Critical Brownian sheet does not have double points. (English) Zbl 1269.60053

Dalang, Robert C.; Khoshnevisan, Davar; Nualart, Eulalia; Wu, Dongsheng; Xiao, Yimin

Ann. Probab. 40, No. 4, 1829-1859 (2012).

Let \[ T = \{ ( s , t ) ; \in ( 0 , \infty ) ^{2 N } , s _{i } \neq t _{i } , i = 1 , \dotsc , N \}, \] let \(B\) be a \(N\)-parameter \(\mathbb{R}^{d }\)-Brownian sheet, \(\emptyset \neq A \subset \mathbb{R}^{d }\), \(W _{1 }\), \(W _{2 }\) other, independent, \(N\)-parameter \(\mathbb{R}^{d }\) Brownian sheets. The main result states that \[ \operatorname{P}(\text{exists } ( u _{1 } , u _{2 } ) \in T , B ( u _{1 } ) = B ( u _{2 } ) \in A ) > 0 \] is equivalent to \[ \operatorname{P} (\text{exists}\; (u_{1},u_{2})\in T , W _{1 } ( u _{1 } ) = W _{2 } ( u _{2 } ) \in A ) > 0. \] A tool used are “pinned sheets” \(B _{s } ( t ) = B(t)-E ( B ( t ) ; B ( s ) )\).
This yields the following corollaries.
1. If \[ M _{2 } = \{ x : \text{exists } s \neq t \text{ in }( 0 , \infty )^{N }\text{ with } B ( s ) = B(t)=x\} \] (the set of double points of \(B\)), then \(\operatorname{P} ( M _{2 } \cap A \neq \emptyset ) > 0\) holds for all \(A \neq \emptyset\) if \(d<2N\), is equivalent to \(\operatorname{Cap}_{2 ( d - 2 N ) } ( A ) > 0\) if \(d > 2 N\) and, if \(d = 2 N\), is equivalent to the existence of a probability \(\mu\) with compact support \(\subset A\) and \[ \int \int |\log _{+ } ( | x-y | ^{- 1 } ) | ^{2 } d \mu ( x ) d \mu ( y ) < \infty. \] \(\operatorname{Cap}_{\beta } ( A ) ^{- 1 }\) is defined as \[ \inf \int \int k _{\beta } ( x-y ) d \mu ( x ) d \mu ( y ) \] over all probabilities \(\mu\) with compact support \(\subset A\), \(k _{\beta } ( x ) = \| x \| ^{- \beta }\) for \(\beta > 0\), \(\log _{+ } ( \| x \| ^{- 1 } )\) for \(\beta = 0\), \(1\) for \(\beta < 0\).
2. \(B\) has double points if and only if \(d < 4 N\). \(M _{2 }\) has a.s. positive Lebesgue measure if and only if \(d < 2 N\).
3. Define \(s < _{\pi } t\) for \(s , t \in \mathbb{R}^{N }\), \(\pi \subset \{ 1 , \dotsc , n \} \) as \(s _{i } \leq t _{i }\) for \(i \in \pi\) and \(s _{i } \geq t _{i }\) for \(i \notin \pi\), \(s < < _{\pi } t\) if \(s < _{\pi } t\) and \(s _{i } \neq t _{i }\) for all \(i\), \(M _{k }^{ \sim }\) as the set of all \(x \in \mathbb{R}^{d }\) with \(B ( s _{1 } ) = \dotsb = B ( s _{k } ) = x\) for some \(\pi\) and some \(s _{1 } < < _{\pi } \dotsb < < _{\pi } s _{k }\). Then \(P ( M _{k } ^{ \sim } \cap A \neq \emptyset ) > 0\) if and only if \(\operatorname{Cap} _{k ( d - 2 N )} ( A ) > 0\).
4. If the Hausdorff dimension \(\dim _{H } A\) of \(A\) is \(< 2 ( d-2 N )\), then, a.s., \(A\) does not contain double points; in the contrary case, \(\| \dim _{H } ( M _{2 } \cap A ) \| _{\infty } = \dim _{H } A-2 ( d-2 N )\).
The main technical result is as follows.
If \[ \Theta = \prod _{j = 1 }^{N } [ a _{j }, b _{j }] \subset ( 0 , \infty ) ^{N } \] and if \[ \pi \subset \{ 1 , \dotsc , N \},\;s \in ( 0 , \infty ) ^{N } \backslash \Theta,\;s < _{\pi } t \] for all \[ t \in \Theta,\;{\mathcal F} _{\pi} ( s ) = {\mathcal B} ( B ( u ) ; u < _{\pi } s ), \] then, if \(A\) is a bounded random set, with \({\mathcal F} _{\pi } ( s ) \otimes {\mathcal B} (\mathbb{R}^{d })\) measurable \(\chi _{A ( \omega ) } ( x )\) in \(( \omega , x )\), the random sets \((\operatorname{Cap}_{d - 2 N } ( A ) > 0 )\) and the set on which the expectation \(\operatorname{E} (\cdot ; {\mathcal F} _{\pi } ( s ))\) of “there exists \(u\in \Theta\) with \(B ( u ) \in A \)” is non-negative differ by a null set, a.s. A stochastic variant of the definition of \(\operatorname{Cap}_{\beta } A\) (for fixed \(\beta\)) is proved.

