Langevin dynamics for vector variables driven by multiplicative white noise: A functional formalism

We discuss general multidimensional stochastic processes driven by a system of Langevin equations with multiplicative white noise. In particular, we address the problem of how time reversal diffusion processes are affected by the variety of conventions available to deal with stochastic integrals. We present a functional formalism to build up the generating functional of correlation functions without any type of discretization of the Langevin equations at any intermediate step. The generating functional is characterized by a functional integration over two sets of commuting variables, as well as Grassmann variables. In this representation, time reversal transformation became a linear transformation in the extended variables, simplifying in this way the complexity introduced by the mixture of prescriptions and the associated calculus rules. The stochastic calculus is codified in our formalism in the structure of the Grassmann algebra. We study some examples such as higher order derivative Langevin equations and the functional representation of the micromagnetic stochastic Landau-Lifshitz-Gilbert equation.

Figure 1

Generalized Stratonovich prescription.

Figure 2

Sketch of a “time-loop” evolution of Langevin Eq. (1). The forward evolution $t\to t+\Delta t$ is performed with a prescription $\alpha$, while the backward trajectory $t+\Delta t\to t$ evolves with the time reversal conjugate prescription $(1-\alpha )$. Except for the Stratonovich convention $\alpha =1/2$, $\Delta x{\left(t\right)}_{\alpha }\ne 0$.

Figure 3

Retarded Green's function.

Figure 4

Advanced Green's function.