Kolmogorov numbers of Riemann–Liouville ... - Semantic Scholar

Kolmogorov numbers of Riemann–Liouville operators over small sets and applications to Gaussian processes

Werner Linde Friedrich-Schiller-Universit¨ at Jena

Summary. We investigate compactness properties of the Riemann–Liouville operator Rα of fractional integration when regarded as operator from L2 [0, 1] into C(K), the space of continuous functions over a compact subset K in [0, 1]. Of special interest are small sets K, i.e. those possessing Lebesgue measure zero (e.g. fractal sets). We prove upper estimates for the Kolmogorov numbers of Rα against certain entropy numbers of K. Under some regularity assumption about the entropy of K these estimates turn out to be two–sided. By standard methods the results are also valid for the (dyadic) entropy numbers of Rα . Finally we apply these estimates for the investigation of the small ball behavior of certain Gaussian stochastic processes, as e.g. fractional Brownian motion or Riemann–Liouville processes, indexed by small (fractal) sets. Keywords. Fractional integration, Kolmogorov numbers, entropy numbers, fractal sets, small deviation, fractional Brownian motion. 2000 Mathematics Subject Classification. 47B06, 26A33, 60G15, 28A80

1

Introduction

The aim of the present paper is to investigate compactness properties of the Riemann–Liouville fractional integration operator Rα when regarded as an operator from L2 [0, 1] into C(K) (the space of continuous functions over K) for certain compact subsets K ⊆ [0, 1]. Here, as usual, the operator Rα is defined by Z t 1 (Rα f )(t) := (t − s)α−1 f (s) ds , t ∈ [0, 1] . (1.1) Γ(α) 0

1

Properties of Rα as operator from L2 [0, 1] into C(K) are of special interest for ”small” sets K, i.e. those with Lebesgue measure zero. To our opinion those questions are interesting in their own right, although our main motivation for their investigation came from the theory of stochastic processes. Recall that Rα is tightly related to the fractional Brownian motion BH of Hurst index H = α − 1/2 as well as to the so–called Riemann–Liouville process WH (cf. [13], [17], [18] or section 6 below). Thus our results lead to a deeper insight into the structure of BH and WH when indexed by ”small” subsets K in [0, 1] (e.g. fractal sets). From a probabilistic point of view similar questions were recently treated in [19] and led to new properties for a large class of L´evy processes. Let us also mention some related results in [3] where the authors investigate compactness properties of integral operators in dependence of the entropy numbers of the underlying compact set. We shall use two different quantities to measure the degree of compactness of Rα , namely Kolmogorov and (dyadic) entropy numbers. Let us shortly recall their definition. If S is a compact operator from a Banach space E into a Banach space F its Kolmogorov numbers dn (S) are defined by ( ) dn (S) = dn (S : E → F ) := inf

sup dF (Sx, Fn ) : Fn ⊆ F , dim(Fn ) < n

(1.2)

kxkE ≤1

where, as usual, dF (y, Fn ) := inf {ky − zkF : z ∈ Fn } denotes the distance of y ∈ F to the subspace Fn (w.r.t. the norm in F ). The (dyadic) entropy numbers of S are given by   n−1 2[   en (S) = en (S : E → F ) := inf ε > 0 : S(BE ) ⊂ (yj + ε BF ) .   j=1

Here BE and BF denote the (closed) unit balls in E and F , respectively. In other words, en (S) is the infimum over all ε > 0 such that S(BE ) can be covered by at most 2n−1 balls of radius ε > 0 in F . We refer to [24], [25], [6] and [26] for more information about Kolmogorov and entropy numbers.

As shown in [5], [23] and [7] these two numbers are tightly related. For example, if an operator S maps a Hilbert space H into a Banach space E, then it holds dn (S : H → E) ≈ n−γ · (log n)β for some γ > 1/2 and β ∈ R iff en (S : H → E) ≈ n−γ · (log n)β . Here we have used the following notation. Given two sequences (an )n≥1 and (bn )n≥1 of positive real numbers we write an  bn provided that an ≤ c · bn for a certain c > 0. If, furthermore, also bn  an , then we write an ≈ bn .

2

Let us come back to Rα as defined in (1.1). First note that Rα maps L2 [0, 1] into Lq [0, 1] for a certain q ≥ 1 iff α > max {0, 1/2 − 1/q}. Moreover, if 2 ≤ q ≤ ∞, then dn (Rα : L2 [0, 1] → Lq [0, 1]) ≈ en (Rα : L2 [0, 1] → Lq [0, 1]) ≈ n−α

(1.3)

(cf. [11], [1] and [17]). Observe that for α > 1/2 the functions Rα f , f ∈ L2 [0, 1], are continuous, thus in this case we may regard Rα as operator from L2 [0, 1] into C[0, 1]. Of course, the asymptotic in (1.3) remains valid in this case as well. Given a compact subset K ⊆ [0, 1], then for α > 1/2 the operator Rα may be regarded in natural way as operator from L2 [0, 1] into C(K), i.e. we investigate Rα f with respect to the norm kRα f kC(K) = sup |(Rα f )(t)| ,

f ∈ L2 [0, 1] .

t∈K

Intuitively it is clear that the degree of compactness of Rα should increase (i.e. its Kolmogorov and/or entropy numbers tend to zero faster) provided that K becomes smaller. To make this more precise we need some suitable measure for the size of the compact set K. At a first glance one might expect the Hausdorff dimension of K as such a measure. Yet it turns out that this not the right quantity for our purposes. More suited are quantities related with the so–called box dimension of K (cf. [9]), i.e. we describe the size of K by its covering properties. More precisely, an adequate tool for the size of K is the behavior of its entropy numbers εm (K) defined by   m   [ εm (K) := inf δ > 0 : K ⊆ ∆j , ∆j intervals of length < δ . (1.4)   j=1

If 1/2 < α ≤ 3/2, then Rα is known to map L2 [0, 1] into C α−1/2 [0, 1], the space of (α − 1/2)– H¨older continuous functions over [0, 1]. Hence, for those α’s quite general assertions about so–called H¨older operators apply and the results in [4], [6] and [30] lead to the following: Proposition 1.1 Suppose 1/2 < α ≤ 3/2 and εm (K) ≤ h(m) for a regularly varying decreasing function h. Then this implies dm (Rα : L2 [0, 1] → C(K)) ≤ c · m−1/2 · h(m)α−1/2 .

