Moreno Miguel Vera, Arenas Zochil González, Barci Daniel G
Departamento de Física Teórica, Universidade do Estado do Rio de Janeiro, Rua São Francisco Xavier 524, 20550-013 Rio de Janeiro, Rio de Janeiro, Brazil.
Phys Rev E Stat Nonlin Soft Matter Phys. 2015 Apr;91(4):042103. doi: 10.1103/PhysRevE.91.042103. Epub 2015 Apr 6.
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.
我们讨论由具有乘性白噪声的朗之万方程组驱动的一般多维随机过程。特别地，我们解决了时间反转扩散过程如何受到处理随机积分的各种约定影响的问题。我们提出一种泛函形式，无需在任何中间步骤对朗之万方程进行任何类型的离散化，即可构建关联函数的生成泛函。生成泛函的特征在于对两组对易变量以及格拉斯曼变量进行泛函积分。在此表示中，时间反转变换在扩展变量中成为线性变换，从而简化了由处方混合和相关微积分规则引入的复杂性。随机微积分在我们的形式体系中由格拉斯曼代数的结构进行编码。我们研究了一些例子，如高阶导数朗之万方程和微磁随机朗道 - 里夫希茨 - 吉尔伯特方程的泛函表示。
