Packing dimension of the image of fractional Brownian motion

STATISTICS& PROBABILITY LETTERS ELSEVIER

Statistics & Probability Letters 33 (1997) 379 387

Packing dimension of the image of fractional Brownian motion Yimin

Xiao *

Department of Mathematics, The Ohio State University 231 l~K 18th Avenue, Columbus, OH 43210. USA Received 1 April 1996; revised 1 May 1996

Abstract

Let X(t) (t ~ ~N) be a fractional Brownian motion of index ~ in Rd. For any analytic set E C_EN, we show that DimX(E) = 1Dim~a E

a.s.,

where Dim E is the packing dimension of E and Dims E is the packing dimension profile of E defined by Falconer and Howroyd (1995).

AMS classification: Primary 60G15; 60G17 Keywords: Packing dimension; Packing dimension profiles; Fractional Brownian motion; Image

1. I n t r o d u c t i o n

Let X ( t ) = (Xa(t) . . . . . Xd(t)) (t E ~N) be a fractional Brownian motion of index ~ (0 < 7 < 1) in R d (see Kahane, 1985, Ch. 18). If N = 1, ~ = 1, then X ( t ) (t E ~) is the ordinary Brownian motion in ~d. If 1 N > 1, c( = 3, then X ( t ) (t E ~u) is Levy's Brownian motion with N parameters. Kahane (1985) proved that for every Borel set E C_ ~U,

dimX(E)=min(d,

~ dim E )

a.s.,

where dim E is the Hausdorff dimension of E. We refer to Falconer and properties o f Hausdorff measure and Hausdorff dimension. Packing dimension was introduced in the early 1980s as a dual then it has become a very useful tool in analyzing fractal sets (see Saint Raymond and Tricot, 1986; Falconer, 1990; Mattila, 1995).

* E-mail: [email protected].

0167-7152/97/$17.00 @ 1997 Elsevier Science B.V. All rights reserved

PIIS0167-7152(96)00151-4

(1.1) (1990) or Mattila (1995) for the definition concept to Hausdorff dimension and since e.g. Tricot, 1982; Tricot and Taylor, 1985; It is natural to ask whether the following

Y. Xiao/Statistics & Probability Letters 33 (1997) 379-387

380

analogue of (1.1) holds for packing dimension Dim: DimX(E) -:- min ( 1d,D iam E. ) s . -

(1.2)

Since dim~ ~d, Talagrand and Xiao (1996) showed that (1.2) does not hold in general and they also obtained the best lower bound for DimX(E): with probability 1, { { 1 } DimE.d} DimX(E)~>max min d, dimE ; ~ d + D i ~ - - ~ i ~ _ ~ d )

,

(1.3)

and the equality holds for all sets with d i m e = DimE and for some sets with dimE : 0. This raises the obvious question - Can the lower bound in (1.3) be improved by knowing 0 < dimE < DimE? The following example (N = 1) shows that the answer is obviously negative. Let E1 be the compact set constructed in Lemma 3.2 in Talagrand and Xiao (1996) and let E2 be any Borel set of R with dim E2 = Dim E2 ~<

Dim E1 • c~d ~d + DimE1 • (1 - c~d) "

Let E = E1 t_JE2, then d i m e = dimE2, DimE --- DimEl and by Corollary 4.1 in Talagrand and Xiao (1996) and (1.1), with probability 1, DimX(E) = max{DimX(E1); DimX(E2)} Dim E . d c~d + D i m E . (1 - ~d) In particular, we cannot have a general formula for DimX(E) in terms of dimE and DimE. In fact, no previously known formula for DimX(E) is valid for all E C_~U, even if X is Brownian motion (N = d =

½).