(1.5)

The disadvantage of the preceding result is that it does not apply for large α’s. This is somehow surprising because the larger α the smoother the functions Rα f are. Thus our main goal was to extend Proposition 1.1 to arbitrary α > 1/2 and, moreover, to estimate the Kolmogorov numbers of Rα directly by the entropy numbers of the underlying set K. In the latter problem we did not succeed completely, because the case 1/2 < α < 1 is not covered by the following main result of this paper. Theorem 1.2 Let α ≥ 1. Then there is a κ ∈ N such that for all compact sets K ⊆ [0, 1] it follows that dκm (Rα : L2 [0, 1] → C(K)) ≤ c · m−1/2 · εm (K)α−1/2 , m ∈ N. (1.6) For example, one may choose κ = 4 [α] + 11. 3

Applying Carl’s inequality (cf. [5]) to Theorem 1.2 we get the following estimate for the entropy numbers of the Riemann–Liouville integration operator. Theorem 1.3 Let K ⊆ [0, 1] be compact and suppose that εm (K) ≤ h(m), m ∈ N , for some decreasing function h satisfying supm≥1 h(m)/h(2m) := λ < ∞ . Then for α ≥ 1 there is a c > 0 (only depending on λ and α) such that em (Rα : L2 [0, 1] → C(K)) ≤ c · m−1/2 · h(m)α−1/2 .

(1.7)

Remark: Estimate (1.7) is tightly related to results presented in [33]. But observe that there only sets K are investigated which satisfy some kind of self–similarity while our estimates apply to arbitrary compact subsets of the unit interval. Furthermore, we prove that under some regularity assumptions about εm (K) estimate (1.6) is optimal. More precisely, the following will be shown. Theorem 1.4 Suppose α > 1/2 and let K ⊆ [0, 1] be a compact set such that for some λ ≥ 1 we have εm (K) ≤ λ · ε2m (K) , m∈N. (1.8) Then it follows that dm (Rα : L2 [0, 1] → C(K)) ≥ c · m−1/2 · εm (K)α−1/2

(1.9)

where c > 0 only depends on α and λ. Combining the preceding Theorem with Theorem 1.2 (resp. Proposition 1.1 for 1/2 < α < 1) and with the above mentioned relation between the entropy and Kolmogorov numbers for operators defined on Hilbert spaces leads to the following. Corollary 1.5 Suppose that εm (K) ≈ m−θ · (log m)β for some θ ≥ 1 and β ∈ R (note that by K ⊆ [0, 1] we necessarily have β ≤ 0 for θ = 1). Then this implies dm (Rα : L2 [0, 1] → C(K)) ≈ em (Rα : L2 [0, 1] → C(K)) ≈ m−1/2−θ(α−1/2) · (log m)β(α−1/2) . The organization of the paper is as follows. In section 2 we prove Theorem 1.2 first for integer α’s. Then a multiplication formula (Lemma 3.2) allows us to deduce the general case from that of integer α’s. This will be carried out in section 3. In section 4 we prove Theorem 1.4. Again we derive the proof from that for integer α’s. Compact sets K ⊂ [0, 1] with Lebesgue measure zero admit a very special (Cantor like) representation which allows quite precise estimates for εm (K). The representation as well as the two–sided entropy estimates will be subject of section 5. Finally, in section 6 some probabilistic applications will be stated and proved. For example, we determine the small ball behavior of fractional Brownian motions and Riemann–Liouville processes indexed by compact subsets K of [0, 1] in dependence of the size of K. 4

Throughout the paper c with or without subscript always denotes a positive constant (maybe different at each occurrence) which is either universal or depends only on the order of the Riemann– Liouville operator. Acknowledgement: The author is very grateful to Zhan Shi from Universit´e Paris VI for many very fruitful discussions about probabilistic problems which finally initiated the investigation of the presented topic. Without his remarks and ideas this paper never would have been written. Furthermore we thank Eduard Belinsky for careful reading of the first part of the manuscript and providing several suggestions. Finally we are grateful to Martina Z¨ahle for her helpful remarks concerning the structure of ”small” compact sets in [0, 1].

2

Proof of Theorem 1.2 for integer α’s

As already mentioned, when α > 3/2 the results about H¨older continuous operators do no longer apply. Hence some completely different approach is necessary. The basic idea is to cover K in an optimal way by m intervals ∆1 , . . . , ∆m with |∆j | < δ and to prove very precise Sm estimates (in dependence of δ and m) for dn (Rα ) as operator with values in C(∆) where ∆ := j=1 ∆j . Let us fix the notation. Here and later on ∆1S , . . . , ∆m are always intervals in [0, 1] with disjoint interior, say ∆j = [aj , bj ], 1 ≤ j ≤ m, and ∆ := m j=1 ∆j . We may regard now Rα as operator from L2 [0, 1] into C(∆) in the usual way, i.e. Z t 1 (Rα f )(t) := (t − s)α−1 f (s) ds , t∈∆. (2.1) Γ(α) 0 When splitting Rα into m independent pieces we obtain an operator Rα∆ mapping L2 (∆) into C(∆) acting as follows: m Z 1 X t ∆ (Rα f )(t) := (t − s)α−1 f (s) ds · 1∆j (t) , t∈∆. (2.2) Γ(α) aj j=1

Our strategy is to compare the compactness properties of Rα with those of Rα∆ in dependence of m and the length of the intervals. To this end we introduce operators Sαj , 1 ≤ j ≤ m, mapping L2 [0, aj ] into C(∆j ) by Z aj 1 j (Sα f )(t) := (t − s)α−1 f (s) ds , t ∈ ∆j . (2.3) Γ(α) 0 Since Rα − Rα∆ =

m X

Sαj

(2.4)

j=1

it is necessary to investigate properties of the Sαj ’s more thoroughly. Let m = 1, i.e. we have only one interval ∆ = [a, b] and only one operator Sα defined by (2.3), i.e. Z a 1 (t − s)α−1 f (s) ds , a≤t≤b. (2.5) (Sα f )(t) := Γ(α) 0 5

For each α > 0 this operator maps L2 [0, a] into L2 (∆) and if α > 1/2, then Sα is even an operator into C(∆). A first result describes the structure of Sα for integer α’s. Lemma 2.1 If α is an integer, then it follows that rk(Sα ) ≤ α. Proof: Writing Sα as (Sα f )(t) =

Z a α−1  1 X α−1 (−s)α−1−k f (s) ds · tk · 1∆ (t) Γ(α) k 0 k=0

immediately proves the Lemma. We are now in the position to estimate dn (Rα : L2 [0, 1] → C(∆)) in the case of integer α’s. Proposition 2.2 Suppose α ∈ N and let ∆1 , . . . , ∆m be intervals in [0, 1] as before with union ∆. Then for any n ∈ N we have dn+2mα (Rα : L2 [0, 1] → C(∆)) ≤ c · n−α · |∆|α−1/2 .

(2.6)

In particular, if |∆j | < δ, 1 ≤ j ≤ m, then it follows that d(2α+1)m (Rα : L2 [0, 1] → C(∆)) ≤ c · m−1/2 · δ α−1/2 .

(2.7)

Proof: Since (2.4) and Lemma 2.1 imply for integer α’s that rk(Rα − Rα∆ ) ≤ m α , we conclude dmα+1 (Rα − Rα∆ ) = 0 . Using additivity properties of the dn ’s this leads to dn+mα (Rα ) ≤ dn (Rα∆ ) dn+mα (Rα∆ )

as well

≤ dn (Rα ) .

