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Brownian motion in a Brownian crack. (English) Zbl 0937.60081

Let \(Y^\varepsilon(t)\) be the reflected two-dimensional Brownian motion in Wiener sausage \(D^\varepsilon\) of width \(\varepsilon>0\) around two-sided Brownian motion \(X_1(t)\). Here \(X_1(t)=X^+(t)\) if \(t>0\) and \(X_2(t)=X^-(-t)\) if \(t<0\), where \(X^+\) and \(X^-\) are independent Brownian motions. Suppose that \(X_2(t)\) is also a standard Brownian motion independent of \(X_1\). The process \(X(t)=X_1(X_2(t))\), \(t\geq 0\), is called an “iterated Brownian motion” (IBM). The main result of the article states that for some \(c(\varepsilon)\) the components of \(Y^\varepsilon\), i.e. \(\text{Re} Y^\varepsilon(c(\varepsilon)\varepsilon^{-2}t)\) and \(\text{Im} Y^\varepsilon(c(\varepsilon)\varepsilon^{-2}t)\) converge to one-dimensional Brownian motion and IBM, respectively, as \(\varepsilon\to 0\). Since diffusions on fractals are often constructed by an approximation process, this convergence provides the justification for use of IBM as a convenient model for Brownian motion in a Brownian crack other than a “crack diffusion model” introduced by Chudnovsky and Kunin [J. Appl. Phys. 62, 4124-4129 (1987)]. A few open problems are discussed.

MSC:

60J65 Brownian motion
60F99 Limit theorems in probability theory
60J60 Diffusion processes
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