E. Lytvynov
We review some recent developments in white noise analysis and quantum probability. We pay a special attention to spaces of test and generalized functionals of some Lévy white noises, as well as as to the structure of quantum white noise on these spaces.
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The Clark-Ocone formula in the theory of discrete-time chaotic calculus holds only for square integrable functionals of discrete-time normal noises. In this paper, we aim at extending this formula to generalized functionals of discrete-time normal noises. Let $Z$ be a discrete-time normal noise that has the chaotic representation property. We first prove a result concerning the regularity of generalized functionals of $Z$. Then, we use the Fock transform to define some fundamental operators on generalized functionals of $Z$, and apply the above mentioned regularity result to prove the continui…
In this article we consider multiplicative operator-valued white noise functionals related to a stochastic flow. A generalization of the Krylov-Veretennikov expansion is presented. An analog of such expansion for the Arratia flow is derived.
Functional central limit theorem for superdiffusive SDEs with stable noise
本文研究了由α-稳定过程驱动的超扩散随机微分方程的功能稳定中心极限定理。
In this paper we establish a framework for normal approximation for white noise functionals by Stein's method and Hida calculus. Our work is inspired by that of Nourdin and Peccati (Probab. Theory Relat. Fields 145, 75-118, 2009), who combined Stein's method and Malliavin calculus for normal approximation for functionals of Gaussian processes.
Existence, uniqueness and non-explosion of the mild solution are proved for a class of semi-linear functional SPDEs with multiplicative noise and Dini continuous drifts. In the finite-dimensional and bounded time delay setting, the log-Harnack inequality and $L^2$-gradient estimate are derived. As the Markov semigroup is associated to the functional (segment) solution of the equation, one needs to make analysis on the path space of the solution in the time interval of delay.
By constructing a new coupling, the log-Harnack inequality is established for the functional solution of a delay stochastic differential equation with multiplicative noise. As applications, the strong Feller property and heat kernel estimates w.r.t. quasi-invariant probability measures are derived for the associated transition semigroup of the solution. The dimension-free Harnack inequality in the sense of \cite{W97} is also investigated.
A functional Breuer-Major theorem with Poisson noise
本文将针对Gaussians的泛函Breuer-Major定理扩展至Poisson情形，利用Poisson点过程的$L^p$谱隙不等式证明源自该过程的平稳序列的紧性。
By a general shot noise process we mean a shot noise process in which the counting process of shots is arbitrary locally finite. Assuming that the counting process of shots satisfies a functional limit theorem in the Skorokhod space with a locally Hölder continuous Gaussian limit process and that the response function is regularly varying at infinity we prove that the corresponding general shot noise process satisfies a similar functional limit theorem with a different limit process and different normalization and centering functions. For instance, if the limit process for the counting process…
This work is devoted to deriving the Onsager-Machlup action functional for stochastic partial differential equations with (non-Gaussian) Levy process as well as Gaussian Brownian motion. This is achieved by applying the Girsanov transformation for probability measures and then by a path representation. This enables the investigation of the most probable transition path for infinite dimensional stochastic dynamical systems modeled by stochastic partial differential equations, by minimizing the Onsager-Machlup action functional.
We consider the solution of a stochastic differential equation with additive multidimensional fractional noise. In the case $\frac14<H<\frac12$, we compute the Onsager-Machlup functional (with respect to the driving fractional Brownian motion) for the supremum norm and the Hölder norms with exponent $α\in \left(0,H-\frac14\right)$ for any element of the Cameron-Martin space $\mathcal H_H$, extending a previous result of Moret and Nualart. In the more general case $H<\frac12$ and $α\in \left(0,H\right)$, we formulate a condition on $h\in\mathcal H_H$ under which the computation of the Onsager-M…
We prove a functional non-central limit theorem for jump-diffusions with periodic coefficients driven by strictly stable Levy-processes with stability index bigger than one. The limit process turns out to be a strictly stable Levy process with an averaged jump-measure. Unlike in the situation where the diffusion is driven by Brownian motion, there is no drift related enhancement of diffusivity.
This article presents a complete second order theory for a large class of geometric functionals on homogeneous Poisson input. In particular, the results don't require the existence of a radius of stabilisation. Hence they can be applied to geometric functionals of spatial shot-noise fields excursions such as volume, perimeter, or Euler characteristic (the method still applies to stabilising functionals). More generally, it must be checked that a local contribution to the functional is not strongly affected under a perturbation of the input far away. In this case the exact asymptotic variance i…
Geometric functionals of polyconvex excursion sets of Poisson shot noise processes
针对一类游程集几乎必然多凸的 Poisson 散粒噪声过程（含 Boolean 模型），研究增长窗口下内在体积等几何泛函的期望、方差渐近及中心极限定理。
We consider, for a diffusion process in R^n, the Gallavotti-Cohen functional, defined as the empirical power dissipated in a time interval by the non-conservative part of the drift. We prove a large deviation principle in the limit in which the noise vanishes and the time interval diverges. The corresponding rate functional, which satisfies the fluctuation theorem, is expressed in terms of a variational problem on the classical Freidlin-Wentzell functional.
