t)/2
[3] has shown that
continuous function defined for large values
of t, is a one-sided upper function for
that
{W(t):
on B if and only if
0
_<
10+
t <
} with respect
[(t)]nl+2
t
main purpose of this note is to verify this conjecture.
e
notation a(h)~ b(h) means
lid
a(h)
i.
conjecture
to some
-2(t)/2 dt <
.
The
Throughout this note c
will stand for a positive number whose value may change from line to line.
2.
d
t
The
i.
h-0 b(h)
M IN RESULTS
The ol]owing useful estimates have been used repeatedly in [3] and they can
be verified by the argument similar to that in d-dimensional Euclidean space
d
[4, p. 222].
LEMMA 2.
Let
{W(t):
0 < t <
Banach space B having norm
P
sup
0_< t I <
2P
t2<
sup
0< t<h
h
IIW(t) II
II’II-
be Brownian motion in a real separable
Then for all %, h > 0
Iw(t#
> %h
I/2)
W(tl)
_<
4P
> %h
I/2)
<
(I IW(h) ll>%hl/2).
()
M. CHANG
672
We have the following integral test for a two-slded upper function
{W(t): 0 <
for
oo}:
t <
Let {W(t):0 < t <} 5e -Brownian motion in a real
THEOREM 3.
.II
separable Banach space B having norm
denote the index of
lx
such that sup
0+
F and
U
with respect to
II.II I
Let n
II.II I
I
on B
if and only if
e-2(t)/2 dt<.
2
t
PROOF.
,e.
and assume
Then there is an equivalent norm
I
[ (t) ].n.!+.
II" II,
(2)
is due to
The construction of the equivalent norm
which is defined to be
lxll
where
=x
rllll
lQxll),
n
(x)
1
Y
e
j
j=l
Q(x)
x
(xeB)
(x)ej
and
(x),
ej (-)
denotes the linear function
Consider the sequence a
e
m
as m-.
Let u
i
n, i
io{ ’h
Note that for each i, u
n,i
-m/log m m > 2.
(i
an’ Vn, i
+ vn,i
a
n
Then
log n
m
am+I
an’
mn,i
{:
lW(t+vn i)
W(t-u
F
{:
lw(t+vn
W(t-u
n,i
i
n, i)
n, i
i
+ (log m) -i
0 < i < log n.
If u,v > 0 are sufficiently
small we can choose an n sufficiently large such that a
Assume that the integral (2) diverges.
fj (’)/F.
a
n+l
< u+v < a
Define
II i > 21/2an i/2(an)r}
II > 21/2anI/2(an) }
and
n
[3],
ASYMPTOTIC BEHAVIOR FOR MOTION IN BANACH SPACES
673
Since
{W(t) :0 _< t < oo}
I
dimensional Brownian motion in
divergence of the integral (2) implies that
H
B
is standard n
log n
n=2
F(F
i=0l
Fn, i
and infinitely many of
n’ i
Consequently infinitely many of E
n,i
occurr with probability one.
Now assume that the integral (2) converges.
i < log n such that u
n,i
< u < u
occur.
n,i+l
and v < v
Thus
eeL.
Let us choose a suitable
n,i-I
Then
lll > 2112 (u+v) ll2(u+v) F)
l/2W(t-u) lll > 21/2-an+l
(an+l) F)
P(I IW(t+v)
W(t-u)
P(llW(t+v)
<
--< P(O
< t
sup
< t < u
1
2
n, i+l+Vn, i-i
+ P (0 _< t < sup
t <
I
2
{zW(t): 0 <
Since
B
H,
t <
Un, i+l+Vn, i-i
}
is standard n
lz(W(t 2)
W(tl))l I
IQ(W(t2) W(tl))ll
I
> zI/2
112
an+
1 (an+l))
> z.i/2 112
I
an+ (an+l)F).
dimensional Brownian motion in
by the same argument as those in Theorem 3.1 [2], we conclude
that the first term in the right hand side of the above inequality being
zero for infinitely many n and i.
