Eccentric behaviors of the Brownian sheet along lines.(English)Zbl 1019.60047

The Brownian sheet is a centered, continuous Gaussian random field $$W(t),$$ $$t\in \mathbb R^2$$, indexed by the positive quadrant in the plane, with covariance given by $$E W(s_1,s_2)W(t_1,t_2)=(s_1\wedge t_1) (s_2\wedge t_2)$$. It is one of the natural extensions of Brownian motion to higher dimensional time. Distinct excursion intervals of a Brownian motion (that correspond to fixed level) have no common endpoints. What is the situation for distinct excursion sets of Brownian bubbles in the literature, and this interesting paper examines how bubbles from fixed or random levels come into contact with each other, by examining whether or not the Brownian sheet restricted to a specific type curve can have a point of increase. The Brownian bubbles behavior was also studied in the authors’ previous works.

MSC:

 60G60 Random fields 28A80 Fractals 60G15 Gaussian processes
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 [1] BAÑUELOS, R. and SMITS, R. G. (1997). Brownian motion in cones. Probab. Theory Related Fields 108 299-319. · Zbl 0884.60037 [2] BURDZY, K. (1990). On nonincrease of Brownian motion. Ann. Probab. 18 978-980. · Zbl 0719.60086 [3] DALANG, R. C. and MOUNTFORD, T. (1996). Non-differentiability of curves on the Brownian sheet. Ann. Probab. 24 182-195. · Zbl 0861.60058 [4] DALANG, R. C. and MOUNTFORD, T. (1997). Points of increase of the Brownian sheet. Probab. Theory Related Fields 108 1-27. · Zbl 0881.60047 [5] DALANG, R. C. and WALSH, J. B. (1993). Geography of the level sets of the Brownian sheet. Probab. Theory Related Fields 96 153-176. · Zbl 0792.60038 [6] DEBLASSIE, R. D. (1987). Exit times from cones in Rn of Brownian motion. Probab. Theory Related Fields 74 1-29. · Zbl 0586.60077 [7] DVORETSKY, A., ERD OS, P. and KAKUTANI, S. (1961). Non-increase everywhere of the Brownian motion process. Proc. Fourth Berkeley Symp. Math. Statist. Probab. 2 103-116. Univ. California Press, Berkeley. [8] ITÔ, K. and MCKEAN JR., H. P. (1965). Diffusion Processes and Their Sample Paths. Springer, Berlin. · Zbl 0127.09503 [9] KAHANE, J. P. (1985). Some Random Series of Functions, 2nd ed. Cambridge Univ. Press. · Zbl 0571.60002 [10] KHOSHNEVISAN, D. (1995). On the distribution of bubbles on the Brownian sheet. Ann. Probab. 23 786-805. · Zbl 0833.60044 [11] KNIGHT, F. B. (1981). Essentials of Brownian Motion and Diffusion. Amer. Math. Soc., Providence, RI. · Zbl 0458.60002 [12] LANDKOF, N. S. (1972) Foundations of Modern Potential Theory. Springer, Berlin. · Zbl 0253.31001 [13] MEYER, P. A. (1982). Note sur les processus d’Ornstein-Uhlenbeck. Séminaire de Probabilités XVI. Lecture Notes in Math. 920 95-133. Springer, Berlin. · Zbl 0481.60041 [14] MOTOO, M. (1958). Proof of the law of the iterated logarithm through diffusion equation. Ann. Inst. Statist. Math. 10 21-28. · Zbl 0084.35801 [15] MOUNTFORD, T. S. (1990). Double points and the Ornstein-Uhlenbeck process on Wiener space. Illinois J. Math. 34 38-48. · Zbl 0688.60032 [16] MOUNTFORD, T. S. (1993). Estimates of the Hausdorff dimension of the boundary of positive Brownian sheet components. Séminaire de Probabilités XXVII. Lecture Notes in Math. 1557 233-255. Springer, Berlin. · Zbl 0786.60106 [17] OREY, S. and PRUITT, W. E. (1973). Sample functions of the N-parameter Wiener processes. Ann. Probab. 1 138-163. · Zbl 0284.60036 [18] PERKINS, E. (1981). The exact Hausdorff measure of the level sets of Brownian motion.Wahrsch. Verw. Gebiete 58 373-388. · Zbl 0458.60076 [19] REVUZ, D. and YOR, M. (1991). Continuous Martingales and Brownian Motion. Springer, Berlin. · Zbl 0731.60002 [20] ROGERS, L. C. G. and WILLIAMS, D. (1987). Diffusions, Markov Processes and Martingales 2: Itô Calculus. Wiley, New York. · Zbl 0977.60005 [21] SPITZER, F. (1958). Some theorems concerning two-dimensional Brownian motion. Trans. Amer. Math. Soc. 87 187-197. JSTOR: · Zbl 0089.13601 [22] TAYLOR, S. J. and WENDELL, J. G. (1966). The exact Hausdorff measure of the zero set of a stable process.Wahrsch. Verw. Gebiete 6 170-180. · Zbl 0178.52702 [23] WALSH, J. B. (1986). An Introduction to Stochastic Partial Differential Equations. École de Probabilités de St. Flour XIV. Lecture Notes in Math 1180. Springer, Berlin · Zbl 0608.60060
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