Reviewer: Ion Cuculescu (Bucureşti)

MSC:

60G60	Random fields
60G15	Gaussian processes
60J45	Probabilistic potential theory

Keywords:

Brownian sheet; multiple points; capacity; Hausdorff dimension; pinned sheet; conditional potential theory

References:

[1]	Biermé, H., Lacaux, C. and Xiao, Y. (2009). Hitting probabilities and the Hausdorff dimension of the inverse images of anisotropic Gaussian random fields. Bull. Lond. Math. Soc. 41 253-273. · Zbl 1165.60022 · doi:10.1112/blms/bdn122
[2]	Cairoli, R. and Walsh, J. B. (1975). Stochastic integrals in the plane. Acta Math. 134 111-183. · Zbl 0334.60026 · doi:10.1007/BF02392100
[3]	Čencov, N. N. (1956). Wiener random fields depending on several parameters. Dokl. Akad. Nauk SSSR ( N.S. ) 106 607-609.
[4]	Chen, X. (1994). Hausdorff dimension of multiple points of the \((N,d)\) Wiener process. Indiana Univ. Math. J. 43 55-60. · Zbl 0797.60065 · doi:10.1512/iumj.1994.43.43003
[5]	Dalang, R. C., Khoshnevisan, D. and Nualart, E. (2007). Hitting probabilities for systems of non-linear stochastic heat equations with additive noise. ALEA Lat. Am. J. Probab. Math. Stat. 3 231-271. · Zbl 1170.60322
[6]	Dalang, R. C. and Nualart, E. (2004). Potential theory for hyperbolic SPDEs. Ann. Probab. 32 2099-2148. · Zbl 1054.60066 · doi:10.1214/009117904000000685
[7]	de la Peña, V. H. and Giné, E. (1999). Decoupling : From Dependence to Independence. Randomly Stopped Processes. \(U\)-statistics and Processes. Martingales and Beyond . Springer, New York. · Zbl 0918.60021
[8]	Dvoretzky, A., Erdös, P. and Kakutani, S. (1950). Double points of paths of Brownian motion in \(n\)-space. Acta Sci. Math. Szeged 12 75-81. · Zbl 0036.09001
[9]	Dvoretzky, A., Erdös, P. and Kakutani, S. (1954). Multiple points of paths of Brownian motion in the plane. Bull. Res. Council Israel 3 364-371.
[10]	Dvoretzky, A., Erdős, P., Kakutani, S. and Taylor, S. J. (1957). Triple points of Brownian paths in 3-space. Proc. Cambridge Philos. Soc. 53 856-862. · Zbl 0208.44103 · doi:10.1017/S0305004100032989
[11]	Kakutani, S. (1944). On Brownian motions in \(n\)-space. Proc. Imp. Acad. Tokyo 20 648-652. · Zbl 0063.03106 · doi:10.3792/pia/1195572742
[12]	Khoshnevisan, D. (1997). Some polar sets for the Brownian sheet. In Séminaire de Probabilités XXXI. Lecture Notes in Math. 1655 190-197. Springer, Berlin. · Zbl 0886.60039
[13]	Khoshnevisan, D. (2002). Multiparameter Processes : An Introduction to Random Fields . Springer, New York. · Zbl 1005.60005