(2.8) (2.9)

Both estimates are valid for any choice of m disjoint intervals ∆1 , . . . , ∆m in [0, 1]. In particular, the ˜ 1, . . . , ∆ ˜ m with |∆j | = |∆ ˜ j |, remain true when we shift ∆1 , . . . , ∆m to the left, i.e. when passing to ∆ 1 ≤ j ≤ m, and   ˜ := ∆ ˜1 ∪ ··· ∪ ∆ ˜ m = 0, |∆| . ∆ ˜1...,∆ ˜ m . In the latter case the We are going to apply (2.8) for ∆1 , . . . , ∆m and (2.9) for ∆ ˜ operator Rα (which we denote by Rα in order to distinguish it from the operator given by (2.1)) maps L2 [0, |∆|] into C[0, |∆|], hence by the scaling properties of Rα and by (1.3) we obtain ˜ α ) ≤ c · |∆|α−1/2 · n−α , dn (R

n ∈ N.

(2.10)

˜

Since Rα∆ may be isometrically transformed into Rα∆ it follows that ˜

dn (Rα∆ ) = dn (Rα∆ ) ,

n ∈ N.

Hence by (2.8), (2.9), (2.10) and (2.11) we finally arrive at
˜ d2αm+n Rα : L2 [0, 1] → C(∆) ≤ dαm+n (Rα∆ ) = dαm+n (Rα∆ ) ≤ dn (R˜α ) ≤ c · |∆|α−1/2 · n−α 6

(2.11)

as claimed. Estimate (2.7) may be immediately derived from (2.6) by choosing n = m. As consequence of Proposition 2.2 we may now prove Theorem 1.2 for special α’s. Proof of Theorem 1.2 for integer α’s: Given a natural number m we choose a covering of K by m intervals ∆1 , . . . , ∆m such that δ := sup1≤j≤m |∆j | ≤ 2 · εm (K). Let as before ∆ be the union of the ∆j ’s. Then we define an operator Φ : C(∆) → C(K) by Φ(f ) := f and obtain K

[Rα : L2 [0, 1] → C(K)] = Φ ◦ [Rα : L2 [0, 1] → C(∆)] . Consequently, if α ∈ N, by Proposition 2.2 it follows that d(2α+1)m (Rα : L2 [0, 1] → C(K)) ≤ kΦk · d(2α+1)m (Rα : L2 [0, 1] → C(∆)) ≤ c · m−1/2 · δ α−1/2 ≤ c0 · m−1/2 εm (K)α−1/2 . This completes the proof of (1.6) with κ = 2α + 1.

3

Proof of Theorem 1.2 – General case

We turn now to the case of non–integer α’s. Here (2.8) and (2.9) are no longer valid, thus we have to find some substitute for these estimates. We start with introducing another helpful sequence of so–called operator numbers. Let S be an operator from a separable Hilbert space H into a Banach space E. Then S is said to be an l–operator provided that ∞ X XS := ξk Sfk (3.1) k=1

converges a.s. in E for some (each) ONB (fk )k≥1 in H. Here (ξk )k≥1 denotes an i.i.d. sequence of N (0, 1)–distributed random variables. Whenever S is an l–operator, its l–norm is defined by  1/2 l(S) := E kXS k2 . Given an l–operator S : H → E we set ln (S) := inf {l(S − A) : A : H → E , rk(A) < n} . For properties of these numbers we refer to [26], [17] and [13]. Let now S be a compact operator between two Hilbert spaces H1 and H2 . Then S admits a so–called Schmidt representation, i.e. Sh =

∞ X

σn hh, fn i gn

n=1

with σ1 ≥ σ2 ≥ · · · ≥ 0 tending to zero and two orthonormal systems (fk )k≥1 and (gk )k≥1 in H1 and H2 , respectively. The σn ’s are usually called the singular numbers of S. It is known (cf. [24], 7

11.3.3) that then dn (S) = σn for n ∈ N. Furthermore, S is an l–operator iff it is Hilbert–Schmidt, i.e. iff the σn ’s are square summable and, moreover, as easily can be seen (cf. [13]) then we have
P∞ 2 1/2 . In particular, it holds ln (S) = k=n σk √ n d2n−1 (S) ≤ ln (S) . (3.2) It is worthwhile to mention that a deep result due to A. Pajor and N. Tomczak–Jaegermann (cf. [22]) asserts that (3.2) remains valid (with some universal constant on the right hand side) for l–operators with values in Banach spaces. √ The following Lemma is crucial to get rid of a factor m later on. Before formulating it let us fix the notation. Given Hilbert spaces H1 , . . . , Hm the Hilbert space l2 (Hj ) is then defined by

with norm kxkl2 (Hj )

l2 (Hj ) := {x = (x1 , . . . , xm ) : xj ∈ Hj } 1/2 P 2 m . kx k := j j=1

Lemma 3.1 Let S1 , . . . , Sm be l–operators mapping H into some Hilbert spaces H1 , . . . , Hm . Define S : H → l2 (Hj ) by Sh := (S1 h, . . . , Sm h) for h ∈ H. Then for each n ∈ N it follows that  1/2 m X √ nm · d2nm−1 (S) ≤  ln (Sj )2  . (3.3) j=1

Proof: Let Aj : H → Hj be operators of rank < n such that l(Sj − Aj ) ≤ (1 + ε) ln (Sj ) ,

1≤j≤m,

for a given ε > 0. Define now A : H → l2 (Hj ) by Ah := (A1 h, . . . , Am h) for h ∈ H. Then we have rk(A) < nm and in view of m X k(S − A)hk2 = k(Sj − Aj )hk2Hj j=1

one easily gets lmn (S)2 ≤ l(S − A)2 ≤

m X

l(Sj − Aj )2 ≤ (1 + ε)2

j=1

m X

ln (Sj )2 .

(3.4)

j=1

Thus (3.3) follows directly from (3.2) and (3.4). Our next objective is to estimate the degree of compactness of Rα − Rα∆ as defined in (2.1) and (2.2) in the case α ∈ / N. The basic idea is to reduce this case to that of integer α’s. To this end let us introduce another version of Rα . Given ∆ = [a, b] in [0, 1] define Rα0 on L2 [0, b] by ( Rt 1 α−1 f (s) ds : 0 ≤ t < a Γ(α) R0 (t − s) (3.5) (Rα0 f )(t) = t 1 α−1 f (s) ds : a ≤ t ≤ b Γ(α) a (t − s) For α > 0 this is a well–defined operator with values in L2 [0, b] while for α > 1/2 it has even values in C[0, b]. Furthermore let Sα (for ∆ = [a, b] as before) be defined by (2.5). Then the following multiplication formula will play an important role later on. 8

Lemma 3.2 Suppose α > 1/2 and β > 0. Then we have Sα+β = Rα0 ◦ Sβ + Sα ◦ Rβ0 .

