E. Lytvynov
The paper is devoted to construction and investigation of some riggings of the $L^2$-space of Poisson white noise. A particular attention is paid to the existence of a continuous version of a function from a test space, and to the property of an algebraic structure under pointwise multiplication of functions from a test space.
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The priority of Liming Wu for the Poincaré inequality for birth-death processes on a Poisson space is announced.
Given a set of independent Poisson random variables with common mean, we study the distribution of their maximum and obtain an accurate asymptotic formula to locate the most probable value of the maximum. We verify our analytic results with very precise numerical computations.
Randomly scaled scale-decorated Poisson point process is introduced recently in Bhattacharya et al. [2017] where it appeared as weak limit of a sequence of point processes in the context of branching random walk. In this article, we obtain a characterization for these processes based on scaled-Laplace functional. As a consequence, we obtain a characterization for strictly $α$-stable point process (also known as scale-decorated Poisson point process) based on scaled-Laplace functional . a connection with randomly shifted decorated Poisson point process is obtained. The tools and approach used e…
We derive the class of normalized generalized Gamma processes from Poisson-Kingman models (Pitman, 2003) with tempered alfa-stable mixing distribution. Relying on this construction it can be shown that in Bayesian nonparametrics, results on quantities of statistical interest under those priors, like the analogous of the Blackwell-MacQueen prediction rules or the distribution of the number of distinct elements observed in a sample, arise as immediate consequences of Pitman's results.
The main goal of this paper is to determine the Poisson boundary of lamplighter random walks over a general class of discrete groups $Γ$ endowed with a rich boundary. The starting point is the Strip Criterion of identification of the Poisson boundary for random walks on discrete groups due to Kaimanovich. A geometrical method for constructing the strip as a subset of the lamplighter group starting with a smaller strip in the base group $Γ$ is developed. Then, this method is applied to several classes of base groups $Γ$: groups with infinitely many ends, hyperbolic groups in the sense of Gromov…
These are lecture notes on the subject defined in the title. As such, they do not pretend to be really new, probably except for the only section about Poisson equations with potentials. Yet, the hope of the author is that they may serve as a bridge to the important area of Poisson equations "in the whole space" and with a parameter, the latter theme not being presented here. Why this area is so important was explained in many papers and books (see the references [12, 34, 35]): it provides one of the main tools in diffusion approximation in the area stochastic averaging. Hence, the aim of these…
Let $Ξ\subset\mathbb R^d$ be a set of centers chosen according to a Poisson point process in $\mathbb R^d$. Let $ψ$ be an allocation of $\mathbb R^d$ to $Ξ$ in the sense of the Gale-Shapley marriage problem, with the additional feature that every center $ξ\inΞ$ has an appetite given by a nonnegative random variable $α$. Generalizing some previous results, we study large deviations for the distance of a typical point $x\in\mathbb R^d$ to its center $ψ(x)\inΞ$, subject to some restrictions on the moments of $α$.
A note on mixed Poisson distributions
本文讨论混合泊松分布的额外性质，指出尺度参数趋于无穷时其收敛于混合分布，并通过混合随机变量中心化得到中心极限定理及矩收敛结果。
Multiparameter Lévy white noise theory and applications
该研究构建了多参数Lévy sheet及其补偿Poisson随机测度的白噪声理论与演算体系，可应用于受Lévy噪声影响的随机偏微分方程研究。
The goal of this short note is to establish, in complete generality, the representation for the carré du champ operator associated with the Ornstein-Uhlenbeck semi-group on the Poisson space in terms of the add-one and drop-one operators.
In this note we give a proof for the result stated as Theorem 4 in Poisson Approximation in a Poisson Limit Theorem Inspired by Coupon Collecting.
Let $u = \{u(t, x); (t,x)\in \mathbb R_+\times \mathbb R\}$ be the solution to a linear stochastic heat equation driven by a Gaussian noise, which is a Brownian motion in time and a fractional Brownian motion in space with Hurst parameter $H\in(0, 1)$. For any given $x\in \mathbb R$ (resp. $t\in \mathbb R_+$), we show a decomposition of the stochastic process $t\mapsto u(t,x)$ (resp. $x\mapsto u(t,x)$) as the sum of a fractional Brownian motion with Hurst parameter $H/2$ (resp. $H$) and a stochastic process with $C^{\infty}$-continuous trajectories. Some applications of those decompositions ar…
This note is based on Wonham \cite{Wonham}. The differences between this note and [Wonham] are discussed in Section VIII.