[14]	Khoshnevisan, D. and Shi, Z. (1999). Brownian sheet and capacity. Ann. Probab. 27 1135-1159. · Zbl 0962.60066 · doi:10.1214/aop/1022677442
[15]	Khoshnevisan, D., Wu, D. and Xiao, Y. (2006). Sectorial local non-determinism and the geometry of the Brownian sheet. Electron. J. Probab. 11 817-843 (electronic). · Zbl 1111.60020
[16]	Khoshnevisan, D. and Xiao, Y. (2007). Images of the Brownian sheet. Trans. Amer. Math. Soc. 359 3125-3151 (electronic). · Zbl 1124.60037 · doi:10.1090/S0002-9947-07-04073-1
[17]	Lévy, P. (1940). Le mouvement brownien plan. Amer. J. Math. 62 487-550. · Zbl 0024.13906 · doi:10.2307/2371467
[18]	Mountford, T. S. (1989). Uniform dimension results for the Brownian sheet. Ann. Probab. 17 1454-1462. · Zbl 0695.60077 · doi:10.1214/aop/1176991165
[19]	Nualart, D. (2006). The Malliavin Calculus and Related Topics , 2nd ed. Springer, Berlin. · Zbl 1099.60003
[20]	Orey, S. and Pruitt, W. E. (1973). Sample functions of the \(N\)-parameter Wiener process. Ann. Probab. 1 138-163. · Zbl 0284.60036 · doi:10.1214/aop/1176997030
[21]	Peres, Y. (1996). Intersection-equivalence of Brownian paths and certain branching processes. Comm. Math. Phys. 177 417-434. · Zbl 0851.60080 · doi:10.1007/BF02101900
[22]	Peres, Y. (1999). Probability on trees: An introductory climb. In Lectures on Probability Theory and Statistics ( Saint-Flour , 1997). Lecture Notes in Math. 1717 193-280. Springer, Berlin. · Zbl 0957.60001 · doi:10.1007/978-3-540-48115-7_3
[23]	Rosen, J. (1984). Self-intersections of random fields. Ann. Probab. 12 108-119. · Zbl 0536.60066 · doi:10.1214/aop/1176993376
[24]	Ville, J. (1942). Sur un problème de géométrie suggéré par l’étude du mouvement brownien. C. R. Acad. Sci. Paris 215 51-52. · Zbl 0027.41101
[25]	Wiener, N. (1923). Differerential space. J. Math. Phys. 2 131-174.
[26]	Wiener, N. (1938). The homogeneous chaos. Amer. J. Math. 60 897-936. · Zbl 0019.35406 · doi:10.2307/2371268
[27]	Xiao, Y. (1999). Hitting probabilities and polar sets for fractional Brownian motion. Stochastics Stochastics Rep. 66 121-151. · Zbl 0928.60027
[28]	Xiao, Y. (2009). Sample path properties of anisotropic Gaussian random fields. In A Minicourse on Stochastic Partial Differential Equations. Lecture Notes in Math. 1962 (D. Khoshnevisan and F. Rassoul-Agha, eds.) 145-212. Springer, Berlin. · doi:10.1007/978-3-540-85994-9_5