Abstract

We construct Brownian motion on a class of fractals which are spatially homogeneous but which do not have any exact self-similarity. We obtain transition density estimates for this process which are up to constants best possible. 1 Introduction There is now a fairly extensive literature on the heat equation on fractal spaces, and on the spectral properties of such spaces. Most of these papers treat sets F which have exact selfsimilarity, so that there exist 1-1 contractions / i : F ! F such that / i (F ) " / j (F ) is (in some sense) small when i 6= j, and F = [ i / i (F ): (1.1) In the simplest cases, such as the nested fractals of Lindstrøm [18], F ae R d , the / i are linear, and / i (F ) " / j (F ) is finite when i 6= j. For very regular fractals such as nested fractals, or Sierpinski carpets, it is possible to construct a diffusion X t with a semigroup P t which is symmetric with respect to ¯, the Hausdorff measure on F , and to obtain estimates on the density p t (x; y) of P ...

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