(3.6)

Here Sβ and Rβ0 are regarded as operators into L2 (∆) and into L2 [0, b], respectively. In particular, if α ∈ N, then Sα+β = Rα0 ◦ Sβ + Fα (3.7) where Fα is an operator of rank less or equal than α. Proof: To verify (3.7) we first observe that Sα = Rα − Rα0

(3.8)

and Rα+β = Rα ◦ Rβ

0 as well as Rα+β = Rα0 ◦ Rβ0 .

(3.9)

Consequently, by (3.8) Rα+β = Rα ◦ Rβ = [Rα0 + Sα ] ◦ [Rβ0 + Sβ ] 0 + Sα ◦ Rβ0 + Rα0 ◦ Sβ + Sα ◦ Sβ . = Rα+β

(3.10)

Note that Sβ maps into L2 (∆) while Sα is defined on L2 [0, a], thus Sα ◦ Sβ = 0. Since 0 , Sα+β = Rα+β − Rα+β

from (3.10) we derive (3.6). If α ∈ N, by Lemma 2.1 we get rk(Sα ) ≤ α. Thus setting Fα := Sα ◦Rβ0 by (3.6) this immediately leads to (3.7). The following observation about representation (3.6) will be important later on: Since Sβ maps L2 [0, a] into L2 (∆), by the definition of Rα0 (compare (3.5)) the first term in (3.6) may also be written as     Rα0 ◦ Sβ = Rα : L2 (∆) → C(∆) ◦ Sβ : L2 [0, 1] → L2 (∆) . (3.11) Here the first operator at the right hand side of (3.11) has to be understood as the restriction of Rα to functions in L2 [0, 1] having their support in ∆. Consequently, in view of (2.4) we obtain the following result. Corollary 3.3 Let ∆1 , . . . , ∆m be as before and suppose that α ∈ N. Then for any β > 0 we have   m X ∆ Rα+β − Rα+β = Rα∆ ◦  Sβj  + F (3.12) j=1

where F is an operator with rk(F ) ≤ α m. In view of (3.12) it is necessary to get more information about the degree of compactness of the operators Sβj , 0 < β < 1, regarded as mappings into L2 (∆). 9

Lemma 3.4 Define Sβ : L2 [0, a] → L2 (∆), ∆ = [a, b], by Z a 1 (t − s)β−1 f (s) ds . (Sβ f )(t) := Γ(β) 0 If 0 < β < 1, then there are constants c, cβ > 0 (independent of ∆) such that for all n ≥ 2 1/2

dn (Sβ ) ≤ c · e−cβ n

· |∆|β .

(3.13)

Proof: We split the proof into three steps. In a first one we investigate the operator Sβ∞ mapping L2 [1, ∞) into C[0, 1] defined by Z ∞ ∞ [(t + s)β−1 − sβ−1 ]f (s) ds (Sβ f )(t) := 1

and we claim that dn (Sβ∞ : L2 [1, ∞) → C[0, 1]) ≤ c ·

Γ(n + 3 − β) −n ·2 . Γ(n + 2)

(3.14)

To verify this take f ∈ L2 [1, ∞) and let Pn (Sβ∞ f ; t) be the n–th Taylor polynomial of Sβ∞ f taken at the point t0 = 1/2. Then it follows that Z ∞ ∞ (S f )(t) − Pn (S ∞ f ; t) ≤ 1 · (1 − β)(2 − β) . . . (n + 2 − β) s−n−1+β |f (s)| ds β β 2n (n + 1)! 1 Γ(n + 3 − β) −n ≤ c· · 2 kf k2 Γ(n + 2) which proves (3.14). In a second step we fix a number Λ ≥ 1 and define an operator SβΛ : L2 [0, Λ] → L2 [0, 1] by Λ

Z

(SβΛ f )(t) :=

(t + s)β−1 f (s) ds .

(3.15)

0

We are going to prove that for n ≥ 2 1/2

dn (SβΛ : L2 [0, Λ] → L2 [0, 1]) ≤ c · e−cβ n

(3.16)

with c, cβ > 0 independent of Λ. To this end write (1)

(2)

SβΛ = Sβ + Sβ + F where (1)

(Sβ f )(t) := (2)

(Sβ f )(t) :=

Z

1

(t + s)β−1 f (s) ds ,

0

Z

Λ

[(t + s)β−1 − sβ−1 ]f (s) ds

1

10

(3.17)

and the operator F is defined by Λ

Z

sβ−1 f (s) ds .

(F f )(t) := 1

A result of Laptev (cf. [14]) asserts 1/2

(1)

dn (Sβ : L2 [0, 1] → L2 [0, 1]) ≤ c · e−cβ n

(3.18)

and (3.14) lets us conclude (2)

dn (Sβ : L2 [1, Λ] → L2 [0, 1]) ≤ c ·

Γ(n + 3 − β) −n ·2 Γ(n + 2)

(3.19)

with c > 0 independent of Λ. Of course, rk(F ) = 1, hence by (3.17) (1)

(2)

d2n (SβΛ ) ≤ dn (Sβ ) + dn (Sβ ) , 1/2

i.e. by (3.18) and (3.19) we have dn (SβΛ ) ≤ c · e−cβ n as long as n ≥ 2. This proves (3.16). In a last step we verify now (3.13). Thus put δ := |∆|. By isometric transformations (change of variables) it follows that dn (Sβ ) = δ β · dn (S˜β ) (3.20) where S˜β maps L2 [0, 1/δ] into L2 [0, 1] and (S˜β f )(t) :=

Z

1/δ

(t + s)β−1 f (s) ds .

0

Of course, by (3.16) (with Λ = 1/δ) and by (3.20) we finally get (3.13) as asserted. Corollary 3.5 Let σ1 ≥ σ2 ≥ · · · ≥ 0 be the singular numbers of Sβ . Then 1/2

σn ≤ c · e−cβ n

· |∆|β

provided that n ≥ 2. Hence l2 (Sβ ) =

∞ X

!1/2 σn2

≤ c · |∆|β .

(3.21)

n=2

Remark: It is not difficult to see that (3.21) remains true for l1 (Sβ ) = l(Sβ ) provided that 0 < β < 1/2 while it is no longer valid for l(Sβ ) when 1/2 ≤ β < 1. We are now in the position to extend Proposition 2.2 to fractional integration operators with arbitrary index. Proposition 3.6 Let ∆1 , . . . , ∆m be as before intervals in [0, 1] with disjoint interior and with union ∆ and suppose sup1≤j≤m |∆j | ≤ δ. Then for α ≥ 1 there is a natural number κ = κ(α) such that dκm (Rα : L2 [0, 1] → C(∆)) ≤ c · m−1/2 · δ α−1/2 . (3.22) 11

Proof: Given α ≥ 1, in view of Proposition 2.2 we may suppose that α = k + β where k ∈ N and 0 < β < 1. By (3.12) we get Rα − Rα∆ = Rk∆ ◦ S + F P j where S = m j=1 Sβ and rk(F ) ≤ km, consequently, dm(2k+5) (Rα − Rα∆ ) ≤ dm(k+1) (Rk∆ ) · d4m (S) .

(3.23)

We estimate now both terms on the right hand side of (3.23) separately. Because of (2.11), (2.10) and (2.9) we obtain dm(k+1) (Rk∆ ) ≤ c · m−1/2 · δ k−1/2 (3.24) while Lemma 3.1 for n = 2 together with (3.21) yields √

 1/2 m X 2m · d4m (S) ≤  l2 (Sβj )2  ≤ c · m1/2 · δ β .

(3.25)

j=1

Combining (3.23), (3.24) and (3.25) finally gives dm(2k+5) (Rα − Rα∆ ) ≤ c · m−1/2 · δ α−1/2 , thus for each l ∈ N we conclude dm(2k+5+l) (Rα ) ≤ dml (Rα∆ ) + c · m−1/2 · δ α−1/2

and

dm(2k+5+l) (Rα∆ ) ≤ dml (Rα ) + c · m−1/2 · δ α−1/2 .

(3.26) (3.27)

We argue now as in the proof of Proposition 2.2, i.e. we first apply (3.26) with l = 2k + 6, then (2.11), next (3.27) with l = 1 and finally (1.3). Doing so it follows that dm(4k+11) (Rα ) ≤ dm(2k+6) (Rα∆ ) + c1 · m−1/2 · δ α−1/2 ˜

= dm(2k+6) (Rα∆ ) + c1 · m−1/2 · δ α−1/2 ˜ α ) + c2 · m−1/2 · δ α−1/2 ≤ dm (R ≤ c3 · m−1/2 · δ α−1/2 This completes the proof with κ = 4 [α] + 11. Remark: We do not know whether or not (3.22) remains valid for 1/2 < α < 1. At least our methods do not apply for those α’s. Proof of Theorem 1.2: The assertion follows from Proposition 3.6 exactly in the same way as in the case α ∈ N (where we used Proposition 2.2 instead).

12

4

Lower Estimates

The aim of the present section is to prove Theorem 1.4. Again we start with the investigation of integer α’s. Lemma 4.1 Let K ⊆ [0, 1] be a compact set and suppose that there are s1 , . . . , sm ∈ K such that |si − sj | ≥ δ ,

i 6= j .

(4.1)

If I ⊆ [0, 1] is defined by m h [

I :=

j=1

δ δi sj − , sj + , 2 2

then for α ∈ N it follows that dn (Rα : L2 (I) → C(K)) ≥ c · n−1/2 · log

 me 1/2 n

· δ α−1/2 ,

1≤n≤m.

Proof: We choose a function ψ : R → [0, ∞) possessing the following properties: (i)

supp(ψ) ⊂ (0, 1) ,

(ii)

ψ(1/2) = 1

(iii)

ψ is α–times continuously differentiable.

and

Setting ϕ := ψ (α) , we also have supp(ϕ) ⊂ (0, 1) and, moreover, because of α ∈ N it follows that Rα ϕ = ψ. With the help of this function ϕ we construct now functions ϕj , 1 ≤ j ≤ m, by ϕj (s) := ϕ

 s − s + δ/2  j , δ

s ∈ R,

satisfying kϕj k2 = δ 1/2 · kϕk2

and supp(ϕj ) ⊂ [sj − δ/2, sj + δ/2] .

(4.2)

Furthermore,  t − s + δ/2   t − s + δ/2  j j = δα ψ (Rα ϕj )(t) = δ α (Rα ϕ) δ δ leads by property (ii) of ψ to (Rα ϕj )(sj ) = δ α ,

1 ≤ j ≤ m.

Next we define an operator B : l2m → L2 (I) by B(x) :=

m X

xj ϕj ,

x = (x1 , . . . , xm ) ,

j=1

which by (4.2) satisfies kB(x)k2 = δ 1/2 · kϕk2 · kxk2 . m is given by Another operator Φ : C(K) → l∞


m Φ(f ) := f (sj ) j=1 , 13

f ∈ C(K) .

(4.3)

Of course, kΦk ≤ 1 and in view of (4.3) it follows that (Φ ◦ Rα ◦ B)(x) = δ α · x ,

x ∈ l2m ,

i.e. we have Φ ◦ Rα ◦ B = δ α · i2,∞ m denotes the canonical identity map. Consequently, where i2,∞ : l2m → l∞ m δ α · dn (i2,∞ ) ≤ kBk · dn (Φ ◦ Rα : L2 (I) → l∞ ) ≤ δ 1/2 · kϕk2 · dn (Rα : L2 (I) → C(K))

which completes the proof because of a deep Theorem of Garnaev and Gluskin (cf. [10]) asserting dn (i2,∞ ) ≥ c · n−1/2 log

 me 1/2 n

,

1≤n≤m.

(4.4)

As an immediate consequence of Lemma 4.1 we get the following stronger version of Theorem 1.4 in the case of integer α’s. Proposition 4.2 Suppose α ∈ N and regard Rα as operator from L2 [0, 1] to C(K) for a certain compact set K ⊆ [0, 1]. Then it follows that dm (Rα : L2 [0, 1] → C(K)) ≥ c · m−1/2 · εm (K)α−1/2 . Proof: Given m ∈ N we choose a δ with εm (K)/2 ≤ δ < εm (K). Then there are s1 , . . . , sm satisfying (4.1), hence Lemma 4.1 applies with n = m. Note that, of course, dm (Rα : L2 (I) → C(K)) ≤ dm (Rα : L2 [0, 1] → C(K)) . This completes the proof. Before treating the non–integer case we need another lemma for later purposes. Lemma 4.3 Let ∆1 , . . . , ∆m be intervals in [0, 1] with disjoint interior and |∆j | ≤ δ, 1 ≤ j ≤ m. Then for β ∈ (0, 1) there is a κ ∈ N such that dκm (Rβ : L2 [0, 1] → L2 (∆)) ≤ c · δ β where, as before, ∆ =

(4.5)

Sm

j=1 ∆j .

Proof: The proof follows almost exactly as that of Proposition 3.6 and we use the same notation as there. Writing Rβ = Sβ + Rβ∆ by (3.25) it follows that d4m (Sβ ) ≤ c · δ β . Hence, if n ∈ N, we obtain the estimates dn+4m (Rβ ) ≤ c · δ β + dn (Rβ∆ ) (4.6) as well as dn+4m (Rβ∆ ) ≤ c · δ β + dn (Rβ ) .

14

(4.7)

˜ β is an operator from L2 [0, |∆|] into L2 [0, |∆|], In contrast to the proof of Proposition 3.6 here R thus by (1.3) ˜ β ) ≤ n−β · |∆|β . dn (R (4.8) Summing up, by (4.6), (4.7) and (4.8) we finally get ˜

d9m (Rβ ) ≤ c · δ β + d5m (Rβ∆ ) = c · δ β + d5m (Rβ∆ ) ˜ β ) ≤ c00 · δ β . ≤ c0 · δ β + dm (R This proves the assertion with κ = 9. Now we are in position to prove Theorem 1.4. Proof of Theorem 1.4: First observe that it remains to prove (1.9) for non–integer α’s. Set k := [α] + 1

and β := k − α .

(4.9)

Consequently, we have 0 < β < 1. For the proof we need a covering ∆1 , . . . , ∆m of K as well as points s1 , . . . , sM in K with sufficiently large distance. Let us start with the construction of the S covering. For a given m ∈ N we choose intervals ∆1 , . . . , ∆m with disjoint interior such that K ⊆ m j=1 ∆j := ∆ and, moreover, sup1≤j≤m |∆j | ≤ 2·εm (K). Next we choose some well separated points in K. For a number M > 2m which will be specified later on take δ > 0 such that εM (K) ≤ δ < εM (K) . 2

(4.10)

Then there are s1 , . . . , sM ∈ K for which |si − sj | ≥ δ if i 6= j. With these sj ’s we define now intervals Ij possessing disjoint interiors by h δ δi Ij := sj − , sj + , 2 2

1≤j≤M .

Let D ⊆ {1, . . . , M } be the set D := {j ≤ M : Ij ⊆ ∆} . Then it follows that m ¯ := #(D) satisfies m ¯ ≥ M − 2m and, moreover, [ I := Ij ⊆ ∆ . (4.11) j∈D

For a more precise formulation of the following arguments we denote by JI and J∆ the canonical embeddings from L2 (I) and L2 (∆) into L2 [0, 1], respectively. If k is defined by (4.9), then by Lemma 4.1 we obtain for each n ∈ N the estimate dn (Rk ◦ J∆ : L2 (∆) → C(K)) ≥ dn (Rk ◦ JI : L2 (I) → C(K))  me ¯ 1/2 k−1/2 ·δ . ≥ c · n−1/2 · log n

(4.12)

Next we apply the semigroup property of the Riemann–Liouville operators. Recall that k = α + β. Doing so we obtain       Rk ◦ J∆ : L2 (∆) → C(K) = Rα : L2 [0, 1] → C(K) ◦ Rβ ◦ J∆ : L2 (∆) → L2 [0, 1] 15

and, consequently, by (4.12) for any l ∈ N with l + m ≤ m ¯ it follows that  me ¯ 1/2 k−1/2 ·δ l+m ≤ dm (Rα : L2 [0, 1] → C(K)) · dl (Rβ ◦ J∆ : L2 (∆) → L2 [0, 1]) .

c · (l + m)−1/2 · log

(4.13)

We claim now that for 0 < β < 1 there exists a natural number κ with dκm (Rβ ◦ J∆ : L2 (∆) → L2 [0, 1]) ≤ c · εm (K)β .

(4.14)

Assume for a moment that (4.14) has already been proven. Then we may precise the choice of the number M from above as M := (κ + 3)m, hence m ¯ ≥ (κ + 1)m, and using (4.13) with l := κm we derive from (4.14) and (4.10) that c0 · εM (K)k−1/2 · m−1/2 ≤ dm (Rα : L2 [0, 1] → C(K)) · dκm (Rβ ◦ J∆ : L2 (∆) → L2 [0, 1]) ≤ c · dm (Rα : L2 [0, 1] → C(K)) · εm (K)β .

(4.15)

In view of (1.8) we have εm (K)k−1/2 ≤ ρ · εM (K)k−1/2 for a certain ρ ≥ 1 depending on λ in (1.8) and on α. Hence (4.15) leads to the desired estimate dm (Rα : L2 [0, 1] → C(K)) ≥ c · m−1/2 · εm (K)k−β−1/2 = c · m−1/2 · εm (K)α−1/2 . Consequently, to complete the proof it remains to verify (4.14). First note that for any n ∈ N ∗ ∗ dn (Rβ ◦ J∆ : L2 (∆) → L2 [0, 1]) = dn (J∆ Rβ : L2 [0, 1] → L2 (∆))

(4.16)

∗ R∗ acts as where the dual operator J∆ β ∗ ∗ (J∆ Rβ f )(s)

1

Z

1 = Γ(β)

(t − s)β−1 f (t) dt ,

s∈∆.

s

∗ R∗ may be transformed to R ¯ β : L2 [0, 1] → L2 (∆) ¯ with By an easy isometric transformation J∆ β

¯ β f )(u) := (R

1 Γ(β)

Z

u

(u − v)β−1 f (v) dv ,

¯ . u∈∆

0

¯ ¯ ¯ ¯ ¯ Here the set ∆ is defined by ∆ := ∆1 ∪ · · · ∪ ∆m with ∆j := {1 − u : u ∈ ∆j } for 1 ≤ j ≤ m. ¯ Observe that ∆j = |∆j | ≤ 2 · εm (K), hence we are in the situation of Lemma 4.3 and this leads to ¯ β ) ≤ c · εm (K)β dκm (R (4.17) ∗ R∗ ) = d (R ¯ β ) by where, for example, κ may be chosen as κ = 9. In view of (4.16) and using dn (J∆ n β (4.17) we get the desired estimate (4.14). This completes the proof of Theorem 1.4.

Questions: 1. We do not know whether or not the regularity condition (1.8) is indeed necessary for the lower estimate in the non–integer case. Note that for integer α’s Theorem 1.4 holds without any extra 16

assumption about the behavior of εm (K). 2. We believe that Theorem 1.4 is also valid for the entropy numbers of Rα , yet could not verify this because we do not know whether or not (4.5) is true for the entropy numbers as well. But observe that (1.9) holds for integer α’s. Indeed, using instead of (4.4) an estimate for the entropy numbers of the embedding i2,∞ due to C. Sch¨ utt (cf. [28]) by the same arguments as before em (Rα : L2 [0, 1] → C(K)) ≥ c · m−1/2 · εm (K)α−1/2 whenever α ∈ N.

5

Metric Entropy of Fractal Sets

In view of Theorems 1.2 and 1.4 it is important to find precise upper and lower estimates for εm (K) where K ⊂ [0, 1] is a compact set with |K| = 0. Before stating a general representation theorem for those sets let us introduce the following class of functions on [0, 1].   ∞   X X αk for certain τk ∈ (0, 1) , αk ≥ 0 , (5.1) αk = 1 A := A : [0, 1] → [0, 1] : A(t) =   τk ≤t

k=1

Note that A is exactly the set of distribution functions of discrete probability measures on (0, 1). Proposition 5.1 Let K ⊂ [0, 1] be compact with 0∈K,

1∈K

and

|K| = 0 .

(5.2)

Then there is a function A ∈ A such that K = {A(t) : 0 ≤ t ≤ 1} .