Spatial asymptotic behaviors of fractional stochastic heat equations driven by additive Lévy white noise
该研究针对加性Lévy白噪声驱动的分数阶随机热方程，建立了其空间渐近行为的显式积分检验准则，证明轻尾噪声下方程在全维度具有加性物理间歇性质，证明依赖分数阶Laplacian热核估计与泊松泛函精确尾部行为。
In this article, we consider a stochastic partial differential equation (SPDE) driven by a Lévy white noise, with Lipschitz multiplicative term $σ$. We prove that under some conditions, this equation has a unique random field solution. These conditions are verified by the stochastic heat and wave equations. We introduce the basic elements of Malliavin calculus with respect to the compensated Poisson random measure associated with the Lévy white noise. If $σ$ is affine, we prove that the solution is Malliavin differentiable and its Malliavin derivative satisfies a stochastic integral equation.
The aim of this note is to describe the Poisson boundary of the group of invertible triangular matrices with coefficients in a number field. It generalizes to any dimension and to any number field a result of Brofferio concerning the Poisson boundary of random rational affinities.
First double mode of the Poisson distribution of order $k$
本文研究k阶泊松分布在k≥2时的众数行为，确定了首次出现双峰（众数为0和m>0）的最小速率参数λ，并对哪些正整数不能成为众数给出了近乎完整的刻画。
The sample paths of white noise are proved to be elements of certain Besov spaces with dominating mixed smoothness. Unlike in isotropic spaces, here the regularity does not get worse with increasing space dimension. Consequently, white noise is actually much smoother than the known sharp regularity results in isotropic spaces suggest. An application of our techniques yields new results for the regularity of solutions of Poisson and heat equation on the half space with boundary noise. The main novelty is the flexible treatment of the interplay between the singularity at the boundary and the smo…
We study the notions of mild solution and generalized solution to a linear stochastic partial differential equation driven by a pure jump symmetric Lévy white noise. We identify conditions for existence for these two kinds of solutions, and we identify conditions under which they are essentially equivalent. We establish a necessary condition for the existence of a random field solution to a linear SPDE, and we apply this result to the linear stochastic heat, wave and Poisson equations driven by a symmetric $α$-stable noise.
It is shown that for any positive integer k and positive parameter lambda less than 2/k(k+1), the Poisson distribution of order k with parameter lambda has a unique mode, 0. In addition, the Poisson distribution of order 2 has a unique mode, 0, if lambda is less than -1 plus the square root of 3. It has two modes, 0 and 2, if lambda is equal to -1 plus the square root of 3, and it has a unique mode, 2, if lambda is greater than -1 plus the square root of 3 and less than 1.
In this paper, we consider a quasi-linear stochastic heat equation on $[0,1]$, with Dirichlet boundary conditions and controlled by the space-time white noise. We formally replace the random perturbation by a family of noisy inputs depending on a parameter $n\in \mathbb{N}$ such that approximate the white noise in some sense. Then, we provide sufficient conditions ensuring that the real-valued {\it mild} solution of the SPDE perturbed by this family of noises converges in law, in the space $\mathcal{C}([0,T]\times [0,1])$ of continuous functions, to the solution of the white noise driven SPDE.…
In this article, we establish a probabilistic representation for the second-order moment of the solution of stochastic heat equation in $[0,1] \times \bR^d$, with multiplicative noise, which is fractional in time and colored in space. This representation is similar to the one given in Dalang, Mueller and Tribe (2008) in the case of an s.p.d.e. driven by a Gaussian noise, which is white in time. Unlike the formula of Dalang, Mueller and Tribe (2008), which is based on the usual Poisson process, our representation is based on the planar Poisson process, due to the fractional component of the noi…
This note answers one question in [math.PR/0505668], concerning the connected allocation for the Poisson process in R^2. The proposed solution makes use of the Riemann map from the plane minus the minimal spanning forest of the Poisson point process to the halfplane. A picture of a numerically simulated example is included.
