Andrea Posilicano, Tusheng Zhang
We prove convergence of symmetric diffusions on Wiener spaces by using stopping times arguments and capacity techniques. The drifts of the diffusions can be singular, we require the densities of the processes to be neither bounded from above nor away from zero.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
继续探索
We investigate the optimal rate of convergence in the multidimensional normal approximation of vector-valued Wiener-Ito integrals of which components all belong to the same fixed Wiener chaos. Combining Malliavin calculus, Stein's method for normal approximation and method of cumulants, we obtain the optimal rate of convergence with respect to a suitable smooth distance. As applications, we derive the optimal rates of convergences for complex Wiener-Ito integrals, vector-valued Wiener-Ito integrals with kernels of step functions and vector-valued Toeplitz quadratic functionals.
In [NP09a], Nourdin and Peccati established a neat characterization of Gamma approximation on a fixed Wiener chaos in terms of convergence of only the third and fourth cumulants. In this paper, we investigate the rate of convergence in Gamma approximation on Wiener chaos in terms of the iterated Gamma operators of Malliavin Calculus. On the second Wiener chaos, our upper bound can be further extended to an exact rate of convergence in a suitable probability metric $d_2$ in terms of the maximum of the third and fourth cumulants, analogous to that of normal approximation in [NP15] under one extr…
We prove sufficient conditions, ensuring that a sequence of multiple Wiener-Itô integrals (with respect to a general Gaussian process) converges stably to a mixture of normal distributions. Our key tool is an asymptotic decomposition of contraction kernels, realized by means of increasing families of projection operators. We also use an infinite-dimensional Clark-Ocone formula, as well as a version of the correspondence between "abstract" and "concrete" filtered Wiener spaces, in a spirit similar to Üstünel and Zakai (1997).
We consider sequences of random variables living in a finite sum of Wiener chaoses. We find necessary and sufficient conditions for convergence in law to a target variable living in the sum of the first two Wiener chaoses. Our conditions hold notably for sequences of multiple Wiener integrals. Malliavin calculus and in particular the Gamma-operators are used. Our results extend previous findings by Azmoodeh, Peccati and Poly (2014) and are applied to central and non-central convergence situations. Our methods are applied as well to investigate stable convergence. We finally exclude certain cla…
Let L be the class of limiting laws associated with sequences in the second Wiener chaos. We exhibit a large subset L_0 of L satisfying that, for any F_infinity in L_0, the convergence of only a finite number of cumulants suffices to imply the convergence in law of any sequence in the second Wiener chaos to F_infinity. This result is in the spirit of the seminal paper by Nualart and Peccati, in which the authors discovered the surprising fact that convergence in law for sequences of multiple Wiener-Itô integrals to the Gaussian is equivalent to convergence of just the fourth cumulant. Also, we…
We study the convergence to the multiple Wiener-Itô integral from processes with absolutely continuous paths. More precisely, consider a family of processes, with paths in the Cameron-Martin space, that converges weakly to a standard Brownian motion in $\mathcal C_0([0,T])$. Using these processes, we construct a family that converges weakly, in the sense of the finite dimensional distributions, to the multiple Wiener-Itô integral process of a function $f\in L^2([0,T]^n)$. We prove also the weak convergence in the space $\mathcal C_0([0,T])$ to the second order integral for two important famili…
Superconvergence phenomenon in Wiener chaoses
该研究发现维纳混沌上混沌随机变量序列的正态依分布收敛可升级为超收敛，通过Malliavin分析新范式证明密度正则性提升与导数一致收敛，拓展了多元情形并给出多领域应用，去除了此前相关研究的限制性假设。
The problem of approximating/tracking the value of a Wiener process is considered. The discretization points are placed at times when the value of the process differs from the approximation by some amount, here denoted by eta. It is found that the limiting difference, as eta goes to 0, between the approximation and the value of the process normalized with eta converges in distribution to a triangularly distributed random variable.
