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Invariance principles for Brownian intersection local time and polymer measures. (English) Zbl 0717.60043

Summary: The goal of this article is to give a nonstandard representation of the two-dimensional Varadhan-Edwards-Symanzik polymer measure by a hyperfinite Domb-Joyce model. From the standard point of view, our representation contains a new invariance principle for weakly self- avoiding or self-repellent random walks. An important step towards this result is to give a nonstandard construction of Brownian intersection local time in \(d<4\) and of its renormalization in \(d=2\). Again we obtain new invariance principles similar to that in the one-dimensional case which E. Perkins [Z. Wahrscheinlichkeitstheor. Verw. Geb. 60, 437- 451 (1982; Zbl 0465.60065)] deduced from his nonstandard approach to Brownian local time. Besides the new invariance principles, our nonstandard approach recovers the already known existence results for the limiting objects.

MSC:

60F17 Functional limit theorems; invariance principles
60J65 Brownian motion
60J55 Local time and additive functionals
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