Stochastic Processes Driven by Deterministic Scale Interactions
Abstract
We study various solution behaviors of scale equations which are recently proposed in Kim. On the contrary to conventional mathematical tools, scale equations are capable to accommodate various behaviors at different scale levels into one integrated solution. Some solutions of scale equations often retain strong stochastic properties such as fractional Brownian Motion, although constructing those solutions is a deterministic process. We show that imposing a regularity condition on scale equations determines the behaviors of solutions at both small and large scale levels simultaneously, and moreover, the corresponding solutions occur as solutions of differential equations too. This suggests that scale equations provide a potential framework unifying differential equations through fractal, to stochastic processes.