Cubature on Wiener space [Lyons, T.; Victoir, N.; Proc. R. Soc. Lond. A 8 January 2004 vol. 460 no. 2041 169-198] provides a powerful alternative to Monte Carlo simulation for the integration of certain functionals on Wiener space. More specifically, and in the language of mathematical finance, cubature allows for fast computation of European option prices in generic diffusion models. We give a random walk interpretation of cubature and similar (e.g. the Ninomiya--Victoir) weak approximation schemes. By using rough path analysis, we are able to establish weak convergence for general path-depen…
Weak Convergence of Finite Element Approximations of Stochastic Linear Schrödinger equation driven by additive Wiener noise
该研究用标准有限元法空间离散加性 Wiener 噪声驱动的随机线性 Schrödinger 方程，并证明近似解的弱收敛速率为强收敛速率的两倍。
The aim of this paper is to establish some new results on the absolute continuity and the convergence in total variation for a sequence of d-dimensional vectors whose components belong to a finite sum of Wiener chaoses. First we show that the probability that the determinant of the Malliavin matrix of such vectors vanishes is zero or one, and this probability equals to one is equivalent to say that the vector takes values in the set of zeros of a polynomial. We provide a bound for the degree of this annihilating polynomial improving a result by Kusuoka. On the other hand, we show that the conv…
Path integrals developed by Richard Feynman have been an important tool in Physics in studying quantum field theory. In mathematics, it has also been widely used in providing formal proofs in the study of Index theorem and asymptotic behaviors of heat kernels. Finite dimensional approximations to path integral representations give a way to interpret path integrals and make the formal argument rigorous. The central idea is to restrict a path integral to smaller path spaces where everything is well defined and then to interpret the original path integral as a "limit" when smaller path spaces "ex…
We characterize the relaxation of the perimeter in an infinite dimensional Wiener space, with respect to the weak L^2-topology. We also show that the rescaled Allen-Cahn functionals approximate this relaxed functional in the sense of Gamma-convergence.
We prove that the formula which gives the Wiener chaos decomposition of the multiplication of two multiple Wiener integrals with symmetric kernels is a straightforward application of the Leibniz' formula.
Wiener distribution on holonomy groups
本文研究了在holonomy群上通过随机平行运输的度量连接的Wiener测度收敛性问题。
Given a reference random variable, we study the solution of its Stein equation and obtain universal bounds on its first and second derivatives. We then extend the analysis of Nourdin and Peccati by bounding the Fortet-Mourier and Wasserstein distances from more general random variables such as members of the Exponential and Pearson families. Using these results, we obtain non-central limit theorems, generalizing the ideas applied to their analysis of convergence to Normal random variables. We do these in both Wiener space and the more general Wiener-Poisson space. In the former, we study condi…
In this paper by calculating carefully the capacities (defined by high order Sobolev norms on the Wiener space) for some functions of Brownian motion, we show that the dyadic approximations of the sample paths of the Brownian motion converge in the $p$-variation distance to the Brownian motion except for a slim set (i.e. except for a zero subset with respect to the capacity on the Wiener space of any order). This presents a way for studying quasi-sure properties of Wiener functionals by means of the rough path analysis.
Let ${F_n}$ be a sequence of random variables belonging to a finite sum of Wiener chaoses. Assume further that it converges in distribution towards $F_\infty$ satisfying ${\rm Var}(F_\infty)>0$. Our first result is a sequential version of a theorem by Shigekawa (1980). More precisely, we prove, without additional assumptions, that the sequence ${F_n}$ actually converges in total variation and that the law of $F_\infty$ is absolutely continuous. We give an application to discrete non-Gaussian chaoses. In a second part, we assume that each $F_n$ has more specifically the form of a multiple Wiene…
In this paper, we propose a general means of estimating the rate at which convergences in law occur. Our approach, which is an extension of the classical Stein-Tikhomirov method, rests on a new pair of linear operators acting on characteristic functions. In principle, this method is admissible for any approximating sequence and any target, although obviously the conjunction of several favorable factors is necessary in order for the resulting bounds to be of interest. As we briefly discuss, our approach is particularly promising whenever some version of Stein's method applies. We apply our appr…
Consider $F$ an element of the second Wiener chaos with variance one. In full generality, we show that, for every integer $p\ge 1$, there exists $η_p>0$ such that if $κ_4(F)<η_p$ then the Malliavin derivative of $F$ admits a negative moment of order $p$. This entails that any sequence of random variables in the second Wiener chaos converging in distribution to a non--degenerated Gaussian is getting more regular as its distribution is getting close to the normal law. This substantially generalizes some recent findings contained in \cite{hu2014convergence,hu2015density,nourdin2016fisher} where a…
A parallel neighborhood of a path of a Brownian motion is sometimes called the Wiener sausage. We consider almost sure approximations of this random set by a sequence of random polyconvex sets and show that the convergence of the corresponding mean curvature measures holds under certain conditions in two and three dimensions. Based on these convergence results, the mean curvature measures of the Wiener sausage are calculated numerically by Monte Carlo simulations in two dimensions. The corresponding approximation formulae are given.
We prove a stochastic Taylor expansion for SPDEs and apply this result to obtain cubature methods, i. e. high order weak approximation schemes for SPDEs, in the spirit of T. Lyons and N. Victoir. We can prove a high-order weak convergence for well-defined classes of test functions if the process starts at sufficiently regular initial values. We can also derive analogous results in the presence of Lévy processes of finite type, here the results seem to be new even in finite dimension. Several numerical examples are added.
