A Langevin equation with multiplicative noise is an equation schematically of the form $d\mathbf{q}/dt=-\mathbf{F}\left(\mathbf{q}\right)+e\left(\mathbf{q}\right)\mathit{\xi },$ where $e\left(\mathbf{q}\right)\mathit{\xi }$ is Gaussian white noise whose amplitude $e\left(\mathbf{q}\right)$ depends on $\mathbf{q}$ 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 $\left({\mathbf{q}}_t-{\mathbf{q}}_{t-\Delta t}\right)/\Delta t$ and coordinates are $({\mathbf{q}}_t+{\mathbf{q}}_{t-\Delta t})/2.\mathrm{}$ (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 $\theta \mathrm{}(t=0\mathrm{})=\frac{1}{2}.$ I show that this prescription fails when the amplitude $e\left(\mathbf{q}\right)$ is $\mathbf{q}$ dependent.

• Received 1 December 1999

DOI:https://doi.org/10.1103/PhysRevE.61.6099