LOGARITHMIC MULTIFRACTAL SPECTRUM OF STABLE. Department of Mathematics, National Taiwan University. Taipei, TAIWAN. and. S.

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1 LOGARITHMIC MULTIFRACTAL SPECTRUM OF STABLE OCCUPATION MEASURE Narn{Rueih SHIEH Department of Mathematics, National Taiwan University Taipei, TAIWAN and S. James TAYLOR 2 School of Mathematics, University of Sussex Falmer, Brighton, UK ABSTRACT For a stable subordinator Y t of index, 0 < <, the occupation measure (A) = jft 2 [0; ] : Y t 2 Agj is known to have(with probability ) the property that lim inf ln ln r = ; 8x 2 Y [0; ]: In order to obtain an interesting spectrum for the large values of, we consider the set B = fx 2 Y [0; ] : lim sup = g; c r (ln(=r)) where c is a suitable constant. It is shown that B = ; for >, and B 6= ; for 0 ; moreover dimb = DimB = (? ). Running Title: Log-multifractal spectrum AMS Subject Classication(99): 60G7 Key words and phrases: stable subordinator, multifractals, occupation measures. Research supported by a grant NSC 997/8 {25 {M{002{ Research on which this paper is based was done in Taiwan, when this author was supported by Academia Sinica, Taipei, and a grant from National Science Council of Taiwan. 0

2 x. Introduction In this paper, we continue to study the detailed structure of the random probability measure on < dened by( j j denotes the Lebesgue measure) (:) (A) = jft 2 [0; ] : Y t 2 Agj; where Y t is a nice version of a stable subordinator of index, 0 < <. Thus, (A) is the length of time(up to t = ) spent in A by the process Y t. The topological support of is the random set Y [0; ]. In a recent paper by Hu and Taylor(997), they obtained the ordinary multifractal structure of. With probability, for {a.e. x we have (:2) lim ln ln r = ; but there are exceptional sets where (.2) is false. In fact, it was shown in that paper that with probability (:3) lim inf while, if we dene ln ln r = 8x 2 Y [0; ]; C = fx 2 Y [0; ] : lim sup ln ln r g; and D = fx 2 Y [0; ] : lim sup ln ln r = g; then C = ; for > 2, D 6= ; for 2, and in the latter case dimc = dimd = 22? : The purpose of the present paper is to look in more detail at the large values of as r # 0. The result (.3) tells that, in terms of powers of r, the only relevant power is. To obtain an interesting decomposition, we consider the sets A = fx 2 Y [0; ] : lim sup B = fx 2 Y [0; ] : lim sup g; c r (ln(=r)) = g; c r (ln(=r)) and show the existence of a suitable constant c (see Theorem 5.) which ensures that, with probability, A = ; for >, while B 6= ; for 0 : When the sets are non{empty, we compute their Hausdor(and packing) dimensions for each value of. There are several papers in the literature concerning both \fast" and \slow" points for Brownian motions or stable processes. There is a sense in which our exceptional sets A and B can be thought as the \two{sided slow"points for the subordinator Y t, since large values of imply larger than usual rst passage times for Y t+h

