KOMATSU, TAKASHI

TAKASHIMA, KEIZO

Komatsu, Takashi

Takashima, Keizo

Osaka University Knowledge Archive

The Hausdorff dimension of quasi-all Brownian paths

Osaka City University Repository

Komatsu, T. and Takashima, K.
Osaka J. Math.
21 (1984), 613-619
THE HAUSDORFF DIMENSION OF QUASI-ALL
BROWNIAN PATHS
TAKASHI KOMATSU AND KEIZO TAKASHIMA
(Received September 22, 1983)
1. Introduction. Properties of Brownίan paths have been main sub-
jects of many papers and various results have been already well known. The
size of sample path, for example, has been investigated by evaluating Lebesgue
measure and Hausdorίf measure etc. of path. Taylor [7] determined the Haus-
dorff dimension of almost all Brownian paths. On the other hand, Fukushima
[2] recently studied some basic properties of path, such as nowhere-diίferenti-
ability, Lέvy's Holder continuity, the law of iterated logarithm etc. in connec-
tion with the Dirichlet space theory on the Wiener space. He proved that
these properties hold not only almost surely but also quasi-e very where.
In this paper we shall present a refinement of Taylor's result from the
viewpoint of the Dirichlet space theory. It is easily derived from a combina-
tion of the definition of Hausdorίf measure and Fukushima's result correspond-
ing to Lέvy's Holder continuity that the Hausdorίf dimension of path is no
more than 2 quasi-everywhere. We shall prove that the Hausdorίf dimension
is no less than 2 quasi-everywhere, by showing that a specific Wiener functional
used in [7] belongs not only to L2-space but also to the Dirichlet space.
2. Dirichlet space. We first prepare some notations and definitions
on the Wiener space and a Dirichlet space on it (cf. Shigekawa [6] and Fuku-
shima [2]).
Let W be the Banach space of all jβrf-valued continuous functions w(t)
on [0, 1] satisfying zu(Q)=Q, with standard supremum norm, H be the Hubert
space of all absolutely continuous functions of W having square integrable deri-
vatives, with inner product
and norm \\h\\H=\/ζh, /*>#, where h denotes the derivative of h and a b denotes
the inner product in Rd. Let P be the Wiener measure on W. A Wiener
functional is a real (or more generally Hubert space) valued mapping defined
on W, measurable with respect to the Borel field of W. Let L\W) be the
614 T. KOMATSU AND K. TAKASHIMA
Hubert space of real valued Wiener functional, square integrable with re-
spect to Py with inner product
(F, GVort = E[F(w)G(w)] ,
where E denotes the integration with respect to P. Let L\W\H) be the space
of if -valued Wiener functionals F satisfying E[||JF(«;)|||r]<00.
We define a weak derivative of a Wiener functional after [6], A real valued
Wiener functional F is called //-diίferentiable if for every w of W there exists
DF(w) of H such that for any h of H,
F(w+h)-F(w) = <DF(w\ K>H+o(\\h\\H\ as \\h\\H -> 0 .
A real valued Wiener functional F is called to have a weak derivative if there
exists a sequence of real valued smooth (in the sence of [6]) functionals Fk con-
vergent to F in L\W] such that the sequence of ίf-derivatives DFk is a Cauchy
sequence in L\W\H). The weak derivative of -F is defined by L\W\H)-