It is possible to construct Lévy white noises as generalized random processes in the sense of Gel'fand and Vilenkin, or as an independently scattered random measures introduced by Rajput and Rosinski. In this article, we unify those two approaches by extending the Lévy white noise, defined as a generalized random process, to an independently scattered random measure. We are then able to give general integrability conditions for Lévy white noises, thereby maximally enlarging their domain of definition. Based on this connection, we provide new criteria for the practical determination of the doma…
This note is motivated by connections between the online and offline problems of selecting a possibly long subsequence from a Poisson-paced sequence of uniform marks under either a monotonicity or a sum constraint. The offline problem with the sum constraint amounts to counting the Poisson arrivals before their total exceeds a certain level. A precise asymptotics for the mean count is obtained by coupling with a nonlinear pure birth process.
In this paper we study the Poisson and heat equations on bounded and unbounded domains with smooth boundary with random Dirichlet boundary conditions. The main novelty of this work is a convenient framework for the analysis of such equations excited by the white in time and/or space noise on the boundary. Our approach allows us to show the existence and uniqueness of weak solutions in the space of distributions. Then we prove that the solutions can be identified as smooth functions inside the domain, and finally the rate of their blow up at the boundary is estimated. A large class of noises in…
In this paper we develop a white noise framework for the study of stochastic partial differential equations driven by a d-parameter (pure jump) Levy white noise. As an example we use this theory to solve the stochastic Poisson equation with respect to Levy white noise for any dimension d. The solution is a stochastic distribution process given explicitly. We also show that if d\leq 3, then this solution can be represented as a classical random field in L2(μ), where μis the probability law of the Levy process. The starting point of our theory is a chaos expansion in terms of generalized Charlie…
In this paper we treat semilinear stochastic partial differential equations by two methods. First, we extend the framework of [BDR10] from a Hilbert space to a Gelfand triple and as an application we prove the existence of solutions for the Fokker-Planck equations associated to semilinear equations with space-time white noise and both with polynomially growing nonlinearities and Burgers type nonlinearities at the same time. Second we adopt the approximation technique from [BDR10] to obtain existence of unique strong solutions to semilinear stochastic partial differential equations driven by sp…
Factorial moments of the Poisson distribution of order $k$
本文针对$k$阶Poisson分布，推导了其阶乘矩的显式组合和，并阐明了当$k>1$时阶乘矩结构的复杂性质。
We characterize the local smoothness and the asymptotic growth rate of the Lévy white noise. We do so by characterizing the weighted Besov spaces in which it is located. We extend known results in two ways. First, we obtain new bounds for the local smoothness via the Blumenthal-Getoor indices of the Lévy white noise. We also deduce the critical local smoothness when the two indices coincide, which is true for symmetric-alpha-stable, compound Poisson, and symmetric-gamma white noises to name a few. Second, we express the critical asymptotic growth rate in terms of the moment properties of the L…
The paper contains a basic course on classical Risk Theory for a compound Poisson process. It is based on probabilistic proofs using the method of the "Ballot Theorem" introduced by Tackas. This provides elegant and direct proofs. Also large deviation methods are used to study the distribution of the time to ruin.
The distributions of convex body functionals of the Poisson polyhedron
本文研究Poisson多面体及Crofton多面体体积、表面积和面积的分布，提出混合分布模型并引入广义伽马分布，通过模拟估计参数，首次精确给出面数概率。
A simple axiomatic characterization of the general (infinite dimensional, noncommutative) Ito algebra is given and a pseudo-Euclidean fundamental representation for such algebra is described. The notion of Ito B*-algebra, generalizing the C*-algebra is defined to include the Banach infinite dimensional Ito algebras of quantum Brownian and quantum Levy motion, and the B*-algebras of vacuum and thermal quantum noise are characterized. It is proved that every Ito algebra is canonically decomposed into the orthogonal sum of quantum Brownian (Wiener) algebra and quantum Levy (Poisson) algebra. In p…
The main purpose of this work is to prove Theorem 4.4, so-called, the characterization theorem of Hida measures (generalized measures). As examples of such measures, we shall present the Poisson noise measure and the Grey noise measure in Example 4.5 and 4.6, respectively.