3 and Y t?h. The uniform result for \one{sided slow" points of Y t is due to Hawkes(97). however we do not make direct use of his result; instead we include a \two{sided" version of his result as a corollary of our analysis. We believe our results to be true for a general transient stable process of index. However there are independence problems in the construction of the time sets in x4(the construction is crucial for the lower bound of dimensions), so we do not claim to have proved the results beyond the subordinator case. We should remind the reader that, for 0 < =2 our occupation measure can be thought of as the local time at zero of a strictly stable process of index, with = =(? ); in particular, for = =2 the measure is the Brownian local time at 0. Our paper is organized as follows. In x2 we collect denitions and the main probability estimates used in the later proofs. In x3 we obtain the range of for which A is non{empty and calculate upper bounds for dimensions by standard rst moment arguments. In x4 we give a construction of a Cantor{like random time set T for which each point in Y (T ) is \two{sided slow" for Y t. In the nal x5 we bring together the results to complete the analysis. We will use the notations c or c 0 to denote a nite positive constant, whose value is unimportant(perhaps unknown) and may change from place to place. Special constants with known values will be denoted by c ; c 2 ;. The notation x y for two positive numbers x; y means that x?!. y x2. Preliminaries We will only be concerned with an increasing stable process Y t taking values in <. This process has the index satisfying 0 < <. It has stationary independent increments and we normalize it so that the Laplace transform (2:) E expf?u(y t+h? Y t )g = e?hu u > 0 holds for each t 2 < and h > 0. As usual we can assume that Y t is right continuous with left limits everywhere. The process satises the scaling property that, for any constant u > 0, u?= (Y ut? Y 0 ) is another version of Y t? Y 0. Note that we do not tie down Y 0. We dene two associated processes from each xed t 0 by Y t0 (t) = Y t0+t? Y t0 ; Y 2 t0 (t) = Y t0? Y t0?t: Both Yt0 and Y t0 2 are versions of Y, with Yt0 (0) = Y t0 2 (0) = 0, and they are independent. In looking at the local behaviour of Y t near x = Y t0, we will use Y, resp. Y 2, to describe the behaviour of Y t for t > t 0, resp. t < t 0. We will consider for the moment the?nite occupation measure ~ instead of, ~(A) = jft 2 < : Y t 2 Agj: Note that is the restirction of ~ to the interval [Y 0 ; Y ], so that, for 0 < t <, the local behaviour of ~ and are identical at x = Y t. The usuage of ~ rather than is to avoid end eects when 6 [Y 0 ; Y ]. Now, since the path is monotone increasing, for x = Y t0 ~(x; x + r) = inffs : Y t0 +s x + rg; 2

4 hence (2:2) f~(x; x + r) ug = fyt0 (u) rg: Similar relation holds for ~(x? r; x) and Yt0 2. By the scaling property mentioned above, the probability P f~(x; x + r) r g is the same for all r > 0, for each t 0 2 < and > 0. Lemma 2.. If ~ is the?nite occupation measure of a stable subordinator of index, then, for x = Y t0, P f~(x; x + r) r g c? 2() exp (? c 2 ); as " ; where c = [2(? ) 2() ]?=2 ; c 2 = (? ) : Proof. In view of (2.2), the assertion is equivalent to Hawkes(97, Lemma ). 2 Now, ~(x? r; x) and ~(x; x + r) are independent with the same distribution, so that ~ is the sum of two independent random variables for which we know the asymptotics for the large tail. It is ought to be possible to deduce the asymptotics for the large tail of, but we content ourselves with the estimate which we require for the subsequent analysis. Lemma 2.2. If ~ is the?nite occupation measure of a stable subordinator of index, and x = Y t0, then, for a given > 0, there exist and 2 (which depend on ) such that, for all > 0, (i) P f~ r g exp (? c 3 (? ) ); > ; (ii) P f~ r g c 4 (? )? exp (? c3 ); > 2 ; where c 3 = 2 c2 ; c 4 = 2?(+ ) c : Proof. For the rst assertion, we use the inclusion h f~ r g [ k i i= f~(x? r; x) k r g \ f~(x; x + r) k? i k r g]; for large k(the choice of k depends on ). The dominating term in the above is the middle term; we then apply Lemma 2.. For the second assertion, we use the inclusion 2 f~ r g h f~(x? r; x) 2 r g \ f~(x; x + r) 2 r g]: In the formulation of Hawkes(97, Theorem 2), he used the local time of a strictly stable process of index, < 2. This restricts the corresponding subordinator to be of index, 0 < =2. However, his proof is valid for 0 < < indeed. We restate his diplay (6) as 3

