Distribution of the norm of a multidimensional Wiener process
Let B be a separable Banach space. For a B-valued Wiener process \(W_ t\) (0\(\leq t\leq 1)\) three sufficient conditions for existence of a constant c independent of Cov \(W_ 1\) (covariance) and \((B,\| \|)\) satisfying \[ P(u\leq \| W\|_{\infty}\leq u+\epsilon)\leq c\epsilon \quad (\forall u,\epsilon >0) \] are proved. They are: 1. dim B\(<\infty.\) 2. B convex, and there is \(p\geq 2\), \(\beta >0\), such that \[ \sup \{\| x+y\| /2:\;\| x\| =\| y| =1,\quad \| x-y\| \geq \epsilon,\quad x,y\in B\}\leq 1-\beta \epsilon^ p\quad (0<\epsilon <2). \] 3. \(B=\ell_ 1\) and the first entry of \(W_ 1\) is independent of the others.
scientific article; zbMATH DE number 4192834