MOUNTFORD, T.S.

Mountford, T. S.

Osaka University Knowledge Archive

An extension of a result of Komatsu and Takashima

Osaka City University Repository

Mountford, T.S.
Osaka J. Math.
26 (1989), 597-603
AN EXTENSION OF A RESULT OF KOMATSU
AND TAKASHIMA
T.S. MOUNTFORD
(Received October 18, 1988)
Introduction
It has been shown by Komatsu and Takashima (1984) [7] that in dimensions
higher than one the Brownian path has Hausdorff dimension 2 for quasi-all paths.
This result extends the result of Taylor (1953) [10] who proved it held a.s. with
respect to Wiener measure. In this paper we prove:
Theorem.
Let A be any time set of dimension a, and let {B(t)\ t>Q} be a d-dimensional
Brownian motion. The image of A by {B(t): t>0} has Hausdorff dimension equal
to the minimum of 2a and d quasi-everywhere.
This result extends the following result found in Kahane (1968) [5]:
The image of a time set of Hausdorff dimension a by d-dimensional Brownian
motion has dimension equal to the minimum of 2a and d a.s..
Fukushima (1984) [2] proved that Levy's modulus of continuity held quasi-
everywhere rather than just a.s.. Consequently it is easy to see that for a fixed
set A of Hausdorff dimension a, the dimension of the image of A is bounded
by 2a quasi-everywhere. Hence, the Theorem will be established once we
have shown that for fixed A, the dimension of the image of A by B has dimen-
sion greater than the minimum of 2a and d quasi-everywhere. We shall in fact
explicitly perform the calculations only for the case d=\ and a<dβ, but the
method we exhibit will also work for higher dimensions.
In Section 1 we give a general plan of the method; in the Section 2 we
collect together the building blocks for the final proof which appears in Section
3.
1. Pre-requisites
For a>0 one defines the function φ8
Λ
 on a set A by
where the infimum is taken over all sequences of intervals A{ such that
598 T.S. MOUNTFORD
Ad U Ai and | A4 | < δ (1-4,1 denotes the length of the interval A{).
We define the set function φ
Λ
 by taking the (increasing) limit of φl as δ tends to
zero:
φ
Λ
{A) - limit φi(A).
δ->o
The Hausdorff dimension of a set A is defined to be inf: φ
Λ
(A)=Q. With
Λ
these definitions, it is easy to see that the Hausdorff dimension could also have
been defined as sup: φ
Λ
(A)= oo.
Λ
The Hausdorff dimension of a set A is equal to its capacίtory dimension^ which
is defined as the supremum over β such that
3 a probability measure μ supported on A such that sup 1
 Λ
 < ° ° .
y J \x—y\β
This condition is equivalent to
3 a probability measure μ supported on A such that \\ M * ) M y ) < o o
JJ I x — y I p
The equality of these two dimensions (Hausdorff and capacitory) implies
that if there exists a measure μ supported on a set A such that for each n
[[μ(dx)μ(dy)
)) |*_3,|--vι.^ '
then the Hausdorff dimension of A is greater than or equal to α. This is how
Kahane proved his result: it was shown that for a positive measure μ supported
on A (which can be assumed to be closed) and a standard rf-dimensional Brownian
motion {B(t): t>0}, then the (random) measure v defined by
is supported by the Brownian image of A and that (for 2β<d)
t[μ(dx)μ(dy)
i) \x-y\>
implies that a.s.
Our method of proving the Theorem is showing that a.s. the local time
integral (1) is finite for all paths visited by the Ornstein-Uhlenbeck process.
Let us fix the set A. Without loss of generality, we may assume that A is a
closed subset of the interval [0, 1] and that 3 a probability measure μ supported
by A such that 3 C such that Vy, r, μ{(y—r, y+r)}<Cr". This last condi-
EXTENSION OF A RESULT OF KOMATSU AND TAKASHIMA 599
tion of course implies
V β«x, sup ( PV*)
 < M < O Q .
y J \x—y\p
The existence of such a measure μ does not follow merely from the fact that
