PACKING-DIMENSION PROFILES AND FRACTIONAL BROWNIAN MOTION

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1 PACKING-DIMENSION PROFILES AND FRACTIONAL BROWNIAN MOTION DAVAR KHOSHNEVISAN AND YIMIN XIAO Abstract. In order to compute the packing dimension of orthogonal projections Falconer and Howroyd 997) introduced a family of packing dimension profiles Dim s that are parametrized by real numbers s > 0. Subsequently, Howroyd 200) introduced alternate s-dimensional packing dimension profiles P-dim s and proved, among many other things, that P-dim s E = Dim s E for all integers s > 0 and all analytic sets E R N. The goal of this article is to prove that P-dim s E = Dim s E for all real numbers s > 0 and analytic sets E R N. This answers a question of Howroyd 200, p. 59). Our proof hinges on a new property of fractional Brownian motion.. Introduction Packing dimension and packing measure were introduced in the early 980s by Tricot 982) and Taylor and Tricot 985) as dual concepts to Hausdorff dimension and Hausdorff measure. Falconer 990) and Mattila 995) contain systematic accounts. It has been known for some time now that some Hausdorff dimension formulas such as those for orthogonal projections and those for image sets of fractional Brownian motion do not have packing dimension analogues; see Järvenpää 994) and Talagrand and Xiao 996) for precise statements. This suggests that a new concept of dimension is needed to compute the packing dimension of some random sets. In order to compute the packing dimension of orthogonal projections Falconer and Howroyd 997) introduced a family of packing dimension profiles {Dim s s>0 that we recall in Section 2 below. Falconer and Howroyd 997) proved that for every analytic set E R N and every integer m N,.) dim P P V E) = Dim m E for γ n,m -almost all V G n,m, Date: November 0, Mathematics Subject Classification. Primary 60G5, 60G7, 28A80. Key words and phrases. Packing dimension, dimension profiles, fractional Brownian motion. Research partially supported by NSF grant DMS

2 2 D. KHOSHNEVISAN AND YIMIN XIAO where γ n,m is the natural orthogonally-invariant measure on the Grassman manifold G n,m of all m-dimensional subspaces of R N, and P V E denotes the projection of E onto V. Subsequently, Howroyd 200) introduced a family {B-dim s s>0 of box-dimension profiles, together with their regularizations {P-dim s s>0. The latter are also called packing dimension profiles; see Section 2. Howroyd 200) then used these dimension profiles to characterize the [traditional] box and packing dimensions of orthogonal projections. In addition, Howroyd 200, Corollary 32) proved that for all analytic sets E R N : i) P-dim s E Dim s E if s > 0; and ii) if s 0, N) is an integer then.2) P-dim s E = Dim s E. Finally, P-dim s E and Dim s E agree for arbitrary s N, and their common value is the packing dimension dim P E. The principle aim of this note is to prove that.2) holds for all real numbers s 0, N). Equivalently, we offer the following. Theorem.. Equation.2) is valid for all s > 0. This solves a question of Howroyd 200, p. 59). Our derivation is probabilistic, and relies on properties of fractional Brownian motion fbm). In order to explain the connection to fbm let X := {Xt) t R N be a d-dimensional fbm with Hurst parameter H 0, ). That is, Xt) = X t),..., X d t)) for all t R N, where X,..., X d are independent copies of a real-valued fbm with common Hurst parameter H Kahane, 985, Chapter 8). Xiao 997) proved that for every analytic set E R N,.3) dim P XE) = H Dim HdE a.s. Here we will derive an alternative expression. Theorem.2. For all analytic sets E R N,.4) dim P XE) = H P-dim HdE a.s. Thanks to.3) and Theorem.2, Dim Hd E = P-dim Hd E for all integers d and all H 0, ). Whence follows Theorem.. We establish Theorem.2 in Section 3, following the introductory Section 2 wherein we introduce some of stated notions of fractal geometry in greater detail. Also we add a Section 4

