Renormalization of stochastic differential equations with multiplicative noise using effective potential methods

TL;DR

This paper introduces a novel renormalization method for stochastic differential equations with multiplicative noise, utilizing effective potential techniques from high energy physics, and demonstrates its application on a toy chemical model.

Contribution

It develops a general one-loop effective potential formula for multiplicative noise SDEs and applies it to renormalize a chemical reaction model with power-law noise.

Findings

Derived a general formula for the one-loop effective potential.

Applied the method to a toy chemical model with Gaussian power-law noise.

Computed the scale dependence of model parameters under noise.

Abstract

We present a new method to renormalize stochastic differential equations subjected to multiplicative noise. The method is based on the widely used concept of effective potential in high energy physics, and has already been successfully applied to the renormalization of stochastic differential equations subjected to additive noise. We derive a general formula for the one-loop effective potential of a single ordinary stochastic differential equation (with arbitrary interaction terms) subjected to multiplicative Gaussian noise (provided the noise satisfies a certain normalization condition). To illustrate the usefulness (and limitations) of the method, we use the effective potential to renormalize a toy chemical model based on a simplified Gray-Scott reaction. In particular, we use it to compute the scale dependence of the toy model's parameters (in perturbation theory) when subjected to a Gaussian power-law noise with short time correlations.