Functional inequalities for a family of infinite-dimensional diffusions with degenerate noise
本文通过在有限维投影上构造修正的 $\Gamma$ 演算，为退化噪声无穷维扩散建立可提升至无穷维的显式函数不等式，并用于分析准不变性与平稳分布唯一性。
This paper is devoted to the study of the properties of entropy as a function of the Hurst index, which corresponds to the fractional Gaussian noise. Since the entropy of the Gaussian vector depends on the determinant of the covariance matrix, and the behavior of this determinant as a function of the Hurst index is rather difficult to study analytically at high dimensions, we also consider simple alternative entropy functionals, whose behavior, on the one hand, mimics the behavior of entropy and, on the other hand, is not difficult to study. Asymptotic behavior of the normalized entropy (so ca…
We consider renewal shot noise processes with response functions which are eventually nondecreasing and regularly varying at infinity. We prove weak convergence of renewal shot noise processes, properly normalized and centered, in the space $D[0,\infty)$ under the $J_1$ or $M_1$ topology. The limiting processes are either spectrally nonpositive stable Lévy processes, including the Brownian motion, or inverse stable subordinators (when the response function is slowly varying), or fractionally integrated stable processes or fractionally integrated inverse stable subordinators (when the index of …
A Small-Noise Analysis of Controlled Functional Differential Equations with Gaussian Noise
研究由高斯噪声驱动的受控泛函微分方程的小噪声渐近性，通过布埃 - 迪皮伊变分表示等方法，证明小噪声对数价值函数的博弈论上下界，给出博弈有值条件，得到完整小噪声拉普拉斯原理。
In this paper, we consider a system of forward-backward stochastic differential equations (FBSDEs) with monotone functionals. We show the existence and uniqueness of such a system by the method of continuation similarly to Peng and Wu (1999) for classical FBSDEs and obtain estimates under conditional probability. As applications, we prove the well-posedness result for a mean field FBSDE with conditional law and show the existence of a decoupling function. In addition, we show that mean field games with common noise are uniquely solvable under a linear controlled process with convex and weak-mo…
We present an innovating sensitivity analysis for stochastic differential equations: We study the sensitivity, when the Hurst parameter~$H$ of the driving fractional Brownian motion tends to the pure Brownian value, of probability distributions of smooth functionals of the trajectories of the solutions $\{X^H_t\}_{t\in \mathbb{R}_+}$ and of the Laplace transform of the first passage time of $X^H$ at a given threshold. Our technique requires to extend already known Gaussian estimates on the density of $X^H_t$ to estimates with constants which are uniform w.r.t. $t$ in in the whole half-line $\R…
Onsager-Machlup functional for stochastic lattice dynamical systems driven by time-varying noise
该研究将Onsager-Machlup泛函推广到时变噪声驱动的无穷维随机格点动力系统，证明了解的存在唯一性，通过多种随机分析技术推导了泛函形式，并以数值例子验证了理论结论。
In this paper, we establish concentration inequalities both for functionals of the whole solution on an interval [0, T ] of an additive SDE driven by a fractional Brownian motion with Hurst parameter H $\in$ (0, 1) and for functionals of discrete-time observations of this process. Then, we apply this general result to specific functionals related to discrete and continuous-time occupation measures of the process.
Onsager--Machlup Functional for Fractional Stochastic Newton Dynamics with Time-Dependent Noise Intensities
本文推导了带有时间依赖噪声强度的分数随机牛顿动力学的奥萨格-马赫卢普函数，并通过数值模拟验证其适用性。
Onsager-Machlup functional for stochastic differential equations with time-varying noise
本文研究了具有α-Hölder时变噪声的随机微分方程的Onsager-Machlup泛函，通过引入新可测范数和Girsanov变换推导出该泛函，并数值验证了其在多尺度随机波动率模型中的应用。
Onsager--Machlup Functionals for Generalized Newtonian Equations of Motion with Time-Varying Fractional Noise
研究一类由分数布朗运动驱动的退化随机微分方程的昂萨格-马赫卢普泛函，结合高斯相关不等式等方法控制耦合项，经正则性估计等得到泛函表达式及相关应用。
By means of white noise analysis, we prove some limit theorems for nonlinear functionals of a given Volterra process. In particular, our results apply to fractional Brownian motion (fBm) and should be compared with the classical convergence results of the 1980s due to Breuer, Dobrushin, Giraitis, Major, Surgailis and Taqqu, as well as the recent advances concerning the construction of a Lévy area for fBm due to Coutin, Qian and Unterberger.
