A Family of Processes Interpolating the Brownian Motion and the Self-avoiding Process on the Sierpi Nski Gasket and R

We construct a one-parameter family of self-repelling processes on the Sierpi nski gasket, by taking scaling limits of self-repelling walks on the pre-Sierpi nski gaskets. We prove that our model interpolates between the Brownian motion and the self-avoiding process on the Sierpi nski gasket. Namely, we prove that the process is continuous in the parameter in the sense of convergence in law, and that the order of HH older continuity of the sample paths is also continuous in the parameter. We also establish a law of the iterated logarithm for the self-repelling process. Finally we show that this approach yields a new class of one-dimensional self-repelling processes. To illustrate our questions, rst let us consider the Euclidean lattice, Z d and a random walk on it. The simple random walk (RW) is a walk that jumps to one of its nearest neighbor points with equal probability. On the other hand, a self-avoiding walk (SAW) is a walk that is not allowed to visit any point more than once. If you take the scaling limit, that is, the limit as the lattice spacing (bond length) tends to 0, the RW converges to the Brownian motion (BM) in R d. The scaling limit of a SAW is far more diicult. It is because a SAW must remember all the points it has once visited. In short, it lacks Markov property. For the 1-dimensional lattice, that is, a line, it is trivial { the scaling limit is a constant speed motion to the right or to the left. For 4 or more dimensions, the scaling limit is the Brownian motion. Since the space is large enough, the RW is not much diierent from the SAW. However, for the 2 and 3-dimensional lattice, the scaling limit is not known. >From this viewpoint, the Sierpi nski gasket is a rare example of a low dimensional space, where the scaling limit of a SAW is known. The SAW on the pre-Sierpi nski gasket converges to a non-trivial self-avoiding process, which is not a straight motion along an edge, nor deterministic, and moreover, whose path Hausdorr dimension is greater than 1. It implies that the path spreads in the Sierpi nski gasket, has innnitely ne creases and is self-avoiding. Let us emphasize here that in a low-dimensional space the existence of a non-trivial self-avoiding process itself is "something." On the other hand, the Brownian motion on the …