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Symmetric path integrals for stochastic equations with multiplicative noise

¹Department of Physics, University of Virginia, Charlottesville, Virginia 22901, USA.

Physical Review. E, Statistical Physics, Plasmas, Fluids, and Related Interdisciplinary Topics

|November 23, 2000

Summary

This study details converting multiplicative noise Langevin equations into path integrals using a time-symmetric discretization. It corrects a common assumption about Stratonovich Langevin equations with dependent noise amplitudes.

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Area of Science:

Physics
Stochastic Processes
Mathematical Physics

Background:

Langevin equations with multiplicative noise are crucial for modeling systems where noise amplitude depends on the system's state.
Converting these equations into path integrals is essential for applying powerful tools of quantum field theory and statistical mechanics.
Existing methods for this conversion often rely on approximations or specific discretization conventions.

Purpose of the Study:

To derive the correct path integral representation for Langevin equations with multiplicative noise.
To investigate the impact of time discretization conventions on the path integral formulation.
To identify and correct a common misconception regarding the conversion of Stratonovich Langevin equations to path integrals.

Main Methods:

Derivation of path integral from a Langevin equation using a time-symmetric discretization scheme.
Analysis of the coordinate and time derivative definitions in the discretization process.
Comparison of the derived path integral with existing methods, particularly for the Stratonovich convention.

Main Results:

A correct path integral formulation is derived for Langevin equations with multiplicative noise using a time-symmetric discretization.
The derivation highlights the critical role of the discretization convention in defining the path integral.
It is demonstrated that the common shortcut for Stratonovich Langevin equations fails when the noise amplitude is state-dependent.

Conclusions:

The time-symmetric discretization provides a robust method for converting multiplicative noise Langevin equations to path integrals.
Accurate path integral formulation requires careful consideration of discretization choices, especially for state-dependent noise.
The findings necessitate a re-evaluation of previous analyses that employed simplified conversion techniques for Stratonovich Langevin equations.