Very recently, Falconer and Howroyd (1995) introduced the concept of packing dimension profiles for finite Borel measures and sets in ~N, which carries more information about local and global geometric properties of the measures and sets than packing dimension does. By using packing dimension profiles, Falconer and Howroyd (1995) solved the problem of the packing dimension of projections. The objective of this note is to give a complete solution to the problem of finding a general formula for DimX(E). We will show that for any analytic set E C ~N, DimX(E) = -

1

Dim~E

a.s.,

(1.4)

where Dims E is the packing dimension profile of E defined by using s-dimensional kernel. We remark that when N<.ad, Dim~dE = DimE, and hence (1.4) reduces to (1.2). If N > ~d and DimE = dimE, then Dim~dE = min{ad; DimE} and (1.4) also reduces to (1.2). This note is organized as follows. In Section 2, we recall the definitions and some basic properties of packing dimension and packing dimension profiles. In Section 3, we consider the packing dimension of the image measure /tx of a finite Borel measure # on [~N under X(t) (t E ~N). Then in Section 4, we deduce a

Y. XiaolStatistics & Probability Letters 33 (1997) 379-387

381

similar result for the packing dimension of the image of an analytic set under X(t) (t E ~N) by examining the packing dimension of the finite Borel measures supported on the set. We will use K to denote an unspecified positive constant which may differ from line to line.

2. Preliminaries In this section we recall briefly the definitions and some basic properties of packing dimension and packing dimension profiles. Let (b be the class of functions q5 : (0, 6) ---, (0, 1) which are right continuous, increasing with qS(0+) = 0 such that there exists a finite constant K > 0 for which qS(2s)~
q~(s)

f o r 0 < s < 16.

For ~b E q), Taylor and Tricot (1985) defined the set function

c~-P(E) on RN by

O-P(E)=limsup{~igP(2ri)'B(xi'ri)aredisj°int'xiEE'~:-~o

ri


where B(x,r) denotes the open ball of radius r centered at x. The set function q~-P is not an outer measure because it fails to be countably subadditive. But it gives rise to a metric outer measure q~-p on ~N as follows

@p(E) is called the ~b-packing measure of E. If c~(s) = measure of E. The packing dimension of E is defined by DimE=inf{e

> 0:

s ~, s~-p(E) is called the e-dimensional packing

s~-p(E)=O}.

The packing dimension of a Borel measure /~ on R u (or lower packing dimension as it is sometimes called) is defined by D i m # = inf{DimE : g(E) > 0 and EC_ R u is a Borel set}.

(2.1)

The upper packing dimension of # is defined by Dim* # = inf{DimE : ~ ( N ~ \ E ) = 0 and E C NN is a Borel set}.

(2.2)

For a finite Borel measure # on NN and for any s > 0, let

F~(x,r) = ~ , mint 1, r~ly - xl-S } d~(y), where I" I is the usual Euclidean norm. The following equivalent definitions of Dim # and Dim*# in terms of the potential F~(x, r) are given by Falconer and Howroyd (1995). D i m # = sup{t>~0: liminf r--+0

r-tF~(x,r) = 0 for /t-a. a. x C ~ N ) ,

Dim* ~ = inf{t > 0: lim inf

r-tF~(x,r) > 0 for #-a. a. x E ~N}.

(2.3) (2.4)

Falconer and Howroyd (1995) also defined the packing dimension profile of/~ by using the s-dimensional potential F~(x, r) by Dims/~ = sup{t>~0: lim inf r--+0

r-tF](x,r) = 0 for #-a. a. x E ~N},

(2.5)

Y. Xiao/Statistics & Probability Letters 33 (1997) 379-387

382

and the upper packing dimension profile of # by inf {t > 0 " lim inf r - t F ~ ( x , r ) > 0 for #-a. a. x E ~N r--+0 t

Dim* #

,

(2.6)

It is easy to see that 0 ~.N, then Dims # = Dim #,

Dim, # = Dim* # ,

see Falconer and Howroyd (1995). For any analytic set E C ~N, we define M + ( E ) to be the family of finite Borel measures on E with compact support in E. Then D i m E = sup{Dim# : # E Mc+(E)},

(2.7)

see Hu and Taylor (1994) for a proof. Motivated by this, Falconer and Howroyd (1995) defined the packing dimension profile of E C_ ~N by DimsE = sup{Dims p : # E M + ( E ) } .

