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Multifractal Analysis for the Sample Paths and Occupation Measure of Brownian Sheet

Multifractal Analysis for the Sample Paths and Occupation Measure of Brownian Sheet. Fujian Normal University Lin Huonan.

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Multifractal Analysis for the Sample Paths and Occupation Measure of Brownian Sheet

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  1. Multifractal Analysis for the Sample Paths and Occupation Measure of Brownian Sheet Fujian Normal University Lin Huonan

  2. The multifractal analysis of stochastic processes has been paid much attention in recent years. Some scholars have started the observation about it. For example: . .

  3. Based on the results of the iterated logarithm law and the uniform modulus of continuity for Brownian motion,Orey and Taylor discussed the multifractal decom- position of one dimensional white noise in 1974. .

  4. Some Results about Multifractal Analysis for the Sample Paths and Occupation Measure of Brownian Motion: . Let B= B(t) ; t > 0 be d- dimensional Brownian motion . . A. d=1 , the iterated logarithm law of Brownian motion: for any fixed t >0, a.s.

  5. Lévy’s uniform modulus of continuity for Brownian motion: 1974, Orey and Taylor determined the dimension of certain exceptional fast points : for any ,

  6. B. d > 3, Ciesielski and Taylor (1962) discussed the limsup asymptotic behavior of the occupation measure of Brownian motion and obtained : for any 0< t < T < +∞ ,

  7. whereqdis the first positive root of the Bessel function Jd/2-2(x) . Based on it , they proved that there exists a positive constant c, such that a.s. φ-m( B(s) ;0< s < t ) = c t, for any t >0 where φ(r) = r2 log log(1/r),and φ-m is the Hausdorff measure of φ(r). Before1987, the best result about the occupation measure of transient Brownian motion is as follows:

  8. there exist two finite constants 0<c1<c2, such that a.s. for any 0<t<T, and any , In 2000, Dembo, Peres, Rosen and Zeitouni proved the following :

  9. and

  10. Remark 1: When a =qd2 / 4 , the Hausdorff dimension of the “thick points’’set described as above is 0, but its packing dimension is 1. Remark 2: The key to the proof of their results as above is the property so called “ a localization phenomenon for Brownian motion in d >2 ’’. But this phenomenon breaks down in d=2.

  11. The Localization Phenomenon for Transient Brownian Motion: There exist two positive constants c1,c2, such that when small enough , and . ----------------------------------------------- ----------------------------------------------------------------------------------------------------------------------------------------------------------------- -----------------------------------------------------------------------------------------------------------------------------------------------------------------

  12. C.d=2, Ray,D. (1963) Based on it , Taylor (1964) proved that there exists a positive constant c, such that φ-m( B(s):0< s < t ) = c t, for any t >0 a.s. where φ(r) = r2 log(1/r)logloglog(1/r),andφ-m is the Hausdorff measure of φ( r). Dembo , Peres, Rosen and Zeitouni (2001):

  13. Some Results about the Sample Paths and Occ-upation Measure of Brownian Sheet: Let W= W(t1,t2) , t1>0, t2 >0 be N=2 parameter Wiener process (or Brownian sheet ) in Rd. A. The iterated logarithm law of Brownian sheet (Orey and Pruitt ,1973 ) : for any fixed s1>0, s2 >0, s=(s1, s2 ), where | E| is the Lebesgue measure of E.

  14. Lévy’s uniform modulus of continuity for Brownian sheet An application of Fubini theorem shows that the random set

  15. has Lebesgue measure zero almost surely. In fact, the set is everywhere dense with power of the continuum a.s. Because of the partially- ordered nature in the multi-parameter index space, the discussion about the multifractal analytic properties of high dimensional white noise is much more complex than that of one parameter, and there exist several forms of the multifractal decomposition of themulti-

  16. parameter sets. Recently, Lin and Huang determined the Hausdorff and packing dim-ension of the “fast point ’’sets of Brownian sheet in three forms of increments as follows: (I) The Hausdorff and packing dimension of “fast point’’ sets determined by the incre-ments in the direction of coordinate: for any α∈ [0,N1/2],a.s.

  17. But only for any α∈ [0,1], a.s. Dim . (II) The Hausdorff and packing dimension of “fast point’’ sets determined by the local increments: for any α∈ [0,N1/2],a.s. dim =N – α2

  18. and for any α∈ [0,1],a.s. Dim =N.

  19. (III) The Hausdorff and packing dimension of “fast point’’ sets determined by the rectangle increments: if a.s. for any T>0, But only for , a.s. for any T>0, Dim N.

  20. B.It is well known that when d > 2N , N-parameter d-dimensional Brownian sheet W is transient . Question :Is there some localization phenomenon for a transient Brownian sheet, just as that for transient Brownian motion? What kind of limit asymptotic properties of the occupation measure of Brownian sheet might exist? By now, the best result about the asymptotic behavior of the occupation measure of transient Brownian sheet is as follows:

  21. where b denotes some positive finite constant , t∈Q, andQ is a cube in R+N with δ(Qt ) >0. (Ehm,W., 1981) The next three theorems(Lin and Wang 2004) partly answer the question above .

  22. Remark 3: Because of the partially-ordered nature in the multi-parameter index space and the distinctness between Brownian motion and Brownian sheet, Theorem A can't determine the accurate constants . Using Theorem A one can obtain the asymptotic laws and multifractal decomposition for the occupation measure of Brownian sheet : . Theorem B. Let W beN-parameter d-dimensional Brownian sheet with d>2N , and Q be a cube in R+N with δ(Qt )>0, then

  23. and Remark 4:Theorem B indicates that is the right scaling function for the occupation measure of transient Brownian sheet . When N=1, Theorem B is just Theorem 1.3 in [2] . Theorem C. Let W and Q be as above. If then

  24. and ifthen Remark 5:

  25. Thank you !

  26. The localization phenomenon for Transient Brownian motion: There exist two positive constant c1,c2, such that when small enough , and . ----------------------------------------------------------------------------------------------------------------------------------------------------------------- ----------------------------------------------- -----------------------------------------------------------------------------------------------------------------------------------------------------------------

  27. Some results about Multifractal Analysis for the sample Paths and Occupation Measure of Brownian motion: Let B = B(t) ; t > 0 be d-dimensional Brownian motion A. d=1 , the iterated logarithm law of Brownian motion: for any fixed t >0, a.s. Lévy’s uniform modulus of continuity for Brownian motion: 1974, Orey and Taylor determined the dimension of certain exceptional fast points : for any ,

  28. Some results about Multifractal Analysis for the sample Paths and Occupation Measure of Brownian motion: Let B = B(t) ; t > 0 be d-dimensional Brownian motion A. d=1 , the iterated logarithm law of Brownian motion: for any fixed t >0, a.s. Lévy’s uniform modulus of continuity for Brownian motion: 1974, Orey and Taylor determined the dimension of certain exceptional fast points : for any ,

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