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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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