(2.8)

Clearly, O ~ D i m s E < . s and for any s ~ N , DimsE -- DimE. We use d(s) to denote any one of Dims#, Dim, p, DimsE. Lemma 2.1. Let d: ~+ ~ [0,N] be as above. Then d(s) is continuous. Proof. This is a consequence of Proposition 18 in Falconer and Howroyd (1995). The following lemma contains Theorem 6(a) in Falconer and Howroyd (1995) as a special case. Lemma 2.2. Let I be any cube in ~N and let f : I ~ ~d be a continuous function satisfying a uniform H6lder condition o f order ~. Then for any finite Borel measure tx on ~N with support contained in I, we have Dim #f ~

1 -

-

Dim~a #,

Dim* #f ~< 1 Dim~a #.

(2.9) (2.10)

Proof. (a) To prove (2.9), we take any 7 < Dim#f, then by (2.3), for #f-a. a. u E ~d, we have /,

lim inf r -~' ] ?' ---+0

min{ 1, ra]v - ul -a} dpf(v) = 0,

JR d

that is, for #-a. a. x E ~N, lim inf r -~ f m i n { 1, r a I f ( y ) -- f ( x ) [ - a } d#(y) = 0. r----~0 Jt Since for any x, y E I, I f ( y ) - f(x)[ <.Kiy - xl ~ ,

(2.11)

Y. Xiao/Statistics & Probability Letters 33 (1997) 379-387

383

where K >1 1 is a constant. We have min{1, r d l f ( y ) - f ( x ) l -a} >~K-a min{1, rdly - xl-~d}.

(2.12)

It follows from (2.11) and (2.12) that for p-a. a. x E ~N l i m i n f p-~T f P~O

min{1, p~dly--xl-~d}dt~(y)=O.

J ~ J~¢

This implies D i m ~ p ~>:(7- Since 7 < Dim pf is arbitrary, we have (2.9). (b) For any 7 > Dim~d P, we have l i m i n f r-;'F~d(x,r ) > 0

for p-a. a. x E EU.

(2.13)

r---+0

By (2.12), for u = f ( x ) with x E I

F~ (u,r)>>.g -a

,. m i n { 1 , / l y - x l - ~ a } d p ( y ) .

Hence, for any u = f ( x ) with x E N N satisfying (2.13), we have l i m i n f r-~F~f(u,r)>~K -d l i m i n f r - ~ / ~ f r ---*0

r---*0

min{1, r a l y - x [ - ~ d } d # ( y )

> 0.

JN x

This implies Dim* pf~<~ and hence (2.10).

[]

Corollary 2.3. I f for any e > O, f : I ~ Ea satisfies a uniform H6lder condition of order ~ - e, then (2.9) and (2.10) still hold.

3. Packing dimension of the image measures Let X ( t ) = (Xl(t) . . . . . Xa(t)) (t E ~N) be a fractional Brownian motion of index c( (0 < c( < 1) in gU, defined on some probability space ( ( 2 , f f , ~ ) . We will use 8 to denote the expectation with respect to ~ . In this section, we consider the packing dimension of the image measure #x of /~ under X ( t ) (t E EU) defined by

px(B)=p{tE~N

: X ( t ) CB}

for a n y B o r e l set BC_~d.

The main result is the following theorem.

Theorem 3.1. Let # be any finite Borel measure on ~N. Then with probability 1, 1 Dim Px = - Dim~d p,

(3.1)

1 Dim* #x ---- - Dim*d ~.

(3.2)

Proof. We only prove (3.1); (3.2) can be proved similarly. For any positive integer n and any e > 0,

X ( t ) (t E [-n,n] s ) a.s. satisfies a uniform H61der condition of order c~ - e (see Kahane, 1985, Ch. 18). Let #(n) be the restriction of # on In = [-n, n] s, that is, #(n)(B) = It(BNI~) for any Borel set B in EN. Then by Corollary 2.3, we have D i m / ~ ) ~ < 1 Dim~d# (n)

a.s.,

E Xiao/Statistics & Probability Letters 33 (1997) 379-387

384

which implies 1

Dim #x ~<- Dim~d #

(3.3)

a.s.

