Self-avoiding processes and self-repelling processes on fractals

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We constructed a family of self-repelling walks on the pre-Sierpinski gasket and on the 1-dimensional Euclidean space, respectively, which continuously interpolates between the simple random walk and a self-avoiding walk It is a one-parameter family with parameter u, and u=0 corresponds to a self avoiding walk, u=1 to the simple random walk and 0<u<1 to self-repelling walks The asymptotic behaviors of the walks have been obtained in terms of displacement exponents and a law of iterated logarithms. The result can further be extended to self-attracting walks, with u>1. Our method is based on renormalization group and we found that we can construct more general stochastic chains, using this method. The asympotitic behaviors are obtained in a parallel manner.We studied also the recurrence of the stochastic chains constructed by renormalization group method and obtained a sufficient condition for recurrence. In particular, we proved the above mentioned family of self-repelling and self-attracting walks are recurrent if u>0. We also proved that there is a positive constant c>1 such that the expected return time to the origin is infinite for 0<u<c. This implies that there is a unique, sigma-finite, ergodic invariant measure on the infinite-length path space on the Sierpinski gasket and the 1-dimensional Euclidean space.