Brownian intersection local times: Upper tail asymptotics and thick points
Let \(p<d/(d-2)\) be positive integers, \(U\subset B\subset {\mathbb R}^{d}\) a domain where \(B=B(0,R)\) is a ball \(\{x;|x|\leq R\}\) (for \(d\geq 3\), \(R\) may be \(\infty\)), \(G\) be the Green function in \(B\), \(({\mathcal U}f)(x)= \int_{_{U}}G(x,y)f(y)dy\), considered as \({\mathcal U}:L^{2p/(2p- 1)}(U)\rightarrow L^{2p}(U)\), and \(l\) be the projected intersection local time of \(p\) independent Brownian motions started in points in \(U\) and stopped at the first exit from \(B\) (the rigorous constructions are reviewed in the paper). First result: \[ \lim_{a\to \infty }a^{-1/p}\log P(l(U)>a)= -p/\varrho^{\ast}, \] where \(\varrho^{\ast }=\sup\langle g^{2p-1},{\mathcal U}g^{2p-1}\rangle\) over \(g\in L^{2p}(U)\), \(\|g\|_{2p}=1\). Second result: The \(\sup\) is attained, every \(g\) realizing it has \(\sup_{U}<\infty\) and \(\inf_{U}>0\), is in \({\mathcal C}^{2}(U)\) and in \({\mathcal C}^{1}(\overline {U})\), satisfies \({\mathcal U}g^{2p-1}= \varrho^{\ast }g\) on \(U\), extends continuously on \(B\), null on \(\partial B\), with \((1/2)\Delta g= -\varrho^{\ast -1} g^{2p-1}1_{U}\) on \(B\backslash \partial U\). If \(p=1\), then \(g\) is unique. Third result: If \(d\geq 3\), \(R=\infty\) and \(U=B(0,1)\), then \(g\) is unique, rotationally symmetric around \(0\), \(g(x)=cz(a|x|)\), \(z\) is a solution of \(z''(x)+(d-1)x^{-1}z'(x)+z(x)^{2p-1}=0\), \(z(0)=1\), \(z'(0)=0\), on a maximal \([0,\xi)\) and \(a\) is the minimal (in \((0,\xi)\)) with \((d-2)z(a)= -az'(a)\). Also \(p^{2}/\varrho^{\ast 2}\) is \(a^{4}/4\) for \(p=1\), \(d\geq 3\) and \(4\pi a\int_{0}^{a}z^{4}(s)s^{2}ds\) for \(p=2\), \(d=3\). Fourth result: \[ \sup_{x\in {\mathbb R}^{d}}\limsup_{r\downarrow 0}l(B(x , r))r^{-1}\log^{-2}(1/r)= \limsup_{r\downarrow 0}\sup_{x\in S}l(B(x,r))r^{-1}\log^{-2}(1/r)=\varrho^{\ast 2}/4 \] and the Hausdorff dimension of the set of all \(x\in S\) with \(\limsup_{r\downarrow 0}l(B(x,r))r^{-1}\log^{- 2}(1/r)=a\) is \(1-2a^{1/2}\varrho^{\ast -1}\); \(S\) is the intersection of the sample paths of the \(p\) considered Brownians. The thick points \(x\) are defined by \(\lim \sup_{r\downarrow 0}l(B(x,r))r^{-1}\log^{- 2}(1/r)>0\). The paper contains proofs of these results. The authors announce also, but without proof: \(\lim_{a\rightarrow \infty }P_{(l(U)>a)}(\delta >\varepsilon)=0\), where \(\delta\) is the distance from \(l/l(U)\) to the set of all measures with densities \(g^{2p}\), \(g\) as in the first result.
scientific article; zbMATH DE number 1906096