To prove the reverse inequality, we observe that for any s E NN, by Fubini's theorem,

gF~X(X(s), r) = ~¢J ~ min{ 1, r a Iv - X(s)l-d} d m ( v )

(3.4)

= f~,,, d~min{ 1, r a IX(t) - X(s) t-d} dp(t). We consider min{ 1, r a IX(t) - X(s)l-a} = ~{ Ix(t) - X(s)l ~ r} + e { r d IX(t) - X(s)l-a. 1{IX(t)-X(s)l >~r}}

{

~
Kr d } 1, it _ s[=a

~
{

-Kra ~

( I "~d/2L + ~k2-~ J

rd 1 ( [u[2_ "~ du [>jr lula It - sl ~a exp - 2]t _ sl2=j

Krd L > ~

1, [t_ s[~d j + [t_ s[~

1 exp(--~)

P

dp.

Hence, for any 0 < ¢ < 1, min{1, rd[X(t)--X(s)[-d}<~K min

1, [ t - srd-~ [ = ( a - o J"~ "

(3.5)

For any ? < Dim~d p, by Lemma 2.1, there exists ~ > 0 such that V < Dim=(d_e)#. It follows from (2.5) that liminfr~0r-'~'/~ fRx min{1, rd-~lt --s[-~(d-~)}d#(t) = 0

for p-a.a, s E ~N.

(3.6)

By (3.4)-(3.6) we have that for p-a.a, s C Nu

~(liminf r-~;/~ F~x(X(s),r)) <~K liminf~o r-~'/~ f~x min{1, rd--E[t-- s[--~(d--O}d#(t) ~0.

By using Fubini's argument again, we see that with probability 1, liminf r -7/~ F~x(X(s),r) = 0 r---*O

for p-a.a, s E NU.

Hence, Dim #x ~>7_ a.s. Since, 7 can be arbitrarily close to Dim~d #, we have Dim #x/> 1Dim~d g

a.s.

(3.7)

Y. Xiao / Statistics & Probability Letters 33 (1997) 379-387

Combining (3.3) and (3.7), we complete the proof of (3.1).

385

[]

If N < ~ d , Xiao (1993) proved that with probability 1, DimX(E) = -

1

DimE

(3.8)

for every Borel set EC_ ~U. It is easy to deduce from (2.1), (2.2) and (3.8) that a result similar to (3.8) holds for the packing dimension of the image measures. Theorem 3.2. I f N ~ etd, then with probability 1, 1 Dim#x = - D i m # , Dim* Px = 1Dim* # for every finite Borel measure # on ~N.

Remark. In Theorem 3.1, the exceptional null probability set depends on #. But in Theorem 3.2, the exceptional null probability set does not depend on #. 4. Packing dimension of the image set We see from (2.7) and (2.8) that the packing dimension of an analytic set E can be expressed in terms of the packing dimension of the finite Borel measures supported on E. This allows us to deduce from Theorem 3.1 an analogous result for DimX(E). Theorem 4.1. Let X ( t ) (t E ~ N ) be a fractional Brownian motion o f index ~ in ~a. Then for any analytic set E C ~N, with probability 1, DimX(E) = 1Dim~aE.

(4.1)

The proof of (4.1) is similar to the proof of Theorem 10 in Falconer and Howroyd (1995). We need several lemmas. Lemma 4.1. Let E C_ ~N be an analytic set and let f : ~N ~ ~d be a continuous function. I f 0 <<.t < D i m f ( E ) , then there exists a compact set F C E such that t < D i m f ( F ) . Proof. The proof is the same as that of Lemma 7 in Falconer and Howroyd (1995) with f replacing the orthogonal projection Pv. The following lemma is Theorem 1.20 in Mattila (1995). Lemma 4.2. Let F C_ ~N be a compact set and let f • ~N ____+~d be a continuous function. I f v is a finite Borel measure on ~a with support in f ( F ) , then there exists a finite Borel measure # on ~N such that v = #f and the support o f # is contained in F. Lemma 4.3. Let E C_~N be an analytic set. Then for any continuous function f : ~N ____+~d Dim f ( E ) = sup{Dim # f " p E M+(E)}.

(4.2)

Y. Xiao/Statisties & Probability Letters 33 (1997) 379-387

386

Proof. For any # E

M+(E), we

have #f E M+(f(E)). Then (2.7) implies

D i m f ( E ) ~ > sup{Dim#f : # E M + ( E ) } .

