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

## Abstract

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. Such equations are ambiguous, and depend on the details of one's convention for discretizing time when solving them. I show that these ambiguities 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. I also discuss a simple example where this happens, which is the small frequency limit of Newton's equation qe+e{sup 2}(q)q=-{nabla}V(q)+e{sup -1}(q){xi} with noise and a q-dependent damping term. The resolution does not correspond to simply interpreting naive continuum equations in a standard convention, such as Stratonovich or Ito. (c) 2000 The American Physical Society.

- Authors:

- Department of Physics, University of Virginia, Charlottesville, Virginia 22901 (United States)

- Publication Date:

- OSTI Identifier:
- 20216770

- Resource Type:
- Journal Article

- Journal Name:
- Physical Review. E, Statistical Physics, Plasmas, Fluids, and Related Interdisciplinary Topics

- Additional Journal Information:
- Journal Volume: 61; Journal Issue: 6; Other Information: PBD: Jun 2000; Journal ID: ISSN 1063-651X

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; LANGEVIN EQUATION; STOCHASTIC PROCESSES; GAUSSIAN PROCESSES; THEORETICAL DATA

### Citation Formats

```
Arnold, Peter.
```*Langevin equations with multiplicative noise: Resolution of time discretization ambiguities for equilibrium systems*. United States: N. p., 2000.
Web. doi:10.1103/PhysRevE.61.6091.

```
Arnold, Peter.
```*Langevin equations with multiplicative noise: Resolution of time discretization ambiguities for equilibrium systems*. United States. doi:10.1103/PhysRevE.61.6091.

```
Arnold, Peter. Thu .
"Langevin equations with multiplicative noise: Resolution of time discretization ambiguities for equilibrium systems". United States. doi:10.1103/PhysRevE.61.6091.
```

```
@article{osti_20216770,
```

title = {Langevin equations with multiplicative noise: Resolution of time discretization ambiguities for equilibrium systems},

author = {Arnold, Peter},

abstractNote = {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. Such equations are ambiguous, and depend on the details of one's convention for discretizing time when solving them. I show that these ambiguities 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. I also discuss a simple example where this happens, which is the small frequency limit of Newton's equation qe+e{sup 2}(q)q=-{nabla}V(q)+e{sup -1}(q){xi} with noise and a q-dependent damping term. The resolution does not correspond to simply interpreting naive continuum equations in a standard convention, such as Stratonovich or Ito. (c) 2000 The American Physical Society.},

doi = {10.1103/PhysRevE.61.6091},

journal = {Physical Review. E, Statistical Physics, Plasmas, Fluids, and Related Interdisciplinary Topics},

issn = {1063-651X},

number = 6,

volume = 61,

place = {United States},

year = {2000},

month = {6}

}