Symmetric path integrals for stochastic equations with multiplicative noise

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 that time derivatives are (q_t - q_{t-\Delta t}) / \Delta t and coordinates are (q_t + q_{t-\Delta 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 Stratanovich 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.Comment: 8 page

Action perception is intact in autism spectrum disorder

Date of Acceptance:10/11/2014. Copyright © 2015 the authors 0270-6474/15/351849-09$15.00/0. Copyright of all material published in The Journal of Neuroscience remains with the authors. The authors grant the Society for Neuroscience an exclusive license to publish their work for the first 6 months. After 6 months the work becomes available to the public to copy, distribute, or display under a Creative Commons Attribution 4.0 International (CC BY 4.0) license.Peer reviewedPublisher PD