Convergence Rate Analysis in Limit Theorems for Nonlinear Functionals of the Second Wiener Chaos
本文研究高斯过程平方包络小波变换幅值随机向量的大尺度极限，针对Hurst指数小于1/2和大于1/2两种情形，分别用非Stein方法与Malliavin分析结合多维Stein方法给出Kolmogorov和Wasserstein度量下的收敛速度，并揭示不同Hurst子区间上界的差异。
In this paper we prove that the centered three-dimensional Wiener sausage can be strongly approximated by a one-dimensional Brownian motion running at a suitable time clock. The strong approximation gives all possible laws of iterated logarithm as well as the convergence in law in terms of process for the normalized Wiener sausage. The proof relies on Le Gall's estimates between the Wiener sausage and the Brownian intersection local times.
We prove the chain rule in the more general framework of the Wiener-Poisson space, allowing us to obtain the so-called Nourdin-Peccati bound. From this bound we obtain a second-order Poincare-type inequality that is useful in terms of computations. For completeness we survey these results on the Wiener space, the Poisson space, and the Wiener-Poisson space. We also give several applications to central limit theorems with relevant examples: linear functionals of Gaussian subordinated fields (where the subordinated field can be processes like fractional Brownian motion or the solution of the Orn…
Let $ξ(k,n)$ be the local time of a simple symmetric random walk on the line. We give a strong approximation of the centered local time process $ξ(k,n)-ξ(0,n)$ in terms of a Wiener sheet and an independent Wiener process, time changed by an independent Brownian local time. Some related results and consequences are also established.
The aim of this study is to clarify the consequences of recent theoretical results for the numerical computation of expectation by the shift method, and in particular to yield sufficient criteria for the existence of speed of convergence of the type `iterated logarithm' in several situations. We deepen the case of the Wiener space because it contains many situations useful in applications.
In \cite{n-p-noncentral}, Nourdin and Peccati established a neat characterization of Gamma approximation on a fixed Wiener chaos in terms of convergence of only the third and fourth cumulants. In this paper, we provide an optimal rate of convergence in the $d_2$-distance in terms of the maximum of the third and fourth cumulants analogous to the result for normal approximation in \cite{n-p-optimal}. In order to achieve our goal, we introduce a novel operator theory approach to Stein's method. The recent development in Stein's method for the Gamma distribution of Döbler and Peccati (\cite{d-p}) …
Optimal local central limit theorems on Wiener chaos
本文在韦纳混沌中研究了随机变量序列的局部中心极限定理，证明了密度函数收敛速率由第三和第四累积量的最大绝对值决定。
Using the structure of the Boson-Fermion Fock space and an argument taken from [2], we give a new proof of the triviality of the $L^2$ cohomology groups on an abstract Wiener space, alternative to that given by Shigekawa [9]. We apply some representation theory of the symmetric group to characterise the spaces of exact and co-exact forms in their Boson-Fermion Fock space representation.
We develop a stochastic analysis for a Gaussian process $X$ with singular covariance by an intrinsic procedure focusing on several examples such as covariance measure structure processes, bifractional Brownian motion, processes with stationary increments. We introduce some new spaces associated with the self-reproducing kernel space and we define the Paley-Wiener integral of first and second order even when $X$ is only a square integrable process continuous in $L^2$. If $X$ has stationary increments, we provide necessary and sufficient conditions so that its paths belong to the self-reproducin…
In this paper we present a weak approximation scheme for BSDEs driven by a Wiener process and an (in)finite activity Poisson random measure with drivers that are general Lipschitz functionals of the solution of the BSDE. The approximating backward stochastic difference equations (BSΔEs) are driven by random walks that weakly approximate the given Wiener process and Poisson random measure. We establish the weak convergence to the solution of the BSDE and the numerical stability of the sequence of solutions of the BSΔEs. By way of illustration we analyse explicitly a scheme with discrete step-si…
We prove an asymptotic Cramér's theorem, that is, if the sequence $(X_{n}+ Y_{n})_{n\geq 1}$ converges in law to the standard normal distribution and for every $n\geq 1$ the random variables $X_{n}$ and $Y_{n}$ are independent, then $(X_{n})_{n\geq 1}$ {\it and } $(Y_{n}) _{n\geq 1}$ converge in law to a normal distribution. Then we compare this result with recent criteria for the central convergence obtained in terms of Malliavin derivatives.