5 Lemma 2.3. If is the occupation measure of a stable subordinator of index, then with probability lim sup x2y [0;] (x; x + r) r (ln(=r)) = c 2?; where c 2 is given in Lemma 2.. In our analysis, we want more detailed behaviour about the size of the exceptional B, and will obtain a \two-sided" version of the above Lemma 2.3 as a corollary(see Corollary 5.2). For the moment, we observe that Lemma 2.3 tells us that (r) = r (ln(=r)) is the correct order of the magnitude for the uniform result, as opposed to (r) = r (ln ln(=r)) ; which gives the local upper asymptotic behaviour of, at? a:e: x, and the correct Hausdor measure function for the sample paths, since with probability? my [0; ] = c; see Taylor(986). Our results in this paper have the same avour as those in Orey and Taylor (974) for the \fast" points on a Brownian path. For the relation of our results to multifractal formalism, we refer the reader to Hu and Taylor(997). In the present paper, we are mainly concerned with the set of those x in the range Y [0; ] where the upper asymptotic growth rate of is much larger than (r), the typical rate. The uniform result (.3) implies that, for each > 0, lim = 0; 8x 2 Y [0; ]; r? but we want more precise information, using the measure function (r). Detailed denitions and properties of the dimension indices dened for subset A < d have been described in many papers, including Taylor(986), which is a convenient reference. We use the notations established in that paper, and note that 0 dima DimA (A) for any A <, where dima denotes the Hausdor dimension, DimA the packing dimension, and (A) the upper Minkowski dimension. We will not consider (A ) in the sequel, but note that, since each A is dense in Y [0; ] whenever it is not empty, it follows that (A ) = (Y [0; ]) = then. x3. Upper bound for dimension indices 4

6 For notational convenience we will not work directly with A in xx3,4. We temporarily x > 0; 0 < < ; and a > in the following arguments. We consider the random set E ; = fx 2 Y [0; ] : there exist r n = r n (x; ) # 0; such that (x? r n ; x + r n ) > (? )r n (ln(=r n)) 8ng; and we will prove an exact upper bound for dime ; : We may assume that Y 0 = 0; then it suces to consider x : 0 < x <. For positive integers i; j; k, we set x i;j;k = (i + j=k)a?k ; and I i;j:k = (x i;j;k? ( + =k)a?k ; x i;j;k + ( + =k)a?k ); where k = ; 2; ; i = ; ; [a k ]; j = ; ; k: Let C k be the class of all those I i:j:k such that (3:) (I i;j:k ) > a (? )a?k (ln a k ) : When k is large enough, C k is a cover of E ; \[0; ]. Indeed, whenever a?(k+) < r n < a?k and x satises x i;j?;k < x < x i;j:k (so that x 2 I i;j;k ), then (x? r n ; x + r n ) > (? )r n(ln(=r n )) will imply that I i;j;k has property (3.). Now let N k denote the number of C k, then the expectation of the random variable N k is given by EN k = X i;j P fi i;j;k is hit by the processg P f(3:)g: By Taylor(967, Lemma 4), P i;j P fi i;j;k By Lemma 2.2(i), and ~, Now we impose the assumption is hit by the processg c P [a k ] i=2 k (+=k)a?k (i?)a?k = cka k : P f(3:)g a?k(+(=k))? (3:2) c 3 < : Then, for all a > ; 0 < < ; and > 0,? 2? c3a () a;; :=? c 3 a? (? ) 2? + > : : 5