the Hausdorff dimension of A is equal to a (see Taylor (1961) [11] for details).
Let {O
s
( ): s>0} be the Ornstein-Uhlenbeck process on Wiener space. Define
the measures v
s
 to be
Our result will be established if we can show for each s and every n,
FJS)=[[ ^ K ( ^ ) <oo
for every s and every n a.s., which will follow provided we show {F
n
(s): s>0}
is a.s. continuous.
To show \F
n
(s): s>0} is a.s. continuous we use the following criterion of
Kolmogorov:
A process {Z(t)\ t>0} has a continuous version if3β>0 and an integer k
such that for all s and ty
(2) E[\X(t)-X(s)\']<K\t-s\^
(see e.g. Berman (1970) [1]).
If a process has a version with continuous paths, then any separable version
must also have continuous paths (see Jain and Marcus (1978) [4]). We show
conditon (2) holds for F
n
( ) (Section 3) and that {F
n
(s): s>0} is separable
(Lemma 1 below), thereby establishing the Theorem.
Lemma 1. For each n, the process {F
n
{s)\ s>0} is separable.
Proof. Define Fξ(s) as follows:
FM(s) =
n w
Then by the continuity of the Ornstein-Uhlenbeck process, F%( ) is a.s.
continuous. Letting M go to infinity, we see that the process {F
n
{s)\ s>0}
must be separable.
It only remains to show that condition (2) holds for each F
n
( ) . Meyer
(1982) [8] noted that a realization of the Ornstein-Uhlenbeck process is
{ O f ( . ) ' s>0} = {e-*<2 W(e\ .): s>0} ,
600 T.S. MOUNTFORD
where W denotes the Brownian sheet. Therefore F
n
 has continuous sample
paths if and only if
V (s) = [[ μ(dx)μ(dy)
Λ W
 JJ \W(S.X)-W(S.V)\*\W(s,x)-]
has continuous sample paths, where s>ί. Purely for greater notational simpli-
city we will work with F
w
( ) rather than F
n
( ).
2. Details
The following lemmas will be used in the proof of the Theorem given in
Section 3. The proofs rest on tiresome calculations and are therefore omitted.
Lemma 2. Let X and Y be Gaussian random variables with respective
means m
x
 and my and variances τ
2
x
 and τ2y. Then for a<l and some C,
E
Unfortunately, Lemma 2 gives a poor bound when either \my\ or \τy\ is
greater than \τ
x
\. For these cases we use the following:
Lemma 3. Let X and Y be as in Lemma 2. Then for some finite C,
1 1
\x\»
Lemma 4. Let {B(t): t>0} be a Brownian motion and T={0,x19 x2, •••,
χ
n> y^Ji'i '*•> Vn\ be a set of time points. Then given the values of B in T, the
conditional distribution of B(x)—B(y) is Gaussian with variance greater than
min {(max {d{{x}, T)β,d((y}, T)β}), \x-y\β} .
Here d(E, F) is defined to be equal to the infimum of the distance between a point
in the set E and a point in the set F. The conditional mean of B(x)—B(y) is a
Gaussian variable with mean zero and variance bounded by \ x—y | .
In the succeeding section we consider terms of the form
1 1
π
ί = l I W(s, ut)-W(s9 v,) 12* I W(s+h, Ui)-W(s+h,
Let 2v denote the σ-field generated by W(s, u{), W(s, v{)y W(s+hy wt ), W(s-\-h, z;,-),
i = l , 2, « r. Given £F
r
, the conditional distribution of (W(sy ur+1)—W(s> vr+1),
W(s+h> u
r+1)—W(s+h, vr+1)) is that of (Y, Y+Z), where Y and Z are indepen-
dently distributed as N(c
γ
,s k
r
) and N(cz,h kr) respectively. The conditional
means c
γ
 and cz are immeasurable random variable with means zero and variances
EXTENSION OF A RESULT OF KOMATSU AND TAKASHIMA 601
bounded by s (v
r+1—ur+1) and h-(vr+1—ur+1) respectively. The term kr is
bounded below by
(3) d
r+ι = min {max {d(«r+1, {0,«,, vt) )/2, d{vr+ι, {0, uiy v,} )/2>,
\u
r+1-vr+1\β}yί<,r.
By applying Lemmas 2 and 3 and iterating Lemma 4, we obtain the following:
Lemma 5. Given numbers v^,^, •••,«„ and v
u
v2, ••-,vn, «,<©,- for each
i, and numbers dh i=\, •••, n, defined as in (3). Then for i2>l,
E
\ A 1 1 Ί
L Λι I ίΓ(ί,«,)- W(s, v{) 12(--v»> I W(s+h, wf)-PF(ί+A, p#) 12<--v«) J
A ί C
- M I JίϊK
Once it is noted that the variance of cz is bounded by h, the proof of Lemma 5