As for the second term in the above
inequality, we have
sup
P(0_<tl<t 2 _<Un, i+l+Vn,
_<
4P
<
C
i_l
lQ(W(t2
(I IQW(1) II >(u 2an+I
n, i+l+Vn, i-i
exp{-e
2an+l
W(tl))ll
>
1/2
F2}
n, i+l+Vn, i-i 2 (an+l)
(an+l)F)
1/2
21/2 an+
I (an+ I) F)
P(Ano i)
M. CHANG
674
Where
1/2 sup
B
Q(dx),
A2(x)
n
l]x]t
and
{A }
we choose
(i
be such that
P(An, i)
_<
C exp {-6
n
+ )/2F
+ 3/log
(i
n)an+I,
if
we have
2(an+l)} exp {-2(an+l)}
{-qb2 (an+l)
< C exp
for every xeB.
3, p. 253].
(i + 2/log n)a
Un,i+l+Vn,i_l
Aj(x)I
sup
The last inequality comes from
Since
is a sequence in B* such that
j
}
Thus
log n
7.
Y.
P(An, i)
< c Y. (log n) exp
n--2 i--0
{-2(an+l)}
n=2
{-2(an+l)}
[(an+l)]nl
< c
exp
n=2
<
since the integral (2) converges
,
(see Lemma 2.12 of [2]).
From Lemma 2.15 (i) of [2] we conclude that
P(An,i,
i.o.)
0.
Thus
P(
sup
t-Un, i+l--<tl <t2<t+Vn,
i-ll-- lW(t2) W(tl)l- II
for all i and n sufficiently large.
Thus
< z,,1/2 1/2
an+I (an+l)F)
eU.
In case that B is a real separable Hilbert space, then n
I
equals
the multiplicity of the maximal eigenvalue of the covariance operator
for V.
We have the same result as those of Theorem 3.
THEOREM 4. Let
{W(t): 0 <
t <
} be v-Brownian motion in a real
separable Hilbert space H with norm
II’II,
and suppose
e.
Then
i
675
ASYMPTOTIC BEHAVIOR FOR MOTION IN BANACH SPACES
II’II
is in U with respect to the given norm
[(t) ]nl+2
S04
Where n
e
t
_2(t) / 2
if and only if
dt < oo,
(3)
.
I denotes the multiplicity of the maximal eigenvalue of the
covariance operator for
} and {v
} be the same as those in
n, i
n,i
If u,v > 0 are sufficiently small we choose
Let sequence {a
PROOF.
the proof of Theorem 3.
n
}, {u
n sufficiently large such that
such that u
n,i
< u < u
an+4+/- _< u+v
and v < v
n,i+l
<
an and
then fix i
_<
log n
If the integral (3) diverges,
n,i-I
then we proceed as those in Theorem 3 and conclude that
L.
Now if
the integral (3) converges define
B
n, i
sup
{:
t-Un, i+l<--tl<t2<--tqVn, i IW(t2) W(tl)
21/2an+11/2
>
(an+l) F}.
Then
e(Bn,i)
<
4e(llW(1) ll
<
C[(an+l)
_<
C[(an+I)
2an+I
>
(un, i+l+Vn, i-i
nf2
exp {
(-
(an+1
an+l2 (an+l) F2
ni 2
exp
1/2
(Un, i+l+Vn, i_l) %
_2 (an+l) }
Where % is the maximal eigenvalue of the covarience of
that %
F2
[3].
n,
i+l+Vn ,i-1
and it is known
The last inequality comes from [3] and the fact that
a
u
)
n+l
1+3 (log n)
-1.
Thus
log n
Y.
n=2
i=O
P(Bn,i
< C
log n
Y. Y
[
n=2 i= 0
I)_
(an+
n
C Y.
n=2
exp {-
2 (an+l) }
2(an+1 )} <
since
From Lemma 2.15 (i) we have
P(Bn, i,
[(an+I)]
the integral (3) converges.
I
nl-2
exp
-i-
i.o.)
0.
676
M. CHANG
That is P(B
c
n,
i)
i for sufficiently large i and n.
Thus
U.
REFERENCES
i.
Gross, L. Abstract Wiener Spaces Proc. Fifth Berkeley Symposium on Math
Statistics and Probability, (1965) 31-42.
2.
Jain, N. C. and Taylor, S. J. Local Asymptotic Laws for Brownlan
The Annals of Probabillty Vo.l. i, No. 4 (1973) 527-549.
3.
Kuelks, J. Sample Path Behavior
The Annals of Probability,
4.
Motion.
for Brownlan Motion in Banach Spaces.
No. 2 (1975) 247-261.
Vol. 3
Taylor, S. J. Exact Asymptotic Estimates of Brownlan Path Variation.
D_uke Math. J.. 39, (1972) 219-241.