7 Then, by Chebyshev's inequality(rst moments) we have, for 0, X k P fn k > a k a;; g c X k so that with probability there exists k 0 (!) such that consequently dime ; a;;. Let Then, letting a # ; N k a k a;; 8k k 0 ; E := \ >0 E ; : a?k0 < ; and ; # 0 through countably many positives, we have (3:3) dime? c 3 = (? c 3 ): We remark that the above arguments indeed give the same upper bound estimate for DimE. Now, if we suppose, as opposed to (3.2), that (3:4) c 3 > ; then we can choose a > and > 0 so that a;; < 0(for each xed ). Then, again using a rst moment argument, we can prove that, with probability, there exists k (!) such that the class G k of all those possible I i;j;k which intersect that Y [0; ] is void if k k. The emptiness of G k will imply that E ; = ;. Thus E = ; under (3.4). x4. Lower bound: a Cantor{like set construction We aim to construct a Cantor{like random set T 2 [0; ], where satises (3.2), and for which if x 2 Y (T ), (4:) lim sup : r (ln(=r)) For this purpose, we dene a sequence r n # 0 inductively by (4:2) r = r; 0 < r < ; r n+ = exp (? r? n ): We also set n = 2 r n (ln(=r n)) ; n ~ n = (ln(=r n ) = 2 2 r n(ln(=r n ))?(+) ; = i n ; i = ; 3; 5; ; x i;n = Y ( ); I i;n = [? n ; + n ]; and ~I i;n = [? ~ n ; + ~ n ]: 6

8 We denote the class of all ~ Ii;n by F n. We call I i;n, or equivalently ~ Ii;n, a type{s interval (\S" stands for \slow") for the path Y (!), if the associated Y j (see x2) satisfy Y j ( n )? Y j (~ n ) r n ; j = ; 2; a where a > will be determined later; and in addition, (4:4) Y (~ n ) + Y 2 (~ n ) (? a )r n: Since Y is monotone increasing, condition (4.4) implies that, for each t 2 ~ Ii;n ; x = Y t, so that It follows that A similar argument shows that so we have proved that jx i;n? xj (? a )r n; Y t+n? Y t r n a + (? a )r n = r n : (x; x + r n ) n : (x? r n ; x) n ; (4:5) (x? r n ; x + r n ) 2 n 8x 2 Y ( ~ I i;n ): Now for any xed a >, the distribution of Y (~ n ) + Y 2 (~ n ) is the same as that of Y ti;n +2~ n? Y ti;n, and ~ = n = o(r n ) as n?! : Hence P f(4:4)g?! as n?! : We dene p n = p i;n = P f! : ~ Ii;n is type{s for the path Y (!)g; and obtain a lower bound for p n as follows Lemma 4.. A lower bound for p n is given by p n 2 c 4a?? ( ln( r n ))? ( r n )?c 3a ; for n > n 0 : Proof. By the property of stationary independent increments, p n = P n Y j ( n? ~ n ) < r n a ; j = ; 2o P n (4:4) o = P n (x i;n ; x i;n + r n a ) > n? ~ n o 2 P n (4:4) o = P f(x i;n ; x i;n + r n ) > a ( n? ~ n )g 2 P n (4:4) o P f(x i;n ; x i;n + r n ) > a n g 2 P n (4:4) o : 7

9 The estimate in Lemma 2. completes the proof. 2 Remark. Similar arguments yield p n ( r n )?c 3()a ; for n n ; where 0 < < and a < a. For each ~ I 2 F n, let M n+ ( ~ I) denote the number of those type{s intervals from F n+ which are contained in ~ I. Note that M n+ ( ~ I) is a binomial random variable. Set h j I U n := ~ 0;n? j i 8jI 0;n j p n ; where [] denotes the greatest integer part. Lemma 4.2. Assume that (3.2) holds, namely c 3 =() <. Then with probability there exists n 2 = n 2 (!) such that M n+ (~ I) Un+ 8~ I 2 Fn 8n n 2 : Proof. The denition of U n+ and the fact that M n+ ( I) ~ is binomially distributed together assert that EM n+ ( I) ~ 2U n+ and V arm n+ ( I) ~ cu n+. We remark that I i;n+ and I j;n+ are disjoint time intervals when i 6= j so that the behaviour of Y t on these intervals is independent and M n+ (~ I) counts independent events. By the Chebyshev's inequality(second moments) we have X n c X n c X n P fm n+ (~ I) < Un+ for some ~ I 2 Fn g X V arm n+(~ I) ~I2F n (EM n+ (~ I)) 2? n j ~ I 0;n+ jji 0;n j? p? n+: By Lemma 4., the denition of ~ Ii;n, and dening relations between n+ ; r n+ ; r n ; we can see that whether the series in the above display converges depends on r?c 3a n+ : Under the assumption (3.2) we can nd a suitable a > so that the power in the above is positive. Then we can apply Borel{Cantelli Lemma to obtain the assertion. 2 We also note that U n+ " as n ". Now we construct a Cantor{like random T, under the assumption (3.2), in a procedure adapted from Hu and Taylor(997, x5). We start with F n2, where n 2 is the index in Lemma 4.2. For each ~ I 2 F n2 we pick up U n2 + type{s intervals from F n2 + which are contained in ~ I. Then we form a union F n2 +. Next, for each member ~ I0 in this F n2 + we pick U n2 +2 type{s intervals from F n2 +2 which 8