limit of DFk and denoted by DF (see also Ikeda and Watanabe [3]).
Next we define a Dirichlet space and a Dirichlet form after [2]. Let £F
be the space of Wiener functionals in L\W) having weak derivatives in
L\W;H), and
β(F, G) = ± E[<DF, DGyH] ,
and
(^F, G) = S(Fy G)+(F, G) ΛlΓ>, F, G <= £F .
Then a capacity is defined by
Cap (-4) = inf {tfxCF, F): Fe£F, F>1, P-a.e. on A}
for an open set A of PF, and for any set B of W
Caρ(β) = inf {Cap (.4): A is open,
It is well known that this capacity is a nonnegative, increasing set function
on W satisfying the following properties :
( I ) P(A) < Cap (A) for any Borel set A, in particular Cap (W) = 1 .
(H) Caρ( U
 n
=ιA
n
) = sup {Cap(^4
w
): n> 1}, for an increasing sequence A
n
 .
(HI)
(see also [1]). The term ' 'quasi-every where* or simply 'q.e.' means 'except on
a set of zero capacity'.
The Hausdorίf dimension dim (A) of a set of Rd is defined by
HAUSDORFF DIMENSION OF QUASI-ALL BROWNIAN PATHS 615
= inf{α>0: Λ*(A) = 0} ,
where Λ" denotes the Hausdorff α-measure, i.e. Hausdorff measure with re-
spect to measure function f (cf. Rogers [5]).
Let w[Q, ί]={w(t): 0<f<l} for w of W. Our result is the following.
Theorem. dim(α;[0, 1]) = 2, q.e. (d>2).
3. Proof of Theorem, i) First we shall prove that dim(w[Q, 1])<2 q.e.
Let «;''(/), \<i<d, be the i-th component of w(t). Theorem 3 of [2] tells us
that
lim
δ;o sup {I w'ftj-w'fe) I (2t log 1/*)"1/2:
= 1, q.e.
Let AI be the set of w of W such that wi has the above continuity property
and B be the set of w which has λ-Hϋlder continuity peoperties for all 0<λ
<l/2. Then B includes Π iίιA
έ
 and by (IE) and (Π),
Cap(B c)<Cap(U ί^Af)<ΣMι Cap (Af) = 0 .
On the other hand, by the same argument as in [7], it is easily seen that
for a λ-Hϋlder continuous w, α;[0, 1] has zero Hausdorff /3-measure for β>l/\.
Thus for any w of B, w[Q, 1] has zero Hausdorff /3-measure for any /5>2. This
implies the desired result.
ii) We shall next prove that dim(α;[0, 1])>2 q.e. [7] showed that for
any w of W and l<a<2,
( ( I w(s)—w(t) I -* ds dt< oo
JoJo
implies that 20[0, 1] has positive Hausdorff α-measure. We define Wiener
functionals F(*\ l<α<2, by
F< >(to) = (TI w(s)-w(t) I -Λ ds dt,
JoJo
and F
n
=FV-l/n\ n=l, 2, —. If we can show that for any ra, Cap({F
w
=oo})
—0, then by (Π)
and for any w of Π»=ι{ί1»<00}, ^ [0,1] has positive Hausdorff (2— l/w)-measure.
This shows that dim^O, 1])^2 q.e.
Therefore we have only to prove that Cap({-F(Λ)=oo})=0 for any
<2. Fkα, l<α<2. Let
Λ Λ,
616 T. KOMATSU AND K. TAKASHIMA
Then F
ζ
 is continuous on W and for any w, F
ζ
(w) f F(*\w)—F(w) as 6 I 0.
If we can verify that F^EF and jfiΓ=suρt>051(/'lβ, Fe) is finite, then the next well-
known inequality (cf. [1])
51(Fε, Ff), λ>0 ,
and (H) imply that for λ>0,
Cap({F>λ})<sup
ε>0Cap({ί1ε>λ})<λ-2ί:.
Letting λ->°<>, we have Cap({F=oo})r=0.
The finiteness of K is derived from the following two lemmas and the
proof of Theorem is completed.
Lemma 1. sup
ε
>0(Fe, Fe)
Lemma 2. sup
β>0<?(.Fβ, Fε)<oo .
Proof of Lemma 1. Let φt(x) = ( \ x \ d+£)~*/d and
o Jo
o
Then Δ/
ε
(^)— φg(#), where Δ is the Laplacian on Rd
o Jt
"
In the above we have used the time-homogeneity of Brownian motion. To
