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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

##### MSC:
 60J65 Brownian motion 60H05 Stochastic integrals 60J55 Local time and additive functionals
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