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The construction of Brownian motion on the Sierpinski carpet. (English) Zbl 0691.60070

Let \(F_ n\) denote the n-th term in the decreasing sequence of sets in the construction of the Sierpinski carpet and let \(W^ n_ t\) be a Brownian motion on \(F_ n\) with normal reflection at the internal boundaries. It is shown that for suitably chosen constants \(\alpha_ n\) the sequence \(X^ n_ t=W^ n_{\alpha_ nt}\) is tight. As a limit of this sequence the continuous strong Markov process X whose state space is the Sierpinski carpet \(F=\cap_{n}F_ n\) is obtained. This process X is preserved under certain transformations of the state space, but it is unknown if X is scale-invariant.
The technique of construction is quite different from the method used in other papers for construction of Brownian motions on the Sierpinski gasket because the last in contrast to the Sierpinski carpet is a finitely ramified fractal.
Reviewer: I.Molchanov

MSC:

60J65 Brownian motion
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