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Convergence of the normalized one-dimensional Wiener sausage path measures to a mixture of Brownian taboo processes. (English) Zbl 0697.60032
Summary: For $$T\in [0,\infty)$$ and a Brownian motion path $\beta \in \Omega:=\{\beta \in C([0,\infty),{\mathbb{R}}): \beta (0)=0\}$ denote by $$| C_ T(\beta)|$$ the length of the compact interval $$C_ T(\beta):=\{\beta (t): t\in [0,T]\}$$. Fix $$\nu >0$$ and define $P_ T(A)=E(1_ A\exp (-\nu T| C_ T|))/E(\exp (-\nu T| C_ T|))$ for all Borel sets A of $$\Omega$$. These measures favour those Brownian motion paths which spread over a small interval up to time T without being too improbable.
It is proved that $$(P_ T)_{T\in [0,\infty)}$$ converges weakly to a measure $$P_{\infty}$$ as $$T\to \infty$$. The limiting measure $$P_{\infty}$$ is characterised as a mixture of Brownian motions with a drift, forcing them to remain in an interval of length $$c_ 0:=(\pi^ 2/\nu)^{1/3}$$ for all times. More precisely: $P_{\infty}(A)=\int_{(0,c_ 0)}(\pi /2c_ 0)\sin (\pi b/c_ 0)P_{(b-c_ 0,b)}(A)db$ for all Borel sets A of $$\Omega$$, where $$P_{(b-c_ 0,b)}$$ denotes the path measure of a Brownian taboo process (starting at zero) with taboo set $$\{b-c_ 0,b\}$$.

##### MSC:
 60F10 Large deviations 60F05 Central limit and other weak theorems 60J65 Brownian motion
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