is easy.
The following Lemma is well known, but I have been unable to find a
reference.
Lemma 6. Let μ be a measure on [0, 1] such that V j , r, μ{(y—r,y+r)}
<Cr* for some constant C not depending on y or r. Then for some K not depend-
ing on h or y,
Vy,h,
This lemma gives the following corollary which will be used in Section 3.
Corollary. Given any fixed points {uly vv u2, v2, •••, ury vr} and a measure
μ as in Lemma 6, the integral
is majoήzed by K
rn
hlί2n, where K
rn
 depends on r and n only and is otherwise
independent of the particular {w,} and { t^}.
We are now prepared to assemble the proof of the Theorem.
3. Proof of the Theorem
As was noted in Section 1, the final result will be established if we can
show Vw and s>\
602 T.S. MOUNTFORD
μ(dx)μ(dy)
. ( ' ) = $ • \W(s, x)-W{s,y)\2(«-ιΐn)
is continuous. Let us now consider n to be fixed. By Lemma 1 and Kolmo-
gorov's criterion, it is sufficient to find a k such that
E[\V
Λ
(s+h)-V
n
(s)\*\<Chι+?
for some fixed C and β>0 and all h. Now E[(V.(s+h)- V*(s))u] =
E\ y r n u ι Γ - - t π
J .Jπ-\W{s+h,V,)-W{s+h,:
J f 2k Γ 2k
"Λ Π μ (dUi) μ (dVj) E Π
J ί = l L ί = l
By Lemma 5 this last quantity is majorized by
Γ r 2k < C C
n
hι/2nΛ
\ — \ Π μ{dUi) μ{dVi) mm < ^-i/n, ^ - i ^ } ,
J J ί=l l . ^ r + l Ur + 1 )
which by Lemma 6 is less than or equal to
for some constant Q2ktn n o t depending on h. Taking k greater than n, we
complete the proof.
4. Related Questions
Given this Theorem, it is of interest to further investigate the case 2a^>d.
We know that if the Hausdorff dimension of a set A is greater than 1/2, then
the image of A by one-dimensional Brownian motion a.s. has interior points
(see Kaufmann (1975) [6] and Pitt (1978) [9]). By creating an appropriate
continuous local time for the Ornstein-Uhlenbeck process, we can easily extend
this result from a.s. to q.e.. Kahane (1968) [5] also showed that if a set has
positive capacity w.r.t. £~1/2, then a.s. its image by Brownian motion has positive
Lebesgue measure. A simple martingale argument show s that this result also
holds q.e.. Unfortunately we cannot find an extension of the converse (see
Hawkes (1977) [3]) from a.s. to q.e..
EXTENSION OF A RESULT OF KOMATSU AND TAKASHIMA 603
References
[1] Berman, S.: Gaussian processes with stationary increments: local times and sample
Junction properties, Annals of Mathematical Statistics 41 (1970), 1260-1272.
[2] Fukishima, M.: Basic properties of Brownian motion and a capacity on the Wiener
space, J. Math. Soc. Japan 36 (1984), 147-175.
[3] Hawkes, J.: Local properties of some Gaussian processes, Z. Wahrsch. verw. Gebiete
40 (1977), 309-315.
[4] Jain, N.C. and Marcus, M.B.: Continuity of Subgaussian Processes, In Probability
on Banach Spaces, (J. Kuelbs, ed.) 81-197. Dekker, New York, 1978.
[5] Kahane, J.P.: Some random series of functions, R.D. Heath, Lexington, 1968.
[6] Kaufmann, R.: Fourier analysis and paths of Brownian motion, Bull. Soc. Math.
France 103 (1975), 427-432.
[7] Komatsu, T. and Takahsima, K.: The Hausdorff dimension of quasi-all Brownian
paths, Osaka Journal of Mathematics 21 (1984), 613-619.
[8] Meyer, P.A.: Note sur les processus d'Ornstein-Uhlenbeck, In Seminaire de Pro-
babilites' XVI, Lecture Notes in Mathematics 920, Springer, Berlin, 1982.
[9] Pitt, L.: Gaussian local times, Indiana University J. Math. 27 (1978), 309-330.
[10] Taylor, S.J.: The Hausdorff a-dimension of Brownian path in n-space, Proceedings
of the Cambridge Philosophical Society 49 (1953), 31-39.
[11] Taylor, S.J.: On the connection between Hausdorff measures and generalized capa-
city, Proceedings of the Cambridge Philosophical Society 57 (1961), 524-531.
Department of Mathematics
University of California
Los Angeles
California 90024



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