Joint continuity of the local times of fractional brownian sheets

Ayache, Antoine; Wu, Dongsheng; Xiao, Yimin

Annales de l'I.H.P. Probabilités et statistiques, Tome 44 (2008) no. 4, pp. 727-748.

Désignons par $B^H=\left\{B^H\left(t\right),t\in {ℝ}_{+}^N\right\}$ le $(N,d)$-drap Brownien fractionnaire de paramètre $H=(H_1,...,H_N)\in {(0,1)}^N$ défini par $B^H\left(t\right)=(B_1^H\left(t\right),...,B_d^H\left(t\right))(t\in {ℝ}_{+}^N)$, où $B_1^H,...,B_d^H$ sont des copies indépendantes du drap Brownien fractionnaire à valeurs réelles $B_0^H$. Nous montrons que le temps local de $B^H$ est bicontinu lorsque $d<{\sum }_{\ell =1}^N{H}_{\ell }^{-1}$. Cela résout une conjecture de Xiao et Zhang (Probab. Theory Related Fields 124 (2002)). Nous obtenons aussi des résultats fins concernant la régularité Hölderienne, locale et globale, du temps local. Ces résultats nous permettent d’étudier certaines propriétés analytiques et géométriques des trajectoires de $B^H$.

Let $B^H=\left\{B^H\left(t\right),t\in {ℝ}_{+}^N\right\}$ be an $(N,d)$-fractional brownian sheet with index $H=(H_1,...,H_N)\in {(0,1)}^N$ defined by $B^H\left(t\right)=(B_1^H\left(t\right),...,B_d^H\left(t\right))(t\in {ℝ}_{+}^N)$, where $B_1^H,...,B_d^H$ are independent copies of a real-valued fractional brownian sheet $B_0^H$. We prove that if $d<{\sum }_{\ell =1}^N{H}_{\ell }^{-1}$, then the local times of $B^H$ are jointly continuous. This verifies a conjecture of Xiao and Zhang (Probab. Theory Related Fields 124 (2002)). We also establish sharp local and global Hölder conditions for the local times of $B^H$. These results are applied to study analytic and geometric properties of the sample paths of $B^H$.

[1] R. J. Adler. The Geometry of Random Fields. Wiley, New York, 1981. | MR | Zbl

[2] A. Ayache, S. Leger and M. Pontier. Drap Brownien fractionnaire. Potential Anal. 17 (2002) 31-43. | MR | Zbl

[3] A. Ayache and Y. Xiao. Asymptotic properties and Hausdorff dimension of fractional Brownian sheets. J. Fourier Anal. Appl. 11 (2005) 407-439. | MR | Zbl

[4] D. A. Benson, M. M. Meerschaert and B. Baeumer. Aquifer operator-scaling and the effect on solute mixing and dispersion. Water Resour. Res. 42 (2006) W01415.

[5] S. M. Berman. Local times and sample function properties of stationary Gaussian processes. Trans. Amer. Math. Soc. 137 (1969) 277-299. | MR | Zbl

[6] S. M. Berman. Gaussian sample function: uniform dimension and Hölder conditions nowhere. Nagoya Math. J. 46 (1972) 63-86. | MR | Zbl

[7] S. M. Berman. Local nondeterminism and local times of Gaussian processes. Indiana Univ. Math. J. 23 (1973) 69-94. | MR | Zbl

[8] A. Bonami and A. Estrade. Anisotropic analysis of some Gaussian models. J. Fourier Anal. Appl. 9 (2003) 215-236. | MR | Zbl

[9] J. Cuzick and J. Dupreez. Joint continuity of Gaussian local times. Ann. Probab. 10 (1982) 810-817. | MR | Zbl

[10] M. Dozzi. Occupation density and sample path properties of N-parameter processes. Topics in Spatial Stochastic Processes (Martina Franca, 2001) 127-166. Lecture Notes in Math. 1802. Springer, Berlin, 2002. | MR | Zbl