10 are contained in ~ I0 ; then we form a second union F n2 +2. We proceed inductively and have a chain F n2 + F n2 +2, then we set T = \ n>n2 F n : For each x 2 Y (T ), (4.5) shows that (4.) holds. We can also construct a Borel measure supported by T by a procedure adapted from Hu and Taylor(997, x5) too. We dene to be a multiple of Lebesgue measure on F n2 + such that (F n2 +) = and each member in F n2 + has the equal?mass. Then we dene 2 from, keeping the total 2?mass on F n2 +2 to be and each member in F n2 +2 to have equal 2?mass too. The limiting mesaure (I) = lim i! (I) is a Borel measure supported by T, and ( I) ~ = j ( I) ~ whenever I ~ is a member in F n2 +i and j i. Under (3.2), the index := c 3 < : Let the measure function h(s) be dened by h(s) = s ( ln(=s)) b ; wher b > will be determined later. We will prove that with probability the Hausdor measure (4:6) h? m(t ) = : We prove this assertion by showing that the energy integral I := Z Z T T is nite a.s.. For this purpose, we set where (ds)(dt) js? tj ( ln =(s? t)) b A = [ n>n2 [ mn j= G j;n ; G j;n = f(s; t) 2 T T : 2 j n < js? tj 2 j+ n g; and 2 mn n n? 2 mn+ n : In the above energy integral, it suces to consider the integration over A. For each ~I 2 F n, the expected number of type{s J ~ 2 F n satisfying 2 j n < js? tj 2 j+ n for all s 2 I; ~ t 2 J ~ is at most [2 j p n ] +. This means that if K n is the number of all I ~ i;n in F n ; n > n 2 then the expected number of squares I ~ J, ~ with both I; ~ J ~ type{s intervals in F n, needed to cover G j;n is not more than K n ([2 j+ p 0;n ] + ). Hence, E Z Z G j;n (ds)(dt) js? tj ( ln =(s? t)) b 9 c K 2 n K n 2 j+ p n (2 j n ) ( ln(2 j n )? ) b :

11 We observe that Moreover Therefore, (4:7) E Z Z Xm n j= [ mn j= G n;j 2 j X mn ( ln n?? j) b j= c n? 2 2j m =2 X n j= n ln? =2 ( ln n?? j) 2b n? (2b)=2 : K n U n c n? ( ln(=r n? ))?2? n p n : (ds)(dt) js? tj ( ln =(s? t)) b c? n?( ln? n? ) (2b)=2 ( ln(=r n? )) 2 : Using the dening relations between n ; n? ; r n and r n?, we see that the right hand side of (4.7) is 2b? 2() cr (?) n? rn? ( ln(=r n? )) 2?()() : We can choose b > large enough so that the power of r n? in the above display is positive, say > 0. Then, for a xed 0 < Z Z (ds)(dt) X E A js? tj ( ln =(s? t)) c r 0 b n? < : Consequently I () < a.s., and thus we have proved (4.6). In view of (4.6) and Perkins and Taylor(987, Theorem 3.), we have Lemma 4.3. Suppose that T is the Cantor{like random set constructed above, under the assumption (3.2). Dene the measure function n>n ; (s) = s () ( ln(=s)) ()()+b ; where is dened above and b is large enough(determined as above) and xed. Then, with probability, ;? m(y (T ) = : x5. Conclusion: the log{multifractal spectrum We are ready to state and prove the conclusion of our analysis in this section. Let Y t be a stable subordinator of index, 0 < <. Let be the occupation measure of Y. Dene A = n x 2 Y [0; ] : lim sup B = n x 2 Y [0; ] : lim sup c r (ln(=r)) o ; c r (ln(=r)) = o ; where c = 2 (? )?: We note that Y (T ) A since x 2 Y (T ) implies (4.5), for all large n. 0

