Critical dimensions for the existence of self-intersection local times of the \(N\)-parameter Brownian motion in \(R^d\)
Let \(W=\{W_t, t\in {\mathbb R^N_+}\}\) be an \(N\)-parameter Wiener process in \({\mathbb R}^d\). Given two rectangles \(A, B\) in \({\mathbb R^N_+}\), the authors consider the self-intersection local time of \(W\) defined as the limit as \(\varepsilon\) tends to zero of \[ \alpha_\varepsilon(x)=\int_B\int_A p_\varepsilon^d(W_t-W_s-x) ds dt, \] where \(p_\varepsilon^d(z)=(2\pi \varepsilon)^{-d/2}\exp(-|z|^2/2\varepsilon)\), and \(x\in {\mathbb R}^d\). If \(A=B\), the critical dimension below which \(\alpha_\varepsilon(x)\) does not explode is \(d=2N\). More precisely, if \(d<2N\), the limit exists in the Sobolev space \({\mathbb D}_{2,\rho}\), for any \(\rho<2N-d/2\). If \(d\geq 2N\) a similar type of convergence holds for the renormalization \(\alpha_\varepsilon(x)/ f(\varepsilon)\), where \[ f(\varepsilon)=\begin{cases} \log 1/\varepsilon &\text{ for } d=2N,\\ e^{(2N-d)/2} &\text{ for } d>2N.\end{cases} \] If the intersection \(A\cap B\) has dimension \(p\) with \(0\leq p<N\), the critical dimension turns out to be \(4N-2p\), and the renormalization function is \(f(\varepsilon)=\log 1/\varepsilon \) in the case \(d=4N-2p\) and \(f(\varepsilon)= \varepsilon^{(4N-d-2p)/2} \) if \(d>4N-2p\). In the case \(A\cap B=\emptyset\) there is no explosion. The convergence of the renormalization \((\alpha_\varepsilon(x)- E(\alpha_\varepsilon(x)))/ \sqrt{f(\varepsilon)}\) is also investigated in all cases. The proof of these results is based on the Wiener chaos expansion and on the techniques introduced by the authors [in: Seminar on stochastic analysis, random fields and applications. Prog. Probab. 36, 151-168 (1995; Zbl 0827.60036)] where they treated the case of the Brownian sheet (\(N=2\)).
scientific article; zbMATH DE number 1339515