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

We present a study on multidimensional stochastic processes described by a Langevin equation 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. For this reason we used a tool called formulation of functional integral, where we build a functional generator of correlation functions without reference to any discretization in the Langevin equations. This type of formalism is characterized by a functional integration over two sets of commuting variables, called Grassmann variables. In this sense, the stochastic process is represented in a manner similar to a quantum theory of fields with contents "bosonic" and "fermionic"..The usual prescriptions to define the stochastic integral arise in the formalism by the definitions of Green functions in the sector of Grassmann variables in the field theory. The stochastic calculus is codified in our formalism in the structure of the Grassmann algebra. In particular we are interested in the study of systems that exhibit a noisy behavior multiplicative way, which means that the intensity of stochastic fluctuations depends on the system state. We study some examples such as higher order derivative Langevin equations and the functional representation of the micromagnetic stochastic Landau-Lifshitz-Gilbert equation.