The Langevin Approach: a simple stochastic method for complex phenomena

TL;DR

The paper introduces a parameter-free Langevin stochastic method for extracting evolution equations from data, applicable to time and scale processes, demonstrated on turbulence data and extendable to various fields.

Contribution

It presents a novel, simple Langevin approach that can analyze complex stochastic phenomena without parameters, applicable across multiple disciplines.

Findings

Successfully retrieved energy cascade in turbulence data

Validated method against computational simulations

Extended approach to diverse fields like finance and medicine

Abstract

We describe a simple stochastic method, so-called Langevin approach, which enables one to extract evolution equations of stochastic variables from a set of measurements. Our method is parameter-free and it is based on the nonlinear Langevin equation. Moreover, it can be applied not only to processes in time, but also to processes in scale, given that the data available shows ergodicity. This chapter introduces the mathematical foundations of the Langevin approach and describes how to implement it numerically. A specific application of the method is presented, namely to a turbulent velocity field measured in the laboratory, retrieving the corresponding energy cascade and comparing with the results from a computational simulation of that experiment. In addition, we describe a physical interpretation bridging between processes in time and in scale. Finally, we describe extensions of the method for time series reconstruction and applications to other fields such as finance, medicine, geophysics and renewable energies.