A note on Tricomi-type partial differential equations with white noise initial condition
本文推广了 Tricomi 型偏微分方程解的表示公式，发现在高斯白噪声初始条件下引入低阶项会破坏随机场解的良定义性，但对于 Hurst 参数为 $H\in(1/2,1)$ 的分数高斯白噪声，适定性可恢复。
On the local linearization of the one-dimensional stochastic wave equation with a multiplicative space-time white noise forcing
针对乘性时空白噪声驱动的一维随机波动方程，研究建立了其双参数线性局部化结论，为该类方程的局部性质分析提供支撑。
Lecture notes for a master-level applied mathematics course on stochastic processes and applications, held at the University of Orléans, France. Contents: Markov chains, Poisson point processes, Markovian jump processes, queueing theory. ----- Notes d'un cours de Master 2 professionnel en mathématiques appliquées, donné à l'Université d'Orléans. Contenu: Chaînes de Markov, processus ponctuel de Poisson, files d'attente.
Nonlinear random vibration under excitations of both Gaussian and Poisson white noises is considered. The model is based on stochastic differential equations, and the corresponding stochastic integrals are defined in such a way that the energy conservation law is satisfied. It is shown that Stratonovich integral and Di Paola-Falsone integral should be used for excitations of Gaussian and Poisson white noises, respectively, in order for the model to satisfy the underlining physical laws (e.g., energy conservation). Numerical examples are presented to illustrate the theoretical results.
A note on shifting distributions via Poisson races
该注记针对从提议分布$P$模拟目标分布$Q$的问题，基于泊松竞赛方法给出了模拟器$I$的期望对数代价的上下界，完善了分布转移模拟的相关理论。
Fokker-Planck equations describe time evolution of probability densities of stochastic dynamical systems and play an important role in quantifying propagation and evolution of uncertainty. Although Fokker-Planck equations can be written explicitly for nonlinear dynamical systems excited by Gaussian white noise, they are not available in general for nonlinear dynamical systems excited by multiplicative non-Gaussian white noise. Marcus stochastic differential equations are often appropriate models in engineering and physics for stochastic dynamical systems excited by non-Gaussian white noise. In…
The theory of sparse stochastic processes offers a broad class of statistical models to study signals. In this framework, signals are represented as realizations of random processes that are solution of linear stochastic differential equations driven by white Lévy noises. Among these processes, generalized Poisson processes based on compound-Poisson noises admit an interpretation as random L-splines with random knots and weights. We demonstrate that every generalized Lévy process-from Gaussian to sparse-can be understood as the limit in law of a sequence of generalized Poisson processes. This …
Sub-Poissonian estimates for exponential moments of additive functionals over pairs of particles with respect to determinantal and symplectic Pfaffian point processes governed by entire functions
本文研究了由整体函数支配的行列式和辛普费芬点过程中粒子对加法函数的指数矩的亚泊松估计。
In this short note we capitalize on and complete our previous results on the regularity of the homogenized coefficients for Bernoulli perturbations by addressing the case of the Poisson point process, for which the crucial uniform local finiteness assumption fails. In particular, we strengthen the qualitative regularity result first obtained in this setting by the first author to Gevrey regularity of order~2. The new ingredient is the independence of Poisson point processes, in a form recently used by Giunti, Gu, Mourrat, and Nitzschner.
In this note we consider a class of neutral stochastic functional differential equations with finite delay driven simultaneously by a fractional Brownian motion and a Poisson point processes in a Hilbert space. We prove an existence and uniqueness result and we establish some conditions ensuring the exponential decay to zero in mean square for the mild solution by means of the Banach fixed point principle.
A quantitative central limit theorem for Poisson horospheres in high dimensions
本研究针对d维双曲空间中平稳超球面泊松过程，分析其在半径R渐增的球中诱导的总表面积，证明了当R与d同时趋于无穷时该随机变量满足定量非标准中心极限定理。
In this note we study the optimal dividend problem for a company whose surplus process, in the absence of dividend payments, evolves as a generalized compound Poisson model in which the counting process is a generalized Poisson process. This model including the classical risk model and the Polya-Aeppli risk model as special cases. The objective is to find a dividend policy so as to maximize the expected discounted value of dividends which are paid to the shareholders until the company is ruined. We show that under some conditions the optimal dividend strategy is formed by a barrier strategy.
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