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

Area of Science:

• Physics
• Mathematics
• Statistical Mechanics

Background:

• Stochastic processes are fundamental in modeling complex systems.
• Langevin equations with multiplicative white noise describe many physical phenomena.
• Time reversal of diffusion processes is complicated by various stochastic integral conventions.

Purpose of the Study:

• To develop a unified functional formalism for multidimensional stochastic processes.
• To address the complexities of time reversal in diffusion processes under different stochastic calculus rules.
• To simplify the analysis of stochastic differential equations with multiplicative noise.

Main Methods:

• A functional formalism is presented to construct the generating functional of correlation functions.
• The method avoids intermediate discretization of Langevin equations.
• It utilizes functional integration over commuting and Grassmann variables, codifying stochastic calculus within Grassmann algebra.

Main Results:

• Time reversal transformations become linear in an extended variable space.
• This simplifies the complexity arising from mixed conventions and calculus rules.
• The formalism is demonstrated with examples like higher-order derivative Langevin equations and the Landau-Lifshitz-Gilbert equation.

Conclusions:

• The developed functional formalism offers a powerful and simplified approach to analyzing stochastic processes.
• It provides a consistent framework for handling time reversal in diffusion processes.
• The method has broad applicability, including in micromagnetics and other complex systems.