论文信息 - THE UNIFORM DIMENSION OF THE LEVEL SETS OF A BROWNIAN SHEET

THE UNIFORM DIMENSION OF THE LEVEL SETS OF A BROWNIAN SHEET

Let WN(t) denote the N-parameter Brownian sheet (Wiener process) taking values in R'. For 0 < T < 1, set A(T) = {teRN: 0 < ti ? T, i = 1, * * *, N} and let E(x, T) = {t e A(T): WN(t) = x}, the set of t where the process is at the level x. Then we show that, with probability one, the Hausdorff dimension of E(x, T) equals N I for all 0 < T ? 1 and every x in the interior of the range of WN(t), t e A(T). This provides an answer to a question raised earlier by Pyke.

Robert J. Adler | R. Adler

[1] harald Cramer,et al. Stationary And Related Stochastic Processes , 1967 .

[2] Robert J. Adler,et al. Hausdorff Dimension and Gaussian Fields , 1977 .

[3] Steven Orey,et al. Sample Functions of the $N$-Parameter Wiener Process , 1973 .

[4] John B. Walsh,et al. Stochastic integrals in the plane , 1975 .

[5] S. Berman. Gaussian Sample Functions: Uniform Dimension and Hölder Conditions Nowhere , 1972, Nagoya Mathematical Journal.

[6] S. Berman. Gaussian Processes with Stationary Increments: Local Times and Sample Function Properties , 1970 .

[7] D. Ray. Sojourn times of diffusion processes , 1963 .

[8] L. Tran. On a problem posed by Orey and Pruitt related to the range of the N-parameter wiener process in Rd , 1976 .