Giuseppe Gaeta, Francesco Spadaro
We provide a symmetry classification of scalar stochastic equations with multiplicative noise. These equations can be integrated by means of the Kozlov procedure, by passing to symmetry adapted variables.
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A Resolution of the Ito-Stratonovich Debate in Quantum Stochastic Processes
本文提出了一种量子噪声同质化方案，解决了乘法有色噪声驱动的量子随机过程中Ito-Stratonovich约定的歧义问题，通过重整化系数和修正项实现一致的马尔可夫极限。
Recent experiments on Brownian colloidal particles have been studied theoretically in terms of overdamped Langevin equations with multiplicative white noise using an unconventional stochastic interpretation. Complementary numerical simulations of the same system are well described using the conventional Stratonovich interpretation. Here we address this dichotomy from a more generic starting point: the underdamped Langevin equation and its corresponding Fokker--Planck equation.
Symmetry classification of scalar autonomous Ito stochastic differential equations with simple noise
本文分类了具有简单噪声的标量自治伊藤随机微分方程的对称性，并利用Kozlov替换对标准形式方程进行积分，扩展了先前分类以包含标准随机对称性。
Integrable Ito equations and properties of the associated Fokker-Planck equations
本文分类了具有标准对称性且可通过Kozlov代换直接积分的标量Ito方程，并研究了与这类对称Ito方程相关的扩散（Fokker-Planck）方程的性质。
Intrinsically noisy mechanisms drive most physical, biological and economic phenomena, from stock pricing to phenotypic variability. Frequently, the system's state influences the driving noise intensity, as, for example, the actual value of a commodity may alter its volatility or the concentration of gene products may regulate their expression. All these phenomena are often modeled using stochastic differential equations (SDEs). However, an SDE is not sufficient to fully describe a noisy system with a multiplicative feedback, because it can be interpreted according to various conventions -- in…
Path Integral for Multiplicative Noise: Generalized Fokker-Planck Equation and Entropy Production Rate in Stochastic Processes With Threshold
本文扩展路径积分形式以建模任意乘性噪声的随机过程，导出了广义 Itô 扩散过程的 Fokker-Planck 方程，并首次定义了几何 Levy α-稳定飞行，通过模拟带阈值的随机过程分析了其概率密度与熵产生率。
Integrable Ito equations with multiple noises
本文研究具有两个独立噪声源的自治Ito方程，确定所有允许标准对称性的方程并利用Kozlov构造积分，同时证明三个及以上噪声源时无标准对称性。
Covariance of the resulting probabilities requires the "anti-Ito" sense. The corresponding Fokker-Planck equation is simplified and preserves important features of the case with a constant diffusion. Multiplicative noise can always be removed by a change of the variables, which is specified explicitly.
Infinite ergodicity for geometric Brownian motion
本文研究几何布朗运动及其推广的渐近分布特性，通过无限遍历性方法探讨其在不同离散化参数下的概率分布行为。
On the integration of Ito equations with a random or a W-symmetry
本文澄清了随机标准对称性和W-对称性在通过Kozlov替换积分Ito方程时的理论细节，指出前者可基本无碍扩展，后者则需谨慎且不能用于随机方程积分。
The kernel of the $\mathbb{Z}_2^{n}$ Dirac-Dunkl operator is examined. The symmetry algebra $\mathcal{A}_{n}$ of the associated Dirac-Dunkl equation on $\mathbb{S}^{n-1}$ is determined and is seen to correspond to a higher rank generalization of the Bannai-Ito algebra. A basis for the polynomial null-solutions of the Dirac-Dunkl operator is constructed. The basis elements are joint eigenfunctions of a maximal commutative subalgebra of $\mathcal{A}_{n}$ and are given explicitly in terms of Jacobi polynomials. The symmetry algebra is shown to act irreducibly on this basis via raising/lowering op…
We study the ergodicity of finite-dimensional approximations of the Schrödinger equation. The system is driven by a multiplicative scalar noise. Under general assumptions over the distribution of the noise, we show that the system has a unique stationary measure $μ$ on the unit sphere $S$ in $\C^n$, and $μ$ is absolutely continuous with respect to the Riemannian volume on $S$. Moreover, for any initial condition in $S$, the solution converges exponentially fast to the measure $μ$ in the variational norm.
Singularities in d-dimensional Langevin equations with anisotropic multiplicative noise and lack of self-adjointness in the corresponding Schrodinger equation
本文研究具有各向异性乘性噪声和径向幂律漂移的d维Langevin方程在原点附近的奇异性行为，通过径向-角向解耦、Feller–Van Kampen分类、指数搜寻方法以及与奇异势Schrödinger方程的映射，建立了随机过程边界分类与量子自伴性问题之间的联系。
This paper derives stochastic partial differential equations (SPDEs) for fluid dynamics from a stochastic variational principle (SVP). The Legendre transform of the Lagrangian formulation of these SPDEs yields their Lie-Poisson Hamiltonian form. The paper proceeds by: taking variations in the SVP to derive stochastic Stratonovich fluid equations; writing their Itô representation; and then investigating the properties of these stochastic fluid models in comparison with each other, and with the corresponding deterministic fluid models. The circulation properties of the stochastic Stratonovich fl…
The definition and manipulation of Langevin equations with multiplicative white noise require special care (one has to specify the time discretisation and a stochastic chain rule has to be used to perform changes of variables). While discretisation-scheme transformations and non-linear changes of variable can be safely performed on the Langevin equation, these same transformations lead to inconsistencies in its path-integral representation. We identify their origin and we show how to extend the well-known Itō prescription ($dB^2=dt$) in a way that defines a modified stochastic calculus to be u…
Probability of Transition to Turbulence in a Reduced Stochastic Model of Pipe Flow
本文通过估计亚稳态转变概率，研究不同噪声结构下管道流湍流起始现象，利用层流态附近含乘性噪声的线性化模型建立转变概率下界，并扩展至多种噪声类型，为理解湍流起始及应用 SPDE 理论提供了方法论。
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