3 DIMENSION-PROFILES AND fbm 3 where we derive yet another equivalent formulation for the s-dimensional packing dimension profile Dim s E of an analytic set E R N. We hope to use this formulation of Dim s E elsewhere in order to compute the packing dimension of many interesting random sets. Throughout we will use the letter K to denote an unspecified positive and finite constant whose value may differ from line to line and sometimes even within the same line. 2. Dimension Profiles In this section we recall briefly aspects of the theories of dimension profiles of Falconer and Howroyd 997) and Howroyd 200). 2.. Packing Dimension via Entropy Numbers. For all r > 0 and all bounded sets E R N let N r E) denote the maximum number of disjoint closed balls of radius r whose respective centers are all in E. The [upper] box dimension of E is defined as 2.) B-dim E = lim sup r 0 log N r E) log/r). We follow Tricot 982) and define the packing dimension of E as the regularization of B-dim E. That is, 2.2) dim P E = inf { sup k B-dim F k : E F k. There is also a corresponding notion of the packing dimension of a Borel measure. Indeed, the [lower] packing dimension of a Borel measure µ on R N is 2.3) dim P µ = inf { dim P E : µe) > 0 and E R N is a Borel set. One can compute dim P E from dim P µ as well: Given an analytic set E R N let M + c E) denote the collection of all finite compactly-supported Borel measures on E. Then, according to Hu and Taylor 994), 2.4) dim P E = sup { dim P µ : µ M + c E) The Packing Dimension Profiles of Falconer and Howroyd. Given a finite Borel measure µ on R N and an s 0, ] define 2.5) F s µ x, r) := R N ψ s ) x y µdy), r

4 4 D. KHOSHNEVISAN AND YIMIN XIAO where for finite s 0, ), 2.6) ψ s x) := min, x s) x R N, and ψ := {y R d : y. The s-dimensional packing dimension profile of µ is defined as { F s µ x, r) 2.7) Dim s µ = sup t 0 : lim inf = 0 for µ-a.a. x R N. r 0 r t Packing dimension profiles generalize the packing dimension because dim P µ = Dim s µ for all finite Borel measures µ on R N and for all s N. See Falconer and Howroyd 997, p. 272) for a proof. Falconer and Howroyd 997) also defined the s-dimensional packing dimension profile of a Borel set E R N by 2.8) Dim s E = sup { Dim s µ : µ M + c E) The Packing Dimension Profiles of Howroyd. If E R N and s > 0, then a sequence of triples w i, x i, r i ) i= is called a ψ s, δ)-packing of E whenever w i 0, x i E, 0 < r i δ, and 2.9) sup i j= ) xi x j w j ψ s. r j For all E R N, define { 2.0) P α,s 0 E) := lim sup w i 2r i ) α : w i, x i, r i ) i= δ 0 i= Then the α-dimensional ψ s -packing measure P α,s E) is defined as { 2.) P α,s E) = inf P α,s 0 E k ) : E E k. is a ψ s, δ)-packing of E The s-dimensional packing dimension profile of E can then be defined as 2.2) P-dim s E := inf {α > 0 : P α,s E) = 0. We will make use of the following two lemmas. They are ready consequences of Lemma 20 and Theorem 22 of Howroyd 200), respectively.. Lemma 2.. If E R N every α 0, γ). and P γ,s E) > 0, then E has non-sigma-finite P α,s -measure for

