# Symmetric path integrals for stochastic equations with multiplicative noise

## 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. 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{sub t}-q{sub t-{delta}}{sub t})/{delta}t and coordinates are (q{sub t}+q{sub t-{delta}}{sub t})/2. (This is the convention that permits standard manipulations of calculus on the action, like naive integration by parts.) It has sometimes been assumed in the literature that a Stratonovich Langevin equation can be quickly converted to a path integral by treating time as continuous but using the rule {theta}(t=0)=(1/2). I show that this prescription fails when the amplitude e(q) is q dependent. (c) 2000 The American Physical Society.

- Authors:

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

- Publication Date:

- OSTI Identifier:
- 20216771

- 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; STOCHASTIC PROCESSES; GAUSSIAN PROCESSES; FEYNMAN PATH INTEGRAL; LANGEVIN EQUATION; THEORETICAL DATA

### Citation Formats

```
Arnold, Peter.
```*Symmetric path integrals for stochastic equations with multiplicative noise*. United States: N. p., 2000.
Web. doi:10.1103/PhysRevE.61.6099.

```
Arnold, Peter.
```*Symmetric path integrals for stochastic equations with multiplicative noise*. United States. doi:10.1103/PhysRevE.61.6099.

```
Arnold, Peter. Thu .
"Symmetric path integrals for stochastic equations with multiplicative noise". United States. doi:10.1103/PhysRevE.61.6099.
```

```
@article{osti_20216771,
```

title = {Symmetric path integrals for stochastic equations with multiplicative noise},

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. 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{sub t}-q{sub t-{delta}}{sub t})/{delta}t and coordinates are (q{sub t}+q{sub t-{delta}}{sub t})/2. (This is the convention that permits standard manipulations of calculus on the action, like naive integration by parts.) It has sometimes been assumed in the literature that a Stratonovich Langevin equation can be quickly converted to a path integral by treating time as continuous but using the rule {theta}(t=0)=(1/2). I show that this prescription fails when the amplitude e(q) is q dependent. (c) 2000 The American Physical Society.},

doi = {10.1103/PhysRevE.61.6099},

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}

}