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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