skip to main content
OSTI.GOV title logo U.S. Department of Energy
Office of Scientific and Technical Information

Title: 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:
 [1]
  1. 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}
}