Tanaka's formula and renormalization for intersections of planar Brownian motion
Let \(W_ t\) be a Brownian motion in \({\mathbb{R}}^ 2\). In this article the author investigates a measure of Brownian self-intersections which formally is defined by \[ \alpha(B) = \iint_{B}\delta (W_ t-W_ s)dsdt = \lim_{\epsilon \to 0} \iint_{B}q_{\epsilon} (W_ t-W_ s)dsdt \] where \(\delta\) is ''delta-function'', \(q_{\epsilon}(x) = (2\pi \epsilon)^{-1}\exp (-\epsilon /2)\exp (-| x|^ 2/2\epsilon)\), \(B\subset {\mathbb{R}}^ 2\). The attention to Brownian self-intersections is stimulated by \textit{K. Symanzik}'s ideas to use them in quantum field theory [Euclidean quantum field theory. in: R. Jost (ed.), Local quantum theory. Academic Press (1969)]. If B is disjoint to the diagonal then \(\alpha\) (B) is a value at \(x=0\) of a local time of a random field \(W_ t-W_ s\). When B intersects the diagonal then the right-hand side of \(\alpha\) (B) is infinite. In this case \textit{S. R. S. Varadhan} [Appendix to Symanzik's paper] proposed nonrandom constants \(c(\epsilon)\to \infty\) such that \(\lim_{\epsilon \to 0}\alpha (\epsilon,u)-uc(\epsilon)\), where \(\alpha (\epsilon,u)=\int^{u}_{0}\int^{u}_{0}q_{\epsilon}(W_ t-W_ s)dsdt\), is a well-defined random variable. The author used Ito's formula and calculations with stochastic integrals to give a simpler proof of this fact and to explain Varadhan's renormalization. For other attempts concerning Varadhan's result and to investigate Brownian path intersections see the next review; Zbl 0617.60080.
scientific article; zbMATH DE number 4001146