zbMATH — the first resource for mathematics

A representation for the intersection local time of Brownian motion in space. (English) Zbl 0561.60086
In this paper the author extends his previous results in Commun. Math. Phys. 88, 327-338 (1983; Zbl 0534.60070). Let \(X(s,t)=W_ t-W_ s\) and \(\mu_ B(A)=\lambda_ 2(X^{-1}(A)\cap B)\), where W denotes three dimensional Brownian motion, \(\lambda_ n\) denotes Lebesgue measure on \(R^ n\) and \(B=[a,b]\times [c,d]\subset R^ 2_+\), \(b<c\). Then by definition, \(\alpha (x,B)=d\mu_ B(x)/d\lambda_ 3\) is called the intersection local time relative to B.
The main theorem in the paper is a ”Tanaka-like” representation for \(\alpha\) (x,B), i.e., with probability one, \[ -\alpha (x,B)=G\nu_{[a,b]}(W_ d-x)-G\nu_{[a,b]}(W_ c-x)- \int^{d}_{c}\nabla G\nu_{[a,b]}(W_ t-x)\cdot dW_ t, \] where \(\nu_{[a,b]}(A)=\lambda_ 1(W^{-1}(A)\cap [a,b])\) and \(G\nu\) is the Newtonian potential of \(\nu\). After showing \(G\nu\) is Hölder continuous of any order \(<1\), the procedure of proof is analogous to that in 1- dimension [cf. N. Ikeda and S. Watanabe, Stochastic differential equations and diffusion processes. (1981; Zbl 0495.60005)]. The case of 2 dimensions is also briefly discussed where the logarithmic potential is not bounded at \(\infty\). These results may be very useful in approaches to quantum field theory and polymer statistics with excluded volume.
Reviewer: Ch.Wu

60J65 Brownian motion
60H05 Stochastic integrals
60J55 Local time and additive functionals
Full Text: DOI