Random Renormalization Group Operators Applied to Stochastic Dynamics

Abstract

Let X(t) be a fixed point the renormalization group operator (RGO), Rp,rX(t)=X(rt)/rp. Scaling laws for the probability density, mean first passage times, finite-size Lyapunov exponents of such fixed points are reviewed in anticipation of more general results. A generalized RGO, \(\mathcal{R}_{P,n}\) where P is a random variable, is introduced. Scaling laws associated with these random RGOs (RRGOs) are demonstrated numerically and applied to subdiffusion in bacterial cytoplasm and a process modeling the transition from subdiffusion to classical diffusion. The scaling laws for the RRGO are not simple power laws, but are a weighted average of power laws. The weighting used in the scaling laws can be determined adaptively via Bayes’ theorem.