estimate the right-hand side, we apply the Ito formula to/
s
(#),
where
HAUSDORFF DIMENSION OF QUASI-ALL BROWNIAN PATHS 617
Since \f,(x)\ ^(d-a)-\2~a)-l\x\2-a and |V/,(*)I <(<*-
[((V,Hί))ώ)2]
JO
4E[(ΛK1))-
Γ | V/,K*)) 1 2ds]
Jo
<oo.
o
This completes the proof of Lemma 1.
Corollary. The Ito formula is applicable to \x\2~°>, a<2, i.e.
I
 w
(t) 1 2-* = Γ(2-α) I zσ(ί) | -*a;(ί) </w(s)+ ( V~«) (2~«) I «<
Jo Jo
Furthermore, [ \ w(s) \ -*ds<=L\W) .
Proof of Lemma 2. We use the same notations as in the proof of Lemma
1. By the mean value theorem, for w of W and h of if,
= (1duh(U) (2('dt\1Vφ,(w(S)-zϋ(t))ds)+o(\\h\\ll)Jo Jo J K
as 11^11^-^0. This implies the //-differentiability of FQ and its //-derivative
is given by
Since the boundedness of 110 (^^ )11^ on W is easily derived from that of | Vφ
ε
(x)\9 by the same argument as in Section 2 and 3 of Kusuoka [4], it is verified
that the //-derivative of F
ζ
 coincides with the weak derivative of F
e
 a.s. Thus
2£(F,, F,) =
= 4 (Λ Σ/-ι E[( \"dt\ V.y(w(ί)-
Jθ Jθ Jtt
where ψ^x) is the /-th component of Vφ
ε
(#). Let g
ζj(x) be the j-th com-
ponent of V/
ε
(#). Then g2j(x) is twice continuously differentiable and Δgζj(x)
=i
ψtj(χ) To estimate the above, we again apply the Ito formula to g
e
j(x),
618 T. KOMATSU AND K. TAKASHIMA
Hence we have
As for the first term in the right-hand side, it is less than
16E [ "{gtWV-
since |£8;(#)l <(d— a)'1 M1"*, |# |ΦO. Thus the first term is bounded in-
dependently of 6 by Corollary of Lemma 1. As for the second term, by stand-
ard calculation we have
O Ju
= E[\ttdt\"dv( (
»Ό Jθ Ju
X
Note that
α - * - *
= ((^-α)-1 1 * I -'{( I * I '+ε?- " -&-•"} (-d x, *. I * I
where δ^ is Kronecker's symbol. Hence we have
HAUSDORFF DIMENSION OF QUASI-ALL BROWNIAN PATHS 619
Therefore the second term in the right-hand side of (*) is less than
Jo
and is bounded independently of 6 by Corollary of Lemma 1. This com-
pletes the proof of Lemma 2.
The authors wish to express their hearty gratitude to Prof. N. Ikeda, Prof.
M. Fukushima and Prof. N. Kόno for their valuable advice and suggestion.
References
[1] M. Fukushima: Dirichlet forms and Markov processes, North Holland and
Kodansha, 1980.
[2] M. Fukushima: Basic properties of Brownian motion and a capacity on the Wiener
space, J. Math. Soc. Japan 36 (1984), 161-176.
[3] N. Ikeda and S. Watanabe: Stochastic differential equations and diffusion pro-
cesses, North Holland and Kodansha, 1981.
[4] S. Kusuoka: Dirichlet forms and diffusion processes on Banach spaces, J. Fac.
Sci. Univ. Tokyo, Sec. IA 29 (1982), 79-95.
[5] C.A. Rogers: Hausdorίf measures, Cambridge University Press, 1970.
[6] I. Shigekawa: Derivatives of Wiener functionals and absolute continuity of induced
measures, J. Math. Kyoto Univ. 20 (1980), 263-289.
[7] SJ. Taylor: The Hausdorff a-dimensional measure of Brownian path in n-space,
Proc. Cambridge Philos. Soc. 49 (1953), 31-39.
Department of Mathematics
Osaka City University
Sugimoto 3-3, Sumiyoshi-ku
Osaka 558, Japan



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The Hausdorff dimension of quasi-all Brownian paths

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