(4.3)

To prove the reverse inequality, let t < Dim f(E). Then by Lemma 4.1 there exists a compact set F C_E such that D i m f ( F ) > t. Hence, by (2.7) there exists a finite Borel measure v E M+(f(F)) such that Dimv > t. It follows from Lemma 4.2 that there exists # E M+(F) such that v = #f, this implies sup{Dim #f : ~ E Mc+(E)} > t . Since t < Dim f ( E ) is arbitrary, we have D i m f ( E ) ~ < s u p { D i m # f : /~ E M + ( E ) } .

(4.4)

Eq. (4.2) now follows from (4.3) and (4.4). Proof of Theorem 4.1. Since Dim is a-stable, we may and will assume that E is bounded. Hence, there exists a cube I such that EEL For any e > 0, X(t) (t E I) a.s. satisfies a uniform H61der condition of order ~ - e. Then by Lemma 2.2, for any # E Mc+(E) we have

1

Dim #x ~< - Dim~d #

a.s.

Hence by (2.8) and (4.2) we have D i m X ( E ) ~<1 Dim~d E .

(4.5)

On the other hand, for any t < 1 Dim~aE, by (2.8) there exists # E follows from (3.1) that Dim#x > t

M+(E) such

that at < Dim~d#. It

a.s.

Hence by Lemma 4.3 we have D i m X ( E ) > t a.s.. Since t < ~Dim~aE is arbitrary, the proof is completed. [] Remark. (1) If N<<.~d, then for any analytic set E C_ ~N, Dim~dE = DimE. Hence (4.1) reduces to (1.2). If E c_ ~N, satisfies dim E = Dim E, then Dim~d E = min{~d, DimE} and (4.1) also reduces to (1.2). (2) Let W(t) (t E ~N) be the Brownian sheet in ~a (see Orey and Pruitt, 1973). Then with a little modification the above proofs can show that for any analytic set E C_ RN+, Dim

W(E) =

2 Dima/2 E

a.s.

(3) The Hausdorff dimension of the image set of self-similar processes (including fractional Brownian motion) has been studied by several authors (see Lin and Xiao (1995) and the references therein). It would be interesting to consider the packing dimension of the image set of general self-similar processes, especially self-similar stable processes.

Acknowledgements I am very much indebted to Prof. Falconer who sent me a copy of his recent paper with Dr. Howroyd.

Y. Xiao/Statistics & Probability Letters 33 (1997) 379-387

387

References Falconer, K.J. (1990), Fractal Geometry - Mathematical Foundations And Applications (Wiley, New York). Falconer, K.J. and J.D. Howroyd (1995), Packing dimensions for projections and dimension profiles, Math. Proc. Cambridoe Philos Soc., to appear. Hu, X. and S.J. Taylor (1994), Fractal properties of products and projections of measures in ~d, Math. Proc. Cambridge Philos Soc. 115, 527 544. Kahane, J.-P. (1985), Some Random Series of Functions (Cambridge Univ, Press, Cambridge, 2nd ed.). Lin, Hounan and Yimin. Xiao (1994), Dimension properties of the sample paths of self-similar processes, Acta Math. Sinica 10, 289-300. Mattila, P. (1995), Geometry of Sets and Measures in Euclidean Spaces (Cambridge Univ. Press, Cambridge). Orey, S. and W.E. Pruitt (1973), Sample functions of the N-parameter Wiener processes, Ann. Probab. 1, 138-163. Saint Raymond, X. and C. Tricot, (1988), Packing regularity of sets in n-space, Math. Proc. Cambridge Philos Soc. 103, 133-145. Talagrand, M. and Xiao, Yimin, (1996), Fractional Brownian motion and packing dimension, 9". Theoret. Probab., 9, 579-593. Taylor, S.J. and C. Tricot (1985), Packing measure and its evaluation for a Brownian path, Trans. Amer. Math. Soc. 288, 679-699. Tricot, C. (1982), Two definitions of fractional dimension, Math. Proc. Cambridge Philos Soc. 91, 57-74. Xiao, Yimin (1993), Uniform packing dimension results for fractional Brownian motion, in: A. Badrikian, P.A. Meyer and J.A. Yan, eds., Probability and Statistics - Rencontres Franeo Chinoises en Probabilitks et Statistiques (World Scientific, Singapore) pp. 211-219.