12 Theorem 5.. If is the occupation measure of a stable subordinator of index, 0 < <, then, with probability, (i) A = ;, if >. (ii) A and B are non{empty, if 0 ; moreover dima = dimb = DimA = DimB = (? ): Proof. In xx3,4 we work under the assumption (3.2) and its opposite (3.4), which are equivalent to, respectively, := ( c 3 ) < and >. Moreover, \ " is simply equivalent to \ ( c 3 )? "; the latter one is just the content for A, by denition of c 3 in x2. Thus the results which we have obtained in xx3,4 prove the assertions for A, except for the critical cases = 0;. As for B ; 0 < <, we note that B = A n [ n= A +=n : Let the mesaure function ; (s) now be dened by ; (s) = s ( ) ( ln(=s)) ()( )+b ; where b > is large enough and xed. Then Lemma 4.3(change the index there) implies that, with probability, ;? m(a ) = while ;? m(a +=n ) = 0 for all n, by (3.3). Thus ;? m(b ) = too, which again implies that dimb (? ). Finally the cases = 0; can be treated as follows. For =0, we simply note that dimy [0; ] = DimY [0; ]=; see Taylor(986). For the other extreme, we go back to the critical case in (3.2) of x3, namely c 3 =. In this case, the construction of T in x4 has to be modied. In the denition of type{s intervals, we require that, instead of some xed a >, Y j ( n )? Y j (~ n ) r n ; j = ; 2; a n where a n = + =n. The key display (4.4) holds, with a being replaced by a n. Because r n # 0 very fast, we can still prove (4.5). We have a corresponding result to Lemma 4. and U n+ " too. To construct T ;, in the present case, we pick up all possible type{s intervals from F n+ which are contained in each type{s ~ I 2 Fn, n = ; 2; : We thus form a chain of unions F F 2. Then T = \ n= F n is non{empty since all F n are compact and non{empty(in this case, dimt = 0). 2 Corollary 5.2. With probability, lim sup x2y [0;] r (ln(=r)) = c : Proof. This follows immediately from A = ; for >, while A = B 6= ;; both statements holding with probability. 2

13 References [H] J. Hawkes(97). A lower Lipchitz condition for the stable subordinator. Z. Wahr. verw. Geb. 7 23{32. [HT] Xiaoyu Hu and S. J. Taylor(997). The multifractal structure of stable occupation measure. Stochastic Proc. Appl {299. [OT] S. Orey and S. James Taylor(974). How often on a Brownian path does the law of iterated logarithm fail? Proc. London Math. Soc. 3 74{92. [PT] E. A. Perkins and S. J. Taylor(987). Uniform measure results for the image of subsets under Brownin motion. Probab. Th. Rel. Fields {289. [T] S. J. Taylor(967). Sample path properties of a transient stable processes. J. Math. Mech {246. [T2] S. J. Taylor(986). The measure theory of random fractals. Math. Proc. Camb. Phi. Soc {406. 2

Part V. 17 Introduction: What are measures and why measurable sets. Lebesgue Integration Theory

Part V. 17 Introduction: What are measures and why measurable sets. Lebesgue Integration Theory Part V 7 Introduction: What are measures and why measurable sets Lebesgue Integration Theory Definition 7. (Preliminary). A measure on a set is a function :2 [ ] such that. () = 2. If { } = is a finite