# Symmetric path integrals for stochastic equations with multiplicative noise

@article{Arnold2000SymmetricPI, title={Symmetric path integrals for stochastic equations with multiplicative noise}, author={Arnold}, journal={Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics}, year={2000}, volume={61 6 Pt A}, pages={ 6099-102 } }

A Langevin equation with multiplicative noise is an equation schematically of the form dq/dt=-F(q)+e(q)xi, where e(q)xi is Gaussian white noise whose amplitude e(q) depends on q itself. I show how to convert such equations into path integrals. The definition of the path integral depends crucially on the convention used for discretizing time, and I specifically derive the correct path integral when the convention used is the natural, time-symmetric one whose time derivatives are (q(t)-q(t-Deltat…

## Topics from this paper

## 48 Citations

Langevin equations with multiplicative noise: resolution of time discretization ambiguities for equilibrium systems

- Physics, MedicinePhysical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics
- 2000

It is shown that ambiguities in Langevin equation with multiplicative noise are uniquely resolved if the system has a known equilibrium distribution exp[-V(q)/T] and if, at some more fundamental level, the physics of the system is reversible.

State-dependent diffusion: Thermodynamic consistency and its path integral formulation.

- Mathematics, MedicinePhysical review. E, Statistical, nonlinear, and soft matter physics
- 2007

This work shows that the requirement that a particle's distribution function approach the Boltzmann distribution at long times dictates that a drift term must be added to the Langevin equation, and derives a path integral representation for arbitrary interpretation of the noise.

Rules of calculus in the path integral representation of white noise Langevin equations: the Onsager-Machlup approach

- Physics, Mathematics
- 2017

The definition and manipulation of Langevin equations with multiplicative white noise require special care (one has to specify the time discretisation and a stochastic chain rule has to be used to…

Time-Slicing Path-integral in Curved Space

- Physics, Mathematics
- 2021

Path integrals constitute powerful representations for both quantum and stochastic dynamics. Yet despite many decades of intensive studies, there is no consensus on how to formulate them for dynamics…

Building a path-integral calculus: a covariant discretization approach

- Mathematics, Physics
- 2018

Path integrals are a central tool when it comes to describing quantum or thermal fluctuations of particles or fields. Their success dates back to Feynman who showed how to use them within the…

A Path Integral Method for Coarse-Graining Noise in Stochastic Differential Equations with Multiple Time Scales

- Mathematics, Physics
- 2007

We present a new path integral method to analyze stochastically perturbed ordinary differential equations with multiple time scales. The objective of this method is to derive from the original system…

Generally covariant state-dependent diffusion

- Physics, Mathematics
- 2013

Statistical invariance of Wiener increments under SO(n) rotations provides a notion of gauge transformation of state-dependent Brownian motion. We show that the stochastic dynamics of…

Magnetization dynamics: path-integral formalism for the stochastic Landau–Lifshitz–Gilbert equation

- Physics
- 2014

We construct a path-integral representation of the generating functional for the dissipative dynamics of a classical magnetic moment as described by the stochastic generalization of the…

Summing over trajectories of stochastic dynamics with multiplicative noise.

- Mathematics, PhysicsThe Journal of chemical physics
- 2014

This work develops a novel path integral formulation for the overdamped Langevin equation with multiplicative noise that solves the inconsistency of the previous path integral formulations for the general stochastic interpretation, and can have wide applications in chemical and physical Stochastic processes.

The chemical birth-death process with Gillespie noise

- Physics, Biology
- 2019

An exact path integral calculation of the transition probability corresponding to a one-dimensional system with state-dependent noise using a Martin-Siggia-Rose-Janssen-De Dominicis (MSRJD) path integral is provided.

## References

SHOWING 1-5 OF 5 REFERENCES

Non-Perturbative Dynamics of Hot Non-Abelian Gauge Fields

- Physics
- 2001

The dynamics of high temperature gauge fields, on scales relevant for nonperturbative phenomena such as electroweak baryogenesis, may be described by a remarkably simple effective theory. This…

Physics 95A

- 252 (1979); H. Kawara, M. Namiki, H. Okamoto, and S. Tanaka, Prog. Theor. Phys. 84, 749 (1990); N. Komoike, Prog. Theor. Phys. 86, 575
- 1991

B275 [FS17] 135

- B275 [FS17] 135
- 1986

Nucl

- Phys. B275 [FS17] 135,
- 1986

Physics 95A Theor. Phys

- Physics 95A Theor. Phys
- 1979