Time reversibility for stochastic dynamics with multiplicative noise

Stochastic differential equations and its applications is a subject of great interest for scientific research. In this area, Langevin and Fokker-Planck formalisms are extensively used. Systems modeled by differential stochastic equations with additive noise have been largely studied and are the most popular models. However, the understanding of the stochastic dynamics and the evolution to equilibrium for systems dealing with multiplicative noise is difficult and there is a lack of general tools for its characterization. In particular, for the multiplicative noise case, the Fokker-Planck equation does depend on the chosen prescription for the stochastic integration of the associated Langevin equation. In such case, it is possible for time evolutions to reach non-Boltzmann equilibrium states.

To deal with these systems, we use a general prescription $\alpha$ for considering the stochastic integration. $\alpha$ is defined as a continuous parameter, $ 0 \leq \alpha \leq 1$, and each of its values corresponds with a different discretization rule for the stochastic differential equation. $\alpha = 0$ corresponds with It? prescription at the time that $\alpha = 1/2$ corresponds with Stratonovich one. We also represent the stochastic process in a functional
Grassman formalism\footnote{Zochil Gonz?lez Arenas and Daniel G. Barci, Phys. Rev. {\bf E81}, 051113 (2010); Phys. Rev. {\bf E 85}, 041122 (2012)}, which turns out to be very convenient for handling stochastic trajectories.

In this work\footnote{Zochil Gonz?lez Arenas and Daniel G. Barci, J. Stat. Mech. P12005.(2012)}\footnote{ Zochil Gonz?lez Arenas, PhD Thesis, CDU $53:519.2$, UERJ, 2012.}\footnote{Miguel V. Moreno, Zochil Gonz?lez Arenas and Daniel G. Barci, Phys. Rev. {\bf E91}, 042103 (2015).}, we study equilibrium properties of Markovian multiplicative white-noise processes. For this, we carefully define the time reversal transformation for this kind of processes, taking into account that the asymptotic stationary probability distribution depends on the prescription. In white noise multiplicative processes, stochastic trajectories evolve with different prescriptions in one direction and in the reverse direction. We show that, using a careful definition of equilibrium distribution and taken into account the appropriate time reversal transformation, usual equilibrium properties, such as detailed balance, are satisfied for any prescription.

The Brazilian agencies, Funda??o de Amparo ? Pesquisa do Estado do Rio de Janeiro (FAPERJ), Conselho Nacional de Desenvolvimento Cient?fico e Tecnol?gico (CNPq) and Coordena??o de Aperfei?oamento de Pessoal de N?vel Superior (CAPES) are acknowledged for partial financial support.