For many years, mathematician and phycian know that those problems are related to the second differential operator in divergence form. Nevertheless, the precised discussion in terms of Dirichlet form was just recent. In this paper,we use the notion of diffusion process on a non-smooth Riemannian manifold to discuss the related boundary problem and limiting problem. Given a finite collection of strings of letters from a fixed alphabet, it is of interest, in the contexts of data compression and DNA sequencing, to find the length of the shortest string which contains each of the given strings as a consecutive substring. In order to analyze the average behavior of the optimal superstring length, substrings of specified lengths are considered with the letters selected independently at random.
C be a bounded measurable function.
For these fractals estimates on the transition density for the Brownian motion are obtained using the associated Dirichlet form. The approach differs from previous treatments as it uses only the Dirichlet form and no estimates on the resolvent. The bounds obtained are expressed in terms of the intrinsic resistance metric on the fractal. Here we will be concerned with Markov properties of solutions, when the driving process is a Levy process living in a vector space, more precisely we will compute their infinitesimal generator. As an application we are studying a process called pseudo alpha stable process constructed as the stochastic development on a Riemannian manifold of a vector valued alpha-stable process living in the tangent space of the manifold.