In this work we present expansions of intersection local times of fractional Brownian motions in $\R^d$, for any dimension $d\geq 1$, with arbitrary Hurst coefficients in $(0,1)^d$. The expansions are in terms of Wick powers of white noises (corresponding to multiple Wiener integrals), being well-defined in the sense of generalized white noise functionals. As an application of our approach, a sufficient condition on $d$ for the existence of intersection local times in $L^2$ is derived, extending the results of D. Nualart and S. Ortiz-Latorre in "Intersection Local Time for Two Independent Frac…
SPDEs driven by standard symmetric $α$-stable cylindrical Lévy processes: existence, Lyapunov functionals and Itô formula
本文研究希尔伯特空间中标准对称$\alpha$-稳定柱形噪声驱动的随机偏微分方程，证明温和解存在性、给出解的长期正则性与Itô公式，针对柱形噪声特性发展了新的论证方法。
The lifted functional approach to mean field games with common noise
本文提出一种新的逐路径方法处理具有共同噪声的平均场博弈，在路径层面恢复双重性，显式求解了线性二次数据等困难例子并建立分离原理，同时澄清了路径相关PDE中正则解的概念。
This paper establishes a functional law of large numbers and a functional central limit theorem for marked Hawkes point measures and their corresponding shot noise processes. We prove that the normalized random measure can be approximated in distribution by the sum of a Gaussian white noise process plus an appropriate lifting map of a correlated one-dimensional Brownian motion. The Brownian results from the self-exiting arrivals of events. We apply our limit theorems for Hawkes point measures to analyze the population dynamics of budding microbes in a host.
We aim to generalize the homogenisation theorem in \cite{Gehringer-Li-tagged} for a passive tracer interacting with a fractional Gaußian noise to also cover fractional non-Gaußian noises. To do so we analyse limit theorems for normalized functionals of Hermite-Volterra processes, extending the result in \cite{Diu-Tran} to power series with fast decaying coefficients. We obtain either convergence to a Wiener process, in the short-range dependent case, or to a Hermite process, in the long-range dependent case. Furthermore, we prove convergence in the multivariate case with both, short and long-r…
Breakdown of Adiabatic Scaling and Noise-induced Functional Synchronization in Deeply Quiescent Excitable Systems
研究探讨了在静息生物介质中噪声诱导的正则性评估受能量景观平坦化和非线性动力学影响，提出对数质心提取法以恢复亚阿迪abatic克拉默斯缩放，并识别强噪声极限下亚阿迪abatic近似的物理边界，发现噪声诱导从亚阈值生理震颤到宏观功能同步的转变。
The existence and uniqueness of a global positive solution is proven for the system of stochastic differential equations describing a nonautonomous stochastic density dependent predator-prey model with Holling-type II functional response disturbed by white noise, centered and non-centered Poisson noises. Sufficient conditions are obtained for stochastic ultimate boundedness, stochastic permanence, non-persistence in the mean, weak persistence in the mean and extinction of a population densities in the considered stochastic predator-prey model.
We consider a linear stochastic differential equation with stochastic drift and multiplicative noise. We study the problem of approximating its solution with the process that solves the equation where the possibly stochastic drift is replaced by a deterministic function. To do this, we use a combination of deterministic Pontryagin's maximum principle approach and direct methods of calculus of variations. We find necessary and sufficient conditions for a function $u \in L^1(0,T)$ to be a minimizer of a certain cost functional. To overcome the problem of the existence of such minimizer, we also …
Letting~$N=\left\{N(t), t\geq0\right\}$ be a standard Poisson process, Stroock~ \cite{Stroock-1981} constructed a family of continuous processes by $$Θ_ε(t)=\int_0^tθ_ε(r)dr, \ \ \ \ \ 0 \le t \le 1,$$ where $θ_ε(r)=\frac{1}ε(-1)^{N(ε^{-2}r)}$, and proved that it weakly converges to a standard Brownian motion under the continuous function topology. We establish the functional large deviations principle (LDP) for the approximations of a class of Gaussian processes constructed by integrals over $Θ_ε(t)$, and find the explicit form for rate function. As an application, we consider the following (…
A spectral-compensated scheme for space-parameter Poisson noise functionals: error bounds and complexity estimates
本文针对空间参数泊松噪声泛函提出谱补偿方案，通过替换小振幅噪声提升误差界、降低计算复杂度，经数值实验验证了截断收敛速率的理论预测。
Characterization of the set of zero-noise limits measures of perturbed cellular automata
该研究刻画了扰动元胞自动机的零噪声极限测度集ℳ₀^l的拓扑与可计算性质，证明其紧连通性及Π₃/Π₂可计算性，反向构造满足条件的元胞自动机，并分析了噪声偏置对极限测度集的影响。
The problem of optimal estimation of linear functionals constructed from the unobserved values of a stochastic sequence with periodically stationary increments based on observations of the sequence with stationary noise is considered. For sequences with known spectral densities, we obtain formulas for calculating values of the mean square errors and the spectral characteristics of the optimal estimates of the functionals. Formulas that determine the least favorable spectral densities and the minimax-robust spectral characteristics of the optimal linear estimates of functionals are proposed in …
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