A family of processes interpolating the Brownian motion and the self - avoiding process on the Sierpi nski

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 selfavoiding 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 Holder 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. 1. Our question To illustrate our questions, rst let us consider the Euclidean lattice, Z 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 . The scaling limit of a SAW is far more di cult. 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 di erent 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 Hausdor dimension is greater than 1. It implies that the path spreads in the Sierpi nski gasket, has in nitely 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 Sierpi nski gasket has been constructed by Barlow, Perkins and Kusuoka as the scaling limit of the simple random walk on the pre-Sierpi nski gasket. (See [4], [5].) 1 Our question is : Now that we have two completely di erent processes on the Sierpi nski gasket{ the Brownian motion and the self-avoiding process, can we construct a family of processes that interpolates continuously these two? We construct the interpolating process as the limit of a self-repelling walk. A selfrepelling walk is something between the RW and a SAW. Visiting the same points more than once is not prohibited, but suppressed compared with the RW. We want to construct a one-parameter family of self-repelling walks such that at one end of the parameter it corresponds to the RW, at the other end the SAW. And we take the scaling limit. scaling limit self-repelling walk RW () SAW j j j # # # BM () SA process Here we will further explain what is meant by interpolation. There is a very important exponent that characterizes walks and their scaling limits. The most well-known scene where it appears is the mean square displacement of the walk on an in nite lattice (graph). For a walk starting at O (the origin), let us assume E[jXnj ] n ; n!1; where Xn is the walker's location after n steps, and jXnj denotes the Euclidean distance from the starting point. is our exponent. If you take the scaling limit, this exponent governs the short-time behavior, E[jX(t)j] t ; t # 0: The same determines also other path properties of the scaling limit such as Holder continuity and the law of the iterated logarithm. For comparison, in the case of the one-dimensional integer lattice, Z, for the RW, is known to be 1=2 (the well-known exponent for the BM), and = 1, for the SAW, obviously, because it is a straight motion in one direction. In general, exponents are very resistent to changes. Bolthausen proved for a model of self-repelling walk on Z, that is always 1 regardless of the strength of self-repulsion. T oth constructed a di erent model such that varies from 1=2 to 2=3. There are a few other models, but none of them connects 1=2 to 1. (See [6, 7, 8, 9, 10, 11].) It is interesting enough if we can connect the BM and the self-avoiding process on the Sierpi nski gasket continuously in the sense of weak convergence of path measures. But can we ask for more? So, our question is rephrased as : can we construct an interpolating family of processes that connects the exponent for the RW/BM continuously all the way to for the SAW/SA process? As we have seen above, it's not easy even on the line { the simplest lattice. However, on the Sierpi nski gasket, we give an a rmative answer and the same method works also on the line, R.