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.

We address the calculation of transition probabilities in multiplicative noise stochastic differential equations using a path integral approach. We show the equivalence between the conditional probability and the propagator of a quantum particle with variable mass. Introducing a time reparametrization, we are able to transform the problem of multiplicative noise fluctuations into an equivalent additive one. We illustrate the method by showing the explicit analytic computation of the conditional probability of a harmonic oscillator in a nonlinear multiplicative environment.

-We present a path integral formalism to compute potentials for nonequilibrium steady states, reached by a multiplicative stochastic dynamics. We develop a weak-noise expansion, which allows the explicit evaluation of the potential in arbitrary dimensions and for any stochastic prescription. We apply this general formalism to study noise-induced phase transitions. We focus on a class of multiplicative stochastic lattice models and compute the steady state phase diagram in terms of the noise intensity and the lattice coupling. We obtain, under appropriate conditions, an ordered phase induced by noise. By computing entropy production, we show that microscopic irreversibility is a necessary condition to develop noise-induced phase transitions. This property of the nonequilibrium stationary state has no relation with the initial stages of the dynamical evolution, in contrast with previous interpretations, based on the short-time evolution of the order parameter.Introduction. -Nonequilibrium statistical mechanics is at the stem of important physical phenomena, many of them at the border with other sciences such as biology, chemistry, geology and even social sciences. Differently from equilibrium statistical mechanics, there is no closed theoretical framework to deal with out-of-equilibrium systems, being the theory of stochastic processes [1] one of the natural approaches to describe them. These systems have been traditionally modeled by stochastic differential equations, as well as through Fokker-Planck equations. Moreover, functional path integral approaches have also been introduced [2]. The latter approach is more adaptive to explore symmetries and general formal aspects of stochastic dynamics, such as out-of-equilibrium fluctuations theorems [3][4][5].

We study the stochastic dynamics of a two-dimensional magnetic moment embedded in a threedimensional environment, described by means of the stochastic Landau-Lifshitz-Gilbert (sLLG) equation. We define a covariant generalization of this equation, valid in the "generalized Stratonovich discretization prescription". We present a path integral formulation that allows to compute any n−point correlation function, independently of the stochastic calculus used. Using this formalism, we show the equivalence between the cartesian formulation with vectorial noise, and the polar formulation with just one scalar fluctuation term. In particular, we show that, for isotropic fluctuations, the system is represented by an additive stochastic process, despite of the multiplicative terms appearing in the original formulation of the sLLG equation, but, for anisotropic fluctuations the noise turns out to be truly multiplicative.

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