5 DIMENSION-PROFILES AND fbm 5 Lemma 2.2. Let A R N be an analytic set of non-sigma-finite P α,s -measure. Then there exists a compact set K A such that P α,s 0 K G) = for all open sets G R N with K G. Moreover, K is also of non-sigma-finite P α,s -measure Upper Box Dimension Profiles. Given r > 0 and E R N, a sequence of pairs w i, x i ) k i= is a size-r weighted ψ s -packing of E if: i) x i E; ii) w i 0; and iii) 2.3) max i k k j= ) xi x j w j ψ s. r Define { k 2.4) N r E ; ψ s ) := sup w i : w i, x i ) k i= is a size-r weighted ψ s -packing of E. i= This quantity is related to the entropy number N r E). In fact, Howroyd 200, Lemma 5) has shown that N r E ; ψ ) = N r/2 E) for all r > 0 and all E R N. We will use this fact in the proof of Lemma 3. below. The s-dimensional upper box dimension of E is defined as 2.5) B-dim s E := lim sup r 0 log N r E ; ψ s ), log/r) where log 0 :=. Note in particular that B-dim s =. It is possible to deduce that s B-dim s E is non-decreasing. Define P A E) to be the collection of all probability measures that are supported on a finite number of points in E. For all µ P A E) define 2.6) J s r, µ) := max x supp µ F µ s x, r) and I s r, µ) := For E R N, define 2.7) Z s r ; E) := inf µ P A E) J s r, µ). Howroyd 200) has demonstrated that for all s, r > 0, 2.8) Z s r ; E) = inf µ P A E) I s r, µ) and N r E ; ψ s ) = Consequently, 2.9) B-dim s E = lim sup r 0 log Z s r ; E). log r F µ s x, r) µdx). Z s r ; E).

6 6 D. KHOSHNEVISAN AND YIMIN XIAO According to Howroyd 200, Proposition 8), 2.20) B-dim s E = B-dim E s N, E R N. Howroyd 200) also proved that P-dim s is the regularization of B-dim s ; i.e., { 2.2) P-dim s E = inf B-dim s E k : E E k, sup k This is the dimension-profile analogue of 2.2). 3. Proof of Theorem.2 Recall that X is a centered, d-dimensional, N-parameter Gaussian random field such that for all s, t R N and j, k {,..., d, 3.) Cov X j s), X k t)) = 2 s 2H + t 2H s t 2H) δ ij. Throughout, we assume that the process X is constructed in a complete probability space Ω, F, P), and that t Xt, ω) is continuous for almost every ω Ω. According to the general theory of Gaussian processes this can always be arranged. Our proof of Theorem.2 hinges on several lemmas. The first is a technical lemma which verifies the folklore statement that, for every r > 0 and E R N, the entropy number N r XE)) is a random variable. We recall that Ω, F, P) is assumed to be complete. Lemma 3.. Let E R N be a fixed set, and choose and fix some r > 0. Then N r XE)) and Z r ; XE)) are non-negative random variables. Proof. It follows from 2.8) that Z r ; XE)) = /N r/2 XE)). Hence it suffices to prove N r XE)) is a random variable. Let CR N ) be the space of continuous functions f : R N R d equipped with the norm 3.2) f = 2 k max t k ft) + max t k ft). According to general theory we can assume without loss of generality that Ω = CR N ). It suffices to prove that for all a > 0 fixed, Θ a := {f CR N ) : N r fe)) > a is open and hence Borel measurable. For then {ω Ω : N r XE)) > a = X Θ a ) is also measurable. To this end we assume that N r fe)) > a, and define n := a. There necessarily exist t,..., t n+ E such that ft i ) ft j ) > 2r for all i j n +. Choose and fix

