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Renormalization of stochastic differential equations with multiplicative noise using effective potential methods.
Jean-Sébastien Gagnon1,2, David Hochberg3, Juan Pérez-Mercader2,4
1Department of Physics, Norwich University, Northfield, Vermont 05663, USA.
Physical Review. E
|January 20, 2021
Summary
This study introduces a new method for renormalizing stochastic differential equations with multiplicative noise, drawing from high-energy physics concepts. The approach successfully calculates parameter scale dependence in a chemical model under specific noise conditions.
Area of Science:
- Physics
- Chemistry
- Mathematics
Background:
- Stochastic differential equations (SDEs) are crucial for modeling systems with inherent randomness.
- Renormalization techniques are essential for handling infinities and scale dependence in physical theories.
- Existing methods primarily address additive noise in SDEs.
Purpose of the Study:
- To develop and present a novel renormalization method for SDEs subjected to multiplicative noise.
- To adapt the effective potential concept from high-energy physics for SDE renormalization.
- To derive a general formula for the one-loop effective potential for multiplicative Gaussian noise.
Main Methods:
- Application of the effective potential concept from high-energy physics.
- Derivation of a general one-loop effective potential formula for ordinary SDEs with multiplicative Gaussian noise.
- Renormalization of a simplified Gray-Scott chemical reaction model.
Main Results:
- A general formula for the one-loop effective potential under multiplicative Gaussian noise was derived.
- The method was successfully applied to renormalize a toy chemical model.
- The scale dependence of model parameters under specific noise conditions was computed via perturbation theory.
Conclusions:
- The effective potential method provides a viable approach for renormalizing SDEs with multiplicative noise.
- The study demonstrates the method's utility and limitations in a chemical kinetics context.
- This work extends renormalization techniques to a broader class of stochastic systems.
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