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Strong approximations of three-dimensional Wiener sausages. (English) Zbl 1127.60026

Let \(B_t,\; t\geq 0,\) be a centered three-dimensional Brownian motion, \(K_i\subset\mathbb R^3, i=1,\dots,l,\) compact sets and \(S^{K_i}(0,t)=\bigcup_{s\leq t} (B_s+K_i)\) the associated Wiener sausages. It is shown that the volumes of \(S^{K_i}(0,t)\) have the following asymptotic almost sure representation \(m(S^{K_i}(0,t))={\mathcal C}(K_i)t+{\mathcal C}^2(K_i)\beta(t\log t)/\pi\sqrt{2} +o(t^{1/2}(\log t)^{1/4+\delta})\) for any positive \(\delta\) as \(t\to\infty\), where \(\beta(t)\) is a one-dimensional Brownian motion common for all \(K_i\)’s and \({\mathcal C}(K)\) is the Newtonian capacity of \(K\). Laws of iterated logarithm as well as functional central limit theorem are immediate corollaries. The proof relies on Le Gall’s \(L^2\)-norm estimates between the Wiener sausage and the Brownian intersection local times.

MSC:

60F15 Strong limit theorems
60J55 Local time and additive functionals
60J65 Brownian motion
60F17 Functional limit theorems; invariance principles
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