7 DIMENSION-PROFILES AND fbm 7 η 0, ) such that η < min{ ft i ) ft j ) 2r : i j n +. We can then find an integer k 0 > 0 such that t i k 0 for all i =,..., n +. It follows from our definition of the norm that for all δ 0, η 2 k 0+2) ) and all functions g CR N ) with g f < δ, 3.3) max i n+ gt i) ft i ) < η 2. This and the triangle inequality imply gt i ) gt j ) ft i ) ft j ) η > 2r for all i j n +, and hence N r ge)) > n. This verifies that {f CR N ) : N r fe)) > a is an open set. The following lemma is inspired by Lemma 2 of Howroyd 200). We emphasize that E [Z r ; XE))] is well defined Lemma 3.). Lemma 3.2. If E R N then 3.4) E[Z r ; XE))] K Z Hd r /H ; E ) r > 0. The constant K 0, ) depends only on d and H. Proof. Note that µ X ) P A XE)) whenever µ P A E). Hence, Z r ; XE)) I r, µ X ). Because I r, µ X ) = { Xs) Xt) r µds) µdt) for all r > 0, E [Z r ; XE))] P { Xs) Xt) r µds) µdt) 3.5) ) r d K s t µds) µdt) = K I Hd Hd r /H, µ ), where the last inequality follows from the self-similarity and stationarity of the increments of X, and where K > 0 is a constant that depends only on d and H. We obtain the desired result by optimizing over all µ P A E). Lemma 3.3. For all nonrandom sets E R N, 3.6) B-dim XE) H B-dim Hd E a.s. Proof. Without loss of generality we assume B-dim Hd E > 0, for otherwise there is nothing left to prove. Then for any constant γ 0, B-dim Hd E) there exists a sequence {r n n= of positive numbers such that r n 0 and Z Hd r n ; E) = or γ n) as n. It follows from

8 8 D. KHOSHNEVISAN AND YIMIN XIAO Lemma 3.2 and Fatou s lemma that [ ] Z r ; XE)) 3.7) E lim inf lim inf r 0 r γ/h n E [ Z r H n ; XE) )] r γ n K lim n Z Hd r n ; E) r γ n = 0. Consequently, 2.9) and 2.20) together imply that B-dim XE) γ/h a.s. The lemma follows because γ 0, B-dim Hd E) is arbitrary. The following Lemma is borrowed from Falconer and Howroyd 996, Lemma 5). Lemma 3.4. If a set E R N G R N such that E G. Then dim P E δ. has the property that B-dimE G) δ for all open sets We are ready to prove Theorem.2. Proof of Theorem.2. Since P-dim Hd E Dim Hd E,.3) implies that dim P XE) is almost surely bounded above by P-dim H HdE. Consequently, it remains to prove the reverse inequality. To this end we may assume without loss of generality that P-dim Hd E > 0, lest the inequality becomes vacuous. Choose and fix an arbitrary α 0, P-dim Hd E). Lemma 2. implies that E has non-σ-infinite P α,hd -measure. By Lemma 2.2, there exists a compact set K E such that P α,hd G K) = for all open sets G R N with G K. By separability there exists a countable basis of the usual euclidean topology on R N. Let {G k be an enumeration of those sets in the basis that intersect K. It follows from Lemma 3.3 that for every k =, 2,... there exists an event Ω k of P-measure one such that for all ω Ω k, 3.8) B-dim X ω G k K) H B-dim HdG k K) α H. Therefore, Ω 0 := Ω k has full P-measure, and for every ω Ω 0, 3.9) B-dim X ω K) U ) B-dim X ω K X U) ) α H. The preceding is valid for all open sets U with XK) U because X U) is open and K X U). According to Lemma 3.4 this proves that dim P X ω K) α/h almost surely. Because α 0, P-dim Hd E) is arbitrary this finishes the proof of Theorem.2.