[11] T. Dunker. Estimates for the small ball probabilities of the fractional Brownian sheet. J. Theoret. Probab. 13 (2000) 357-382. | MR | Zbl

[12] W. Ehm. Sample function properties of multi-parameter stable processes. Z. Wahrsch. Verw Gebiete 56 (1981) 195-228. | MR | Zbl

[13] D. Geman and J. Horowitz. Occupation densities. Ann. Probab. 8 (1980) 1-67. | MR | Zbl

[14] D. Geman, J. Horowitz and J. Rosen. A local time analysis of intersections of Brownian paths in the plane. Ann. Probab. 12 (1984) 86-107. | MR | Zbl

[15] G. H. Hardy. Inequalities. Cambridge Univ. Press, 1934. | JFM | Zbl

[16] H. Kesten. An iterated logarithm law for local time. Duke Math. J. 32 (1965) 447-456. | MR | Zbl

[17] D. Khoshnevisan. Multiparameter Processes: An Introduction to Random Fields. Springer, New York, 2002. | MR | Zbl

[18] D. Khoshnevisan, D. Wu and Y. Xiao. Sectorial local non-determinism and the geometry of the Brownian sheet. Electron. J. Probab. 11 (2006) 817-843. | MR | Zbl

[19] D. Khoshnevisan and Y. Xiao. Images of the Brownian sheet. Trans. Amer. Math. Soc. 359 (2007) 3125-3151. | MR | Zbl

[20] D. Khoshnevisan, Y. Xiao and Y. Zhong. Local times of additive Lévy processes. Stoch. Process. Appl. 104 (2003) 193-216. | MR | Zbl

[21] D. M. Mason and Z. Shi. Small deviations for some multi-parameter Gaussian processes. J. Theoret. Probab. 14 (2001) 213-239. | MR | Zbl

[22] T. S. Mountford. A relation between Hausdorff dimension and a condition on time sets for the image by the Brownian sheet to possess interior-points. Bull. London Math. Soc. 21 (1989) 179-185. | MR | Zbl

[23] T. S. Mountford and D. Baraka. A law of the iterated logarithm for fractional Brownian motions. Preprint, 2005. | Zbl

[24] B. Øksendal and T. Zhang. Multiparameter fractional Brownian motion and quasi-linear stochastic partial differential equations. Stochastics Stochastics Rep. 71 (2000) 141-163. | MR | Zbl

[25] L. D. Pitt. Local times for Gaussian vector fields. Indiana Univ. Math. J. 27 (1978) 309-330. | MR | Zbl

[26] C. A. Rogers and S. J. Taylor. Functions continuous and singular with respect to a Hausdorff measure. Mathematika 8 (1961) 1-31. | MR | Zbl

[27] J. Rosen. Self-intersections of random fields. Ann. Probab. 12 (1984) 108-119. | MR | Zbl

[28] M. Talagrand. Hausdorff measure of trajectories of multiparameter fractional Brownian motion. Ann. Probab. 23 (1995) 767-775. | MR | Zbl

[29] D. Wu and Y. Xiao. Geometric properties of fractional Brownian sheets. J. Fourier Anal. Appl. 13 (2007) 1-37. | MR | Zbl

[30] Y. Xiao. Hölder conditions for the local times and the Hausdorff measure of the level sets of Gaussian random fields. Probab. Theory Related Fields 109 (1997) 129-157. | MR | Zbl

[31] Y. Xiao. Properties of local nondeterminism of Gaussian and stable random fields and their applications. Ann. Fac. Sci. Toulouse Math. XV (2006) 157-193. | Numdam | MR | Zbl

[32] Y. Xiao. Sample path properties of anisotropic Gaussian random fields. Submitted, 2007. | Zbl

[33] Y. Xiao and T. Zhang. Local times of fractional Brownian sheets. Probab. Theory Related Fields 124 (2002) 204-226. | MR | Zbl