Critical Brownian sheet does not have double points
Let NEWLINE\[NEWLINET = \{ ( s , t ) ; \in ( 0 , \infty ) ^{2 N } , s _{i } \neq t _{i } , i = 1 , \dotsc , N \},NEWLINE\]NEWLINE 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 NEWLINE\[NEWLINE\operatorname{P}(\text{exists } ( u _{1 } , u _{2 } ) \in T , B ( u _{1 } ) = B ( u _{2 } ) \in A ) > 0NEWLINE\]NEWLINE is equivalent to NEWLINE\[NEWLINE\operatorname{P} (\text{exists}\; (u_{1},u_{2})\in T , W _{1 } ( u _{1 } ) = W _{2 } ( u _{2 } ) \in A ) > 0.NEWLINE\]NEWLINE A tool used are ``pinned sheets'' \(B _{s } ( t ) = B(t)-E ( B ( t ) ; B ( s ) )\).NEWLINENEWLINE This yields the following corollaries.NEWLINENEWLINE1. If NEWLINE\[NEWLINEM _{2 } = \{ x : \text{exists } s \neq t \text{ in }( 0 , \infty )^{N }\text{ with } B ( s ) = B(t)=x\} NEWLINE\]NEWLINE (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 NEWLINE\[NEWLINE\int \int |\log _{+ } ( | x-y | ^{- 1 } ) | ^{2 } d \mu ( x ) d \mu ( y ) < \infty.NEWLINE\]NEWLINE \(\operatorname{Cap}_{\beta } ( A ) ^{- 1 }\) is defined as NEWLINE\[NEWLINE\inf \int \int k _{\beta } ( x-y ) d \mu ( x ) d \mu ( y )NEWLINE\]NEWLINE 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 \(\beta < 0\).NEWLINENEWLINE2. \(B\) has double points if and only if \(d < 4 N\). \(M _{2 }\) has a.s. positive Lebesgue measure if and only if \(d < 2 N\).NEWLINENEWLINE 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 \(\operatorname{Cap} _{k ( d - 2 N )} ( A ) > 0\).NEWLINENEWLINE 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, \(\| \dim _{H } ( M _{2 } \cap A ) \| _{\infty } = \dim _{H } A-2 ( d-2 N )\).NEWLINENEWLINEThe main technical result is as follows.NEWLINENEWLINEIf NEWLINE\[NEWLINE\Theta = \prod _{j = 1 }^{N } [ a _{j }, b _{j }] \subset ( 0 , \infty ) ^{N }NEWLINE\]NEWLINE and if NEWLINE\[NEWLINE\pi \subset \{ 1 , \dotsc , N \},\;s \in ( 0 , \infty ) ^{N } \backslash \Theta,\;s < _{\pi } tNEWLINE\]NEWLINE for all NEWLINE\[NEWLINEt \in \Theta,\;{\mathcal F} _{\pi} ( s ) = {\mathcal B} ( B ( u ) ; u < _{\pi } s ),NEWLINE\]NEWLINE 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.
scientific article; zbMATH DE number 6067457
scientific article