9 DIMENSION-PROFILES AND fbm 9 4. An Equivalent Definition Given a Borel set E R N, we define PE) as the collection all probability measures µ on R N such that µe) = [µ is called a probability measure on E]. Define for all Borel sets E R N and all s 0, ], 4.) Z s r ; E) := inf I s r, µ). µ PE) Thus, the sole difference between Z s and Z s is that in the latter we use all finitely-supported [discrete] probability measures on E, whereas in the former we use all probability measures on E. We may also define Z s using M + c E) in place of PE) in 4.). Our next theorem shows that all these notions lead to the same s-dimensional box dimension. Theorem 4.. For all analytic sets E R N and all s 0, ], 4.2) B-dim s E) = lim sup r 0 log Z s r ; E). log r Proof. Because P A E) PE) it follows immediately that Z s r ; E) Z s r ; E). Consequently, 4.3) lim sup r 0 log Z s r ; E) log r B-dim s E). We explain the rest only when N = ; the general case is handled similarly. Without loss of much generality suppose E [0, ) and µ is a probability measure on E. For all integers n and i {0,,..., n define C i = C i,n to be /n times the half-open interval [i, i + ). Then, we can write I s /n, µ) = T + T 2, where T := ) s µdx) µdy), n x y 4.4) T 2 := 0 i<n j {i,i,i+ 0 i<n j {i,i,i+ C i C i C j C j ) s µdx) µdy). n x y Any interval C j with µc j ) = 0 does not contribute to I s /n, µ). For every j with µc j ) > 0, we choose an arbitrary point τ j E C j and denote w j := µc j ). Then the discrete probability measure ν that puts mass w j at τ j E belongs to P A E). For simplicity of notation, in the following we assume µc j ) > 0 for all j = 0,,..., n.

10 0 D. KHOSHNEVISAN AND YIMIN XIAO If j {i, i, i +, then sup x Ci sup y Cj x y 3 τ j τ i, whence we have 4.5) T 2 ) s w 3 s i w j. n τ j τ i 0 i<n j {i,i,i+ If j {i, i, i +, then a similar case-by-case analysis can be used. This leads us to the bound, 4.6) ) I s n, µ 3 s 0 i,j<n = 3 s /n a b ) s w i w j n τ j τ i ) s νda) νdb). Consequently, the right-hand side of 4.6) is at most 3 s Z s /n ; E). It follows that ) ) ) 4.7) 3 s Z s n ; E Z s n ; E Z s n ; E. If r is between /n and /n+), then Z s r ; E) is between Z s /n ; E) and Z s /n+) ; E). A similar remark applies to Z s. Because log n logn + ) as n, this proves the theorem. References [] Falconer, K. J.990), Fractal Geometry Mathematical Foundations And Applications. Wiley & Sons Ltd., Chichester. [2] Falconer, K. J. and Howroyd, J. D. 996), Projection theorems for box and packing dimensions. Math. Proc. Cambridge Phil. Soc. 9, [3] Falconer, K. J. and Howroyd, J. D. 997), Packing dimensions for projections and dimension profiles. Math. Proc. Cambridge Phil. Soc. 2, [4] Howroyd, J. D. 200), Box and packing dimensions of projections and dimension profiles. Math. Proc. Cambridge Phil. Soc. 30, [5] Hu, X. and Taylor, S. J. 994), Fractal properties of products and projections of measures in R d. Math Proc. Cambridge Phil. Soc. 5, [6] Järvenpää, M. 994), On the upper Minkowski dimension, the packing dimension, and othogonal projections. Annales Acad. Sci. Fenn. A Dissertat. 99. [7] Kahane, J.-P. 985), Some Random Series of Functions. 2nd edition. Cambridge University Press, Cambridge. [8] Mattila, P. 995), Geometry of Sets and Measures in Euclidean Spaces. Cambridge University Press, Cambridge.

11 DIMENSION-PROFILES AND fbm [9] Talagrand, M. and Xiao, Y. 996), Fractional Brownian motion and packing dimension. J. Theoret. Probab. 9, [0] Taylor, S. J. and Tricot, C. 985), Packing measure and its evaluation for a Brownian path. Trans. Amer. Math. Soc. 288, [] Tricot, C. 982), Two definitions of fractional dimension. Math. Proc. Cambridge Phil. Soc. 9, [2] Xiao, Y. 997), Packing dimension of the image of fractional Brownian motion. Statist. Probab. Lett. 33, Department of Mathematics, 55 S. 400 E., JWB 233, University of Utah, Salt Lake City, UT address: davar@math.utah.edu URL: Department of Statistics and Probability, A-43 Wells Hall, Michigan State University, East Lansing, MI address: xiao@stt.msu.edu URL: