A Uniform Dimension Result for Two-Dimensional Fractional Multiplicative Processes

Transcription

1 To appear in Ann. Inst. Henri Poincaré Probab. Stat. A Uniform Dimension esult for Two-Dimensional Fractional Multiplicative Processes Xiong Jin First version: 8 October 213 esearch eport No. 8, 213, Probability and Statistics Group School of Mathematics, The University of Manchester

2 A UNIFOM DIMENSION ESULT FO TWO-DIMENSIONAL FACTIONAL MULTIPLICATIVE POCESSES XIONG JIN Abstract. Given a two-dimensional fractional multiplicative process F t t [,1] determined by two Hurst exponents H 1 and H 2, we show that there is an associated uniform Hausdorff dimension result for the images of subsets of [, 1] by F if and only if H 1 = H Introduction It is well-known that planar Brownian motion doubles the Hausdorff dimension, in the sense that for any Borel set E +, 1.1 P{dim H BE = 2 dim H E} = 1, where B : + 2 is a planar Brownian motion and BE = {Bt : t E} is the image of E through B. This result was first proved by McKean [18] in 1955, following the works of Lévy [17] and Taylor [21] regarding the Hausdorff measure of B +, and was extended to α-stable processes by Blumenthal and Getoor [5]. The result cannot be extended to more general Lévy processes, but one can obtain control such as 1.2 P{β dim H E dim H XE β dim H E} = 1 for certain parameters β and β depending on the process X see [6, 19]. In [6] Blumenthal and Getoor also conjectured that given any Borel set E, there exists a constant λx, E such that P{dim H XE = λx, E} = 1. This conjecture is proved by Khoshnevisan and Xiao [16] in 25, in terms of Lévy exponents. The relation 1.1 involves an exceptional null set N E Ω for each fixed E, and it is natural to ask whether there is a null set N such that N E N holds for uncountably many E. In other words, we would hope for a result like 1.3 P{dim H BE = 2 dim H E for all E O} = 1 for O as large as possible. In the literature results like 1.3 are termed as uniform dimension result, and this was first proved by Kaufman [15] for planar Brownian motion when O is the set of all Borel sets in +. This result was extended to strictly stable Lévy processes by Hawkes and Pruitt [1]. For general Lévy processes the 2 Mathematics Subject Classification. 6G17; 28A8. Key words and phrases. Hausdorff dimension; fractional multiplicative processes; uniform dimension result, level sets. The author held a oyal Society Newton International Fellowship whilst this work was carried out. 1

3 2 XIONG JIN corresponding uniform dimension result may not hold, but for Lévy subordinators one can obtain either a uniform result as 1.3 for smaller family O collection of Borel sets whose Hausdorff dimension and packing dimension coincide or a looser dimension result as 1.2 uniformly for all Borel sets see [1]. For further information regarding the dimension results of stochastic processes, we refer to the survey papers [22, 25]. In this paper we prove a uniform dimension result for two-dimensional fractional multiplicative processes, a class of random continuous functions recently constructed by Barral and Mandelbrot [3]. These processes and their generalisation, multiplicative cascade processes [2], are considered as natural extensions of the classical Mandelbrot measures [14] to functions, or in probabilistic terms, subordinators to general processes. In [12] the author proved a dimension result for two-dimensional multiplicative cascade processes, motivated by the recent works in [7, 4, 2] that proved the Knizhnik-Polyakov-Zamolodchikov formula from quantum gravity for Gaussian multiplicative chaos and Mandelbrot measures. The KPZ formula is a quadratic thus nonlinear relation between dimensions of a given Borel set with respect to the Euclidean metric and the random metric obtained from multiplicative chaos. It is natural to ask whether or not this type of dimension formula holds uniformly for all Borel sets. With the help of multifractal analysis of multiplicative cascades and their graph and range singularity spectra see [1, 11] for example, it can be shown that for the dimension result in [12], as long as the formula is nonlinear, there will be some random sets that break the formula. Thus the only candidate for which the uniform dimension result could possibly hold is the multi-dimensional fractional multiplicative processes in one-dimensional case there is the level set of the process that breaks the formula, which leads to the present study. ecall that as a special case of [12], we have the following dimension result for the two-dimensional fractional multiplicative process F = F 1, F 2 with parameters 1/2 < H 1 H 2 < 1 see section 2 for precise definition: for every Borel set E [, 1], { 1.4 P dim H F E = dim H E 1 + dim } H E H 1 = 1. H 1 H 2 In particular when H 1 = H 2 = H 1/2, 1, we have { 1.5 P dim H F E = 1 } H dim H E = 1. The result 1.4 has exactly the same form as in [24] for Gaussian vector fields, which is shown in [23] to be uniform if and only if the parameters of Gaussian vector fields coincide. In this paper we show that the same phenomenon occurs for two-dimensional fractional multiplicative processes: Theorem 1.1. If H 1 = H 2 = H 1/2, 1, then 1.5 is uniform, that is { 1.6 P dim H F E = 1 } H dim H E for all sets E [, 1] = 1. If H 1 < H 2, then the result 1.4 cannot hold almost surely for all Borel sets. The proof of 1.6 relies on a stopping time technique used in [9] for stable Markov processes, with certain non-trivial modifications due to the fact that fractional multiplicative processes are neither stable nor Markovian. To show that

4 A UNIFOM DIMENSION ESULT FO FACTIONAL MULTIPLICATIVE POCESSES 3 result 1.4 cannot be uniform when H 1 H 2, we use the same trick as in [23] to show that the level set of F 1 breaks the formula. 2. Two-dimensional fractional multiplicative processes d fractional multiplicative processes. Fix two parameters 1/2 < H 1 H 2 < 1. Let ɛ = ɛ 1, ɛ 2 be a random vector such that for j = 1, 2, { +1, with probability H j 1 /2; ɛ j = 1, with probability 1 2 Hj 1 /2. Denote by {, 1} = n 1 {, 1}n the set of finite dyadic words. Let {ɛw = ɛ 1 w, ɛ 2 w : w {, 1} } be a sequence of independent copies of ɛ encoded by {, 1}. Let j {1, 2}. For each w = w 1 w n {, 1} let t w = n w m 2 m m=1 be the corresponding dyadic point in [, 1, and let I w = [t w, t w + 2 n be the corresponding dyadic interval. Then let n ɛ j w = ɛ j w 1 w m m=1 be the random weight on I w. For x [, 1 and n 1 let x n = x 1 x n {, 1} n be the unique word such that x I x n. For n 1 define the piecewise linear function t F j,n : t [, 1] 2 n1 Hj ɛ j x n dx. From Theorem 1.1 in [3] one has that almost surely {F j,n } n 1 converges uniformly to a limit F j, and F j is α-hölder continuous for any α, H j. Then the two dimensional fractional multiplicative process considered in this paper is the mapping F = F 1, F 2 : t [, 1] F 1 t, F 2 t 2. We shall always assume that P{ɛ 1 = ɛ 2 } < 1, to ensure that the process F does not degenerate to one-dimensional case. emark 2.1. If we take the parameter H j, 1/2], then the corresponding sequence {F j,n } n 1 is not bounded in L 2 -norm. Moreover it is shown in [3] that the normalised sequence { Fj,n / 2 X j,n = n1/2 Hj Hj 2 1, if H < 1/2, F j,n / n/2, if H = 1/2 converges, as n, in law to standard Brownian motion restricted on [, 1].

5 4 XIONG JIN 2.2. Statistical self-similarity. For any w {, 1} we can similarly define F [w] j,n : t [, 1] 2n1 Hj t Denote by w the length of w and n ɛ j w x m dx. m=1 g w : t 2 w t t w the canonical mapping from I w to [, 1. Then by definition for any s, t I w and n w one has [F 2.1 F j,n t F j,n s = 2 w Hj [w] ɛ j w j,n w g wt F [w] j,n w g ws ]. Let F [w] j be the limit of {F [w] j,n } n 1. It certainly has the same law as F j, and it is independent of ɛ j w. Moreover for any s, t I w one gets from 2.1 that 2.2 F j t F j s = 2 w Hj ɛ j w [F [w] j g w t F [w] j g w s ] Boundary values and oscillations. Let Z j = F j 1 and Z j w = F [w] j 1 for w {, 1}. Clearly they have the same law. Also from 2.2 one has F j t w + 2 w F j t w = 2 w Hj ɛ j w Z j w, where ɛ j w and Z j w are independent. Let ϕu, v = E e iuz1+vz2 and ϕ j u = E e iuzj be the characteristic functions of Z 1, Z 2 and Z j respectively. The following lemma see Lemma 2 in [12] is essential to our proof: Lemma 2.1. One has ϕ L 1 2 if P{ɛ 1 = ɛ 2 } < 1 and ϕ j L 1. Consequently, Z 1, Z 2 has a bounded joint density function f with f ϕ 1 = ϕu, v dudv <, 2 provided P{ɛ 1 = ɛ 2 } < 1, and Z j has a bounded density function f 1 with f j ϕ j 1 = ϕ j u du <. Let X j = sup s,t [,1 F j t F j s and X j w = sup s,t [,1 F [w] j t F [w] j s for w {, 1}. They have the same law, and from 2.2 one has sup F j t F j s = 2 w Hj X j w. s,t I w Moreover, from Lemma 3.1 in [2] one has that for all q, 2.3 EX q j <.

6 A UNIFOM DIMENSION ESULT FO FACTIONAL MULTIPLICATIVE POCESSES 5 3. Proof of Theorem Proof of 1.6. The idea of the proof is to show that for all n large enough, the number of w {, 1} n/h that the image F I w intersects with a given dyadic square of side length 2 n is bounded above uniformly for all dyadic squares. This will imply the uniform dimension result 1.6. In order to do so we need to control the probability of F t w S, given that F t w1,, F t wk S for w j w, then pass this information to F I w S by using the control of the oscillation of F on I w. Preliminaries. Let H 1 = H 2 = H 1/2, 1 and fix an integer p > 1/2H 1 so that 1 + 1/p/H < 2. For n 1 denote by T n the set of dyadic numbers of generation n, that is { n T n = t w = w j 2 j : w = w 1 w n {, 1} n} {1}. and j=1 Let S n be the collection of all dyadic squares in 2 with side length 2 n. Fix n 1 and S S n. Let m = n1 1/p/H. Define NS := #{w {, 1} m : F t w S} ÑS := #{w {, 1} m : F I w S }. In the following we shall estimate a uniform control of NS for all S S n, and use it to get a uniform control of ÑS, then show that this uniform control gives the uniform dimension result. Uniform control of NS. First notice that For k = 1,, 2 m one can easily get T m = {j 2 m : j =,, 2 m }. 3.1 PNS k EH k S, where H k S := k s 1<s 2< <s k T m We shall control EH k S by iterating. Denote by S the diameter of S. One has 3.2 = H k+1 S s 1<s 2< <s k T m, s k 1 s 1<s 2< <s k T m, s k 1 k k 1 {F sl S} 1 {F sl S} 1 {F sl S}. s k <t T m 1 {F t S} 1 { }. F t F s k S s k <t T m Fix s 1 < < s k < t T m, let N be the smallest integer such that there exists a dyadic word w = w 1 w N such that I w s k, t. This gives 2 N t s k 4 2 N.

7 6 XIONG JIN From 2.2 we may write 3.3 F t F s k = A j wz j w + B j w 2 1/2, j=1,2 where A j w = 2 NH ɛ j w and B j w = F j t F j t w + 2 N + F j t w F j s k. Denote by 3.4 Fw = σ ɛu : u {, 1} \ {w u : u {, 1} }. It is easy to check that the random variables A 1 w, A 2 w, B 1 w, B 2 w and k 1 {F s l S} are measurable with respect to Fw. Also notice that 1 { F t F s k S } F t F sk 1+1/p/H S 1+1/p/H. ecall in Lemma 2.1 the joint density function f of Z 1, Z 2, which is bounded by ϕ 1 <. One gets E F t F s k 1+1/p/H Fw fx, y dxdy = A 2 1 wx + B 1 w 2 + A 2 wy + B 2 w 2 1+1/p/2H f u B1w2 NH 2 N1+1/p ɛ 1w, u B1w2NH ɛ 1w dudv u v 2 1+1/p/2H 2 N1+1/p dudv 1 + ϕ 1 u 2 + v 2 1+1/p/2H := 2 N1+1/p C 1+1/p/H. u 2 +v 2 <1 The finiteness of C 1+1/p/H comes from the fact that we already choose p large enough such that 1 + 1/p/H < 2. This gives k E 1 {F sl S} 1 { } F t F s k S Fw 3.5 S 1+1/p/H k C 1+1/p/H S 1+1/p/H 1 {F sl S} E F t F s k 1+1/p/H Fw k 1 {F sl S} 2 N1+1/p 4 1+1/p C 1+1/p/H S 1+1/p/H For any s k T m we have 3.6 s k <t T m t s k 1+1/p k 1 {F sl S} t s k 1+1/p. 2 m1+1/p l 1+1/p 2 n1 1/p1+1/p/H l 1+1/p.

8 A UNIFOM DIMENSION ESULT FO FACTIONAL MULTIPLICATIVE POCESSES 7 Thus by combining 3.2, 3.5 and 3.6 we get 3.7 EH k+1 S C 2 n1/p 1/p2 /H EH ks, where C = 4 1+1/p C 1+1/p/H 2 1+1/p/2H l 1+1/p < and H ks = k s 1<s 2< <s k T m, s k 1 Obviously H k S H ks, thus from 3.7 we get Together with 3.1 this gives P sup NS k + 1 S S n 1 {F sl S}. EH k+1 S C 2 n1/p 1/p2 /H EH ks C 2 n1/p 1/p2 /H EH k S C k 2 nk1/p 1/p2 /H EH 1 S. C k 2 nk1/p 1/p2 /H S S n E 2 m = C k 2 nk1/p 1/p2 /H E = C k 2 nk1/p 1/p2 /H 2 m m j= 2C k 2 nk1/p 1/p2 /H 2 n1 1/p/H = 2C k 2 n1 1/pk/p 1/H. 1 {F tj S} 1 {F tj S} j= S S n We may choose k = p + 1. Then by the Borel-Cantelli lemma one gets for P- almost every ω Ω that there exists a integer n p ω such that for any n n p ω, 3.8 sup S S n NS p + 2. Uniform control of ÑS. Now we want to pass the information of F t w S to that of F I w S. We start with the oscillation Xw = F [w] j s F [w] j t 2 1/2, w {, 1}. sup s,t [,1] j=1,2 Clearly Xw has the same law as X = sup s,t [,1] F s F t. Then for any n 1, P sup Xw 2 n/p Xw 2 n/p w {,1} n w {,1} n P 2 n 2 n1+1/p EX p+1. From 2.3 it is easy to deduce that EX p+1 <. Then by the Borel-Cantelli lemma one gets for P-almost every ω Ω that there exists a integer n pω such that for any n n pω and w {, 1} n, 3.9 Xw 2 n/p.

