Brownian motion penetrating fractals - An application of the trace theorem of Besov spaces

For a closed connected set F in R-n, assume that there is a local regular Dirichlet form (a symmetric diffusion process) on F whose domain is included in a Lipschitz space or a Besov space on F. Under some condition for the order of the space and the Newtonian 1-capacity of F; we prove that there exists a symmetric diffusion process on R-n which moves like the process on F and like Brownian motion on R-n outside F. As an application, we will show that when F is a nested fractal or a Sicrpinski carpet whose Hausdorff dimension is greater than n - 2, we can construct Brownian motion penetrating the fractal. For the proof we apply the technique developed in the theory of Besov spaces. (C) 2000 Academic Press.