(5.3)

Proof: We may represent the complement K c of K (taken in [0, 1]) in the form c

K =

∞ [

Gk

(5.4)

k=1

with open, disjoint intervals G1 , G2 , . . .. Setting αk := |Gk |, k = 1, 2, . . ., in view of |K| = 0 we obtain ∞ X αk = 1 . (5.5) k=1

By induction we choose now real numbers τk ∈ (0, 1) such that τk < τl for certain 1 ≤ k 6= l < ∞ iff the interval Gk is on the left hand side of Gl . With these αk ’s and τk ’s we define a function A by X A(t) := αk , 0≤t≤1. τk ≤t

Because of (5.5) it follows that A ∈ A and, moreover, by the construction of the τk ’s it holds X X A(τk − 0) = |Gl | and A(τk ) = |Gl | , τl 0 each set Cj may be covered by at most δ −1 · |Cj | + 1 open intervals of length less than δ. Consequently, by (5.8) the set K admits a δ–cover by at most δ −1 ·

m X

|Cj | + m + 1

j=0

18

(5.10)

open intervals. Applying (5.10) with δ = δm

1 = m

∞ X

αk ,

k=m+1

in view of (5.9) the set K may be covered by 2m + 1 intervals of length less than δm . In other words, ∞ 1 X ε2m+1 (K) ≤ δm = αk m k=m+1

as asserted. Estimate (5.7) is even easier to prove. Indeed, fix again m ∈ N and define elements sk ∈ [0, 1], 1 ≤ k ≤ m, by sk := A(τk ). By the construction we have sk ∈ K, and for k 6= l there is either an interval of length αk or of length αl in–between sk and sl . Since the αk ’s are decreasing this implies |sk − sl | ≥ min {αk , αl } ≥ αm . Thus there exist m elements in K with mutually distance of at least αm yielding εm (K) ≥ αm /2 as asserted. Corollary 5.3 Suppose that the weights αk of A ∈ A satisfy αk ≈ k −θ · (log k)β for some θ > 1 and β ∈ R and let K ⊂ [0, 1] be generated by A as in (5.3). Then independent of the choice of the τk ’s we have εm (K) ≈ m−θ · (log m)β . Remark: If either the αk ’s tend to zero very rapidly, e.g. αk = 2k , or very slowly, e.g. αk ≈ k −1 · (log k)−β for some β > 1, then Proposition 5.2 does not lead to sharp estimates for εm (K). Here the τk ’s (more precisely, the way how the τk ’s are ordered) are important for the degree of compactness of K. Example: Let us treat as an example the classical Cantor set C in [0, 1]. As can be seen easily, this set may be generated by a function A where the decreasing weights αk satisfy αk ≈ k − log 3/ log 2 . Consequently, it follows that dm (Rα : L2 [0, 1] → C(C)) ≈ em (Rα : L2 [0, 1] → C(C)) ≈ m−1/2−θ(α−1/2) where θ = log 3/ log 2. Proposition 5.2 allows us to reformulate Theorem 1.2 and Theorem 1.4. Given a function A ∈ A (we always suppose now that the weights αk are in decreasing order) and α > 1/2 we may define an operator RαA : L2 [0, 1] → C[0, 1] as follows. (RαA f )(t)

1 := Γ(α)

Z

A(t)

(A(t) − s)α−1 f (s) ds ,

0

19

0≤t≤1.

(5.11)

Proposition 5.4 For A ∈ A with (decreasing) weights αk let RαA be defined by (5.11). Then the following are valid. (1) For certain κ ∈ N and c > 0 it follows that dκm (RαA

−α

: L2 [0, 1] → C[0, 1]) ≤ c · m

·

∞  X

αk

α−1/2

.

(5.12)

k=m+1

(2) If αk ≈ k −θ (log k)β for some θ > 1 and β ∈ R, then this implies dm (RαA : L2 [0, 1] → C[0, 1]) ≈ em (RαA : L2 [0, 1] → C[0, 1]) −1/2 −θ(α−1/2)

≈ m

β(α−1/2)

· (log m)

(5.13) .

Remark: necessarily normalized, i.e. it P Suppose that the weights αk of the function A are not A maps L [0, d] into C[0, 1]. Yet α = d for some d > 0. In this case the operator R holds ∞ 2 α k=1 k by the scaling properties of Rα estimates (5.12) as well as (5.13) remain valid in this more general situation (without any extra factor depending on d).

6

Probabilistic applications

Given a Hilbert space H and an operator S : H → C(K) for a certain compact metric space K such that ∞ X XS := ξj Sfj (6.1) j=1

converges a.s. (in C(K)) for some (each) ONB (fj )j≥1 in H (here as in (3.1) the ξj ’s are i.i.d. standard normal) we may regard XS as stochastic process indexed by K. More precisely, we set XS (t) :=

∞ X

ξj (Sfj )(t) ,

t∈K.

(6.2)

j=1


Note that XS = XS (t) t∈K is then a centered Gaussian process possessing a.s. continuous paths. For example, if ∞ X m−1/2 em (S) < ∞ (6.3) m=1

by Dudley’s theorem (cf. [8]) combined with a result in [32] the sum in (6.1) converges a.s., hence the process XS over K is well–defined with a.s. continuous paths. A higher degree of compactness of the operator S leads to better small deviation estimates for the process XS . More precisely, the following was proved in [17] and [12]. Proposition 6.1 Given γ > 0 and β ∈ R the following are equivalent. (i)

em (S)  m−1/2 −γ (log m)β  − log P sup |XS (t)| ≤ ε  ε−1/γ log(1/ε)β/γ . 

(ii)

t∈K

Moreover, the above equivalence remains valid for ≈ instead of  in (i) and (ii), respectively. 20

Here we have used the following notation: Given two functions f and g on (0, ∞), then f (ε)  g(ε) means that there is a constant c > 0 such that f (ε) ≤ c · g(ε) for small ε > 0. We write f (ε) ≈ g(ε) provided that f (ε)  g(ε) holds together with g(ε)  f (ε). Let H > 0 be given and let K be a compact subset of [0, 1]. Then we regard the Riemann– Liouville operator RH+1/2 as before as operator from L2 [0, 1] into C(K). In view of (1.3) for all H > 0 this operator satisfies (6.3), hence for any fixed ONB (fj )j≥1 in L2 [0, 1] the process WH (t) :=

∞ X

ξj (RH+1/2 fj )(t)

j=1 ∞

=

X 1 ξj Γ(H + 1/2) j=1

Z

t

(t − s)H −1/2 fj (s) ds ,

t∈K,

0

is a well–defined centered Gaussian process with a.s. continuous paths over K. The process WH = (WH (t))t∈K is usually called (cf. [18]) Riemann–Liouville process with Hurst index H. For H = 1/2 the process WH is the Wiener process while for H = k + 1/2 with k ∈ N we get the k–times (pathwise) integrated Wiener process. A first application of Proposition 6.1 leads to the following result. Theorem 6.2 Suppose the compact set K ⊆ [0, 1] satisfies εm (K)  m−θ (log m)β for some θ ≥ 1 and β ∈ R (again we necessarily have β ≤ 0 for θ = 1). Then this implies   − log P sup |WH (t)| ≤ ε  ε−1/(θH) · log(1/ε)β/θ . (6.4) t∈K