9 8 XIONG JIN To combine 3.8 and 3.9, let Ω = p>1/2h 1 { ω Ω : np ω n pω < }. So PΩ = 1. Fix ω Ω, p > 1/2H 1 and n n p ω n pω large enough such that We are going to control the number n1 1/p/H > n1 2/p/H. ÑS := {w {, 1} n1 1/p/H : F I w S }. First for any w {, 1} n1 1/p/H one has For S S n let sup F s F t = 2 w H Xw 2 n1 1/p/H H 1/p. s,t I w US = { z 2 : distz, S 2 n1 1/p/H H 1/p}. Then for any w {, 1} n1 1/p/H we have 3.1 F I w S F I w US F t w US. On the other hand, there are at most { area z 2 : distz, S 2 n1 1/p/H H 1/p n}/ 2 2n π 2 n1 1/p/H H 1/p n 2 2 2n π n1 1/p/H 2H 1/p 2n/ n1 1/p/H π n1 1/p/H 2H 1/p 2H/1 2/p = π n1 1/p/H 22H/1 2/p+1/p many S S n that intersect US, and for each S S n there are at most p+2 words w {, 1} n1 1/p/H such that F t w S. Together with 3.1, this implies that for n large enough, 3.11 sup S S n ÑS C 2 n1 1/p/H 22H/1 2/p+1/p, where C = p + 2 π Lower bound of dim H F E. For any set E [, 1], let C N be any dyadic square covering of F E such that S C N S dimh F E+1/p 2 N. Denote by ns = log 2 S 1 1/p/H. For N large enough one has that { Iw : w {, 1} ns, F I w S } S C N forms a covering of E. Moreover, due to 3.11, for H 2H / s = dim H F E + 1/p 1 1/p /p + 1 p

10 A UNIFOM DIMENSION ESULT FO FACTIONAL MULTIPLICATIVE POCESSES 9 one has S C N w {,1} ns, F I w S which implies I w s C dim H E dim H F E + 1/p S C N S dimh F E+1/p C 2 N, H 2H / 1 1/p /p + 1 p. Since this holds for all p > 1/2H 1, we get dim H E H dim H F E. Upper bound of dim H F E. Now consider any dyadic interval covering I N such that I dimh E+1/p 2 N. I I N For N large enough one gets from 3.9 that of E sup F s F t = I w H Xw I w H 1/p s,t I w for any I w I N, thus for each I one can use a square of side length 2 I H 1/p to cover F I. We have 2 I H 1/p dimh E+1/p/H 1/p = C I dimh E+1/p C 2 N, I I N I I N where C = 2 dim H E+1/p/H 1/p. This gives dim H F E dim H E + 1/p. H 1/p Since this holds for all p > 1/2H 1, we get dim H E H dim H F E Proof of the fact that a uniform result cannot hold when H 1 < H 2. We shall use the following result. Proposition 3.1. For j = 1, 2 almost surely there exists a Borel set F j [, 1] with positive Lebesgue measure such that for each y, dim H L j y = 1 H j, where L j y = {x [, 1] : F j x = y} is the level set of F j at level y. From Proposition 3.1 one has that almost surely dim H L 1 y = 1 H 1 > for some y F 1 [, 1]. On the other hand, since F 2 is α-hölder continuous for all α, H 2, and F L 1 y = {y} F 2 L 1 y, so dim H F L 1 y = dim H F 2 L 1 y dim H L 1 y H 2 < dim H L 1 y H 1. This shows that the relation in 1.4 cannot hold almost surely for all Borel sets.

11 1 XIONG JIN Proof of Proposition 3.1. It is enough to prove the result for F 1. We will use the same method as used for constructing the local time of fractional Brownian motion to compute the Hausdorff dimension of its level sets, see [13] for example. Let ν be the occupation measure of F 1 with respect to the Lebesgue measure on [, 1], that is the Borel measure defined as νb = 1 { F 1t B} dt for any Borel set B. First we show that almost surely ν is absolutely continuous with respect to the Lebesgue measure. We consider the Fourier transform of ν: ˆνu = e iuf1t dt. We will show that E ˆνu 2 du <. This will imply that almost surely ˆν is in L 2. Therefore almost surely ν is absolutely continuous with respect to the Lebesgue measure on and its density function belongs to L 2. By using Fubini s theorem one has E ˆνu 2 du = E e iu F1t F1s du dsdt. s,t [,1] Fix s < t 1. Let N 1 be the smallest integer such that there exists a dyadic word w = w 1 w N such that I w s, t, thus From 2.2 we may write 2 N t s 4 2 N F 1 t F 1 s = A 1 w Z 1 w + B 1 w, where A 1 w = 2 NH1 ɛ 1 w and B 1 w = F 1 t F 1 t w + 2 N + F 1 t w F 1 s. ecall that Z 1 w is independent of A 1 w and B 1 w. ecall 3.4 and ϕ 1 u = Ee iuz1 the characteristic function of Z 1. One has E e iu F1t F1s du Fw = e iub1w ϕ 1 A 1 w u du. Thus 3.13 E e iu F1t F1s du E From Lemma 2.1 we get ϕ 1 1 <, thus E ˆνu 2 du 4 H1 ϕ 1 1 ϕ 1 A 1 w u du = E A 1 w 1 = 2 NH1 ϕ H1 s t H1 ϕ 1 1. s,t [,1] ϕ 1 u du s t H1 dsdt <.

12 A UNIFOM DIMENSION ESULT FO FACTIONAL MULTIPLICATIVE POCESSES 11 We have proved that almost surely ν is absolutely continuous with respect to the Lebesgue measure. This implies that almost surely for ν-almost every y F 1 [, 1] the following limit 1 1 lim 1 r r { dt F 1t y r} exists and belongs to,, thus yielding a positive finite Borel measure ν y carried by L 1 y = {t [, 1] : F 1 t = y}, defined as 1 gt dν y t = lim r + r 1 { F 1t y r} gt dt, g C[, 1]. Moreover, for any Borel measurable function G : [, 1] + one has Gt, y dν y t dy = Gt, F 1 t dt. y F 1[,1] [,1] Let γ >. By Fatou s lemma and Fubini s theorem we have s t γ dν y sdν y t dy 3.14 = y F 1[,1] y F 1[,1] lim inf r = lim inf r 1 r 1 r [ lim r 1 r y F 1[,1] 1 { ] s t F 1s y r} γ ds dν y t dy 1 { F 1s y r} s t γ dν y t dy ds 1 { F 1s F 1t r} s t γ dtds. Fix s < t 1. ecall 3.12 and the fact that Z 1 w has a bounded density function f 1 with f 1 ϕ 1 1, so E 1 { } Fw = 1 { } F 1s F 1t r B x+ 1 w A 1 w r f1 x dx A 1 w = 1 { } f1 z B 1w dz z r A 1 w A 1 w 2r ϕ 1 1 A 1 w 3.15 = 2 ϕ 1 1 r 2 NH1. Again using Fatou s lemma and Fubini s theorem we get from 3.14 and 3.15 that E s t γ dν y sdν y t dy C s t γ+h1 dsdt, y F 1[,1] where C = 2 ϕ H1. Due to the mass distribution principle we get that for any γ < 1 H 1, almost surely for ν-almost every y F 1 [, 1], dim H L 1 y γ. This gives us the desired lower bound. For the upper bound, we use the fact that almost surely the Hausdorff dimension of the graph of F 1, defined as {t, F 1 t : t [, 1]}, is equal to 2 H 1 Theorem 1.1 in [3]. Then from Corollary 7.12 in [8] we know that there cannot exist a

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14 A UNIFOM DIMENSION ESULT FO FACTIONAL MULTIPLICATIVE POCESSES 13 School of Mathematics, University of Manchester, Oxford oad, Manchester M13 9PL, United Kingdom address: