Brownian Motion Penetrating Fractals: An Application of the Trace Theorem of Besov Spaces

For a closed connected set F in Rⁿ, 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 Rⁿ which moves like the process on F and like Brownian motion on Rⁿ outside F. As an application, we will show that when F is a nested fractal or a Sierpinski 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.