Moreover, if even εm (K) ≈ m−θ (log m)β , then   − log P sup |WH (t)| ≤ ε ≈ ε−1/(θH) · log(1/ε)β/θ . t∈K

Proof: Suppose first εm (K)  m−θ (log m)β . Then by Theorem 1.3 this implies em (RH+1/2 : L2 [0, 1] → C(K))  m−1/2−θH · (log m)βH . An application of Proposition 6.1 with γ = θH and with βH easily gives (6.4) as asserted. If even εm (K) ≈ m−θ (log m)β , this time we may use Corollary 1.5 and obtain em (RH+1/2 : L2 [0, 1] → C(K)) ≈ m−1/2−θH · (log m)βH . Another application of Proposition 6.1 (this time for ≈) completes the proof of (6.5). Example: In the special case C of the Cantor set it follows that   − log P sup |WH (t)| ≤ ε ≈ ε− log 2/(H log 3) . t∈C

21

(6.5)

In particular, for the k–times integrated Wiener process over C this implies that the order of its small ball behavior (in the log–level) is ε−2 log 2/((2k+1) log 3) . In the case 0 < H < 1 the process WH is tightly related with the fractional Brownian motion BH of Hurst index H. Recall that BH is a centered Gaussian process indexed by [0, ∞) with a.s. continuous paths satisfying E BH (t)BH (s) =

 1  2H s + t2H − |t − s|2H , 2

0 ≤ t, s < ∞ .

The following concrete representation of BH over [0, 1] turns out to be very useful. Let the Hilbert space H be given by H := L2 [0, 1] ⊕ L2 [0, ∞) and define SH : H → C[0, 1] by
SH (f ⊕ g) := cH RH+1/2 f + QH g . Here  Z cH := Γ(H + 1/2) (2H)−1 +

∞

(1 + s)H−1/2 − sH−1/2


2

−1/2 ds

0

and the operator QH : L2 [0, ∞) → C[0, 1] is defined by Z ∞ 1 [(t + s)H−1/2 − sH−1/2 ]f (s)ds . (QH f )(t) = Γ(H + 1/2) 0

(6.6)

As shown in [20] (cf. also [27]) the operator SH generates the fractional Brownian motion BH on [0, 1] as stated in (6.2), i.e. we have BH (t) :=

∞ X

ξj (SH fj )(t) ,

t ∈ [0, 1] .

j=1

Of course, regarding SH as operator from H into C(K) for some compact subset K ⊆ [0, 1], this operator generates (BH (t))t∈K in the same way. In particular, Proposition 6.1 applies and relates the behavior of en (SH : H → C(K))) with the small ball behavior of BH over K. Consequently, we get the following version of Theorem 6.2 for the fractional Brownian motion. Theorem 6.3 Suppose the compact set K ⊆ [0, 1] satisfies εm (K)  m−θ (log m)β for some θ ≥ 1 and β ∈ R . Then for 0 < H < 1 this implies   − log P sup |BH (t)| ≤ ε  ε−1/(θH) · log(1/ε)β/θ . (6.7) t∈K

Moreover, if even εm (K) ≈ m−θ (log m)β , then   − log P sup |BH (t)| ≤ ε ≈ ε−1/(θH) · log(1/ε)β/θ . t∈K

22

(6.8)

Proof: As shown in [1] the operator QH defined in (6.6) satisfies em (QH : L2 [0, ∞) → C[0, 1])  2−c m

1/3

(6.9)

with some c > 0 only depending on H. Of course, then also em (QH : L2 [0, ∞) → C(K))  2−c m

1/3

(6.10)

for any compact subset K ⊆ [0, 1]. Suppose now εm (K)  m−θ (log m)β . Then by Theorem 1.3 this implies em (RH+1/2 : L2 [0, 1] → C(K))  m−1/2−θH · (log m)βH . Thus by e2m−1 (SH : H → C(K)) ≤ em (cH RH+1/2 : L2 [0, 1] → C(K)) + em (cH QH : L2 [0, ∞) → C(K)) from (6.10) we derive em (SH : H → C(K))  m−1/2−θH · (log m)βH as well and the proof of (6.7) may now completed as before by an application of Proposition 6.1. Assertion (6.8) follows by similar arguments and thus we omit the proof. Remark: If K = [0, 1], thus θ = 1 and β = 0, then (6.8) was first proved in [29] and [21]. Later on this was sharpened in [16]. Before we state another probabilistic application let us recall some facts about stable subordinators. Let Γ1 < Γ2 < · · · be the arrival times of a Poisson process with intensity 1 and let τ1 , τ2 , . . . be independent, uniformly distributed on [0, 1]. Assume that (Γj )j≥1 and (τj )j≥1 are independent. For some p ∈ (0, 1) define the random function A on [0, 1] via X −1/p A(t) := Γk , 0≤t≤1. τk ≤t

Then A is a L´evy process over [0, 1], non–decreasing and p–stable (usually called p–stable subordinator, cf. [2] for more information). Let now BH be a fractional Brownian motion of Hurst index H ∈ (0, 1) over [0, ∞), independent of the p–stable subordinator A and define XH as XH (t) := BH (A(t)) ,

0≤t≤1.

Note that for H = 1/2 the stochastic process X1/2 is the so–called 2p–stable L´evy motion. The small ball behavior of XH (in the usual way and also conditionally, i.e. for a fixed path of A) may be derived from results for general subordinators in [19] (if H = 1/2 cf. [31] for the non–conditional case). Not covered by the results in [19] is the process YH (t) := WH (A(t)) ,

0≤t≤1,

with H > 1. The deeper reason is that Talagrand’s small ball result (cf. [15], p.257), a basic ingredient in [19], does no longer apply for those H’s. For a precise formulation of the next result let us suppose that WH is modelled over (Ω, P) while A is defined on (Ω0 , P0 ). 23

Proposition 6.4 For 0 < p < 1 let A be a p–stable subordinator independent of WH , H > 0. Then for almost all ω 0 ∈ Ω0 we have   0 − log P sup WH (A(t, ω )) ≤ ε ≈ ε−p/H . (6.11) 0≤t≤1

Proof: By the Strong Law of Large Numbers it follows that limj→∞ Γj /j = 1 a.s. Hence, almost all weights αk of the subordinator A behave like k −1/p and, consequently, by Proposition 5.4 and A( ·,ω 0 ) the Remark following it, for a.s. all ω 0 ∈ Ω0 the entropy numbers of RH+1/2 behave like m−1/2 −H/p . Thus the assertion follows from Proposition 6.1. Remark: Applying Fatou’s Lemma to (6.11) leads to a one–sided estimate for the usual (non– conditional) small ball behavior of YH . More precisely, we then get   0 − log (P × P ) sup |WH (A(t))| ≤ ε  ε−p/H . 0≤t≤1

The corresponding lower estimate will be treated (in a more general context) in a forthcoming paper.

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Werner Linde Faculty of Mathematics and Computer Sciences Friedrich-Schiller-Universit¨ at Jena Ernst Abbe Platz 2 07743 Jena Germany [email protected]

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