Hidden symmetries and equilibrium properties of multiplicative white-noise stochastic processes

TL;DR

This paper investigates the equilibrium properties and hidden symmetries of multiplicative white-noise stochastic processes, introducing a prescription-independent formalism and a supersymmetric framework to unify their analysis.

Contribution

It develops a Grassman formalism and supersymmetric approach to analyze equilibrium and symmetries of multiplicative white-noise processes regardless of prescription.

Findings

Equilibrium properties hold for any prescription when using proper definitions.

A covariant supersymmetric formulation of the process is established.

Hidden symmetries are identified and analyzed through Ward-Takahashi identities.

Abstract

Multiplicative white-noise stochastic processes continuously attract the attention of a wide area of scientific research. The variety of prescriptions available to define it difficults the development of general tools for its characterization. In this work, we study equilibrium properties of Markovian multiplicative white-noise processes. For this, we define the time reversal transformation for this kind of processes, taking into account that the asymptotic stationary probability distribution depends on the prescription. Representing the stochastic process in a functional Grassman formalism, we avoid the necessity of fixing a particular prescription. In this framework, we analyze equilibrium properties and study hidden symmetries of the process. We show that, using a careful definition of equilibrium distribution and taken into account the appropriate time reversal transformation, usual equilibrium properties are satisfied for any prescription. Finally, we present a detailed deduction of a covariant supersymmetric formulation of a multiplicative Markovian white-noise process and study some of the constraints it imposes on correlation functions using Ward-Takahashi identities.