A causal multifractal stochastic equation and its statistical properties

TL;DR

This paper introduces a new continuous causal multifractal stochastic process with a simple generating equation, capable of modeling intermittent turbulence phenomena and long-range correlations in a unified framework.

Contribution

It presents a novel continuous causal multifractal process with a simple generating equation, expanding the modeling tools for turbulence and complex systems.

Findings

Demonstrates multifractal properties of the process

Shows the process's composition rule for scale dependence

Provides stochastic differential equations for evolution

Abstract

Multiplicative cascades have been introduced in turbulence to generate random or deterministic fields having intermittent values and long-range power-law correlations. Generally this is done using discrete construction rules leading to discrete cascades. Here a causal log-normal stochastic process is introduced; its multifractal properties are demonstrated together with other properties such as the composition rule for scale dependence and stochastic differential equations for time and scale evolutions. This multifractal stochastic process is continuous in scale ratio and in time. It has a simple generating equation and can be used to generate sequentially time series of any length.