Stochastic Processes Driven by Deterministic Scale Interactions

TL;DR

This paper explores scale equations that unify differential, fractal, and stochastic processes, demonstrating how regularity conditions influence solution behaviors across scales, with solutions exhibiting properties like fractional Brownian motion.

Contribution

It introduces a framework where scale equations can encompass diverse behaviors at multiple scales, linking deterministic and stochastic solutions.

Findings

Regularity conditions determine solution behaviors at different scales.

Solutions can exhibit stochastic properties such as fractional Brownian motion.

Scale equations unify differential, fractal, and stochastic processes.

Abstract

We study various solution behaviors of scale equations which are recently proposed in \cite{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.