Let {F_n} be a normalized sequence of random variables in some fixed Wiener chaos associated with a general Gaussian field, and assume that E[F_n^4] --> E[N^4]=3, where N is a standard Gaussian random variable. Our main result is the following general bound: there exist two finite constants c,C>0 such that, for n sufficiently large, c max(|E[F_n^3]|, E[F_n^4]-3) < d(F_n,N) < C max(|E[F_n^3]|, E[F_n^4]-3), where d(F_n,N) = sup |E[h(F_n)] - E[h(N)]|, and h runs over the class of all real functions with a second derivative bounded by 1. This shows that the deterministic sequence max(|E[F_n^3]|, E…
The aim of this paper is the rigorous derivation of a stochastic non-linear diffusion equation from a radiative transfer equation perturbed with a random noise. The proof of the convergence relies on a formal Hilbert expansion and the estimation of the remainder. The Hilbert expansion has to be done up to order 3 to overcome some diffculties caused by the random noise.
We develop a theory of Malliavin calculus for Banach space valued random variables. Using radonifying operators instead of symmetric tensor products we extend the Wiener-Ito isometry to Banach spaces. In the white noise case we obtain two sided L^p-estimates for multiple stochastic integrals in arbitrary Banach spaces. It is shown that the Malliavin derivative is bounded on vector-valued Wiener-Ito chaoses. Our main tools are decoupling inequalities for vector-valued random variables. In the opposite direction we use Meyer's inequalities to give a new proof of a decoupling result for Gaussian …
We establish explicit bounds on the convex distance between the distribution of a vector of smooth functionals of a Gaussian field, and that of a normal vector with a positive definite covariance matrix. Our bounds are commensurate to the ones obtained by Nourdin, Peccati and Réveillac (2010) for the (smoother) 1-Wasserstein distance, and do not involve any additional logarithmic factor. One of the main tools exploited in our work is a recursive estimate on the convex distance recently obtained by Schulte and Yukich (2019). We illustrate our abstract results in two different situations: (i) we…
Joint convergence in Wiener chaos via transport hierarchy and Malliavin covariances
研究多重Wiener-Itô积分序列与高斯极限的联合收敛性，发现联合收敛完全由迭代Malliavin协方差控制，并显式确定极限分布结构。
Rate of convergence of Wong-Zakai approximations for stochastic partial differential equations
本文证明了由Wiener过程驱动的随机偏微分方程的Wong-Zakai逼近收敛速率本质上与逼近Wiener过程的驱动过程W_n的收敛速率相同，前提是W_n的面积过程也以该速率收敛到W的面积过程，并考虑了非退化及退化情形。
An upper bound for the Wasserstein distance is provided in the general framework of the Wiener-Poisson space. Is obtained from this bound a second order Poincaré-type inequality which is useful in terms of computations. For completeness sake, is made a survey of these results on the Wiener space, the Poisson space, and the Wiener-Poisson space, and showed several applications to central limit theorems with relevant examples: linear functionals of Gaussian subordinated field (where the subordinated field can be processes like fractional Brownian motion or the solution of the Ornstein-Uhlenbeck …
The Ogawa stochastic integral is shortly reviewed and formulated in the framework of abstract Wiener spaces. The condition of universal Ogawa integrability in the multidimensional case is investigated, proving that it cannot hold in general without the introduction of a "renormalization term". Explicit examples are provided.
We prove a local limit theorem, i.e. a central limit theorem for densities, for a sequence of independent and identically distributed random variables taking values on an abstract Wiener space; the common law of those random variables is assumed to be absolutely continuous with respect to the reference Gaussian measure. We begin by showing that the key roles of scaling operator and convolution product in this infinite dimensional Gaussian framework are played by the Ornstein-Uhlenbeck semigroup and Wick product, respectively. We proceed by establishing a necessary condition on the density of t…
Suppose $B$ is a Brownian motion and $B^n$ is an approximating sequence of rescaled random walks on the same probability space converging to $B$ pointwise in probability. We provide necessary and sufficient conditions for weak and strong $L^2$-convergence of a discretized Malliavin derivative, a discrete Skorokhod integral, and discrete analogues of the Clark-Ocone derivative to their continuous counterparts. Moreover, given a sequence $(X^n)$ of random variables which admit a chaos decomposition in terms of discrete multiple Wiener integrals with respect to $B^n$, we derive necessary and suff…
In the framework of classical probability, we consider the normal product distribution $F_\infty \sim N_1 \times N_2$ where $N_1, N_2$ are two independent standard normal random variable, and in the setting of free probability, $F_\infty \sim \left( S_1 S_2 + S_2 S_1 \right)/\sqrt{2}$ known as {\it tetilla law} \cite{d-n}, where $S_1, S_2$ are freely independent normalized semicircular random variables. We provide novel characterization of $F_\infty$ within the second Wiener (Wigner) chaos. More precisely, we show that for any generic element $F$ in the second Wiener (Wigner) chaos with varian…
We study Markovian symmetry and non-symmetry random evolutions in $\mathbf{R}^n$. Weak convergence of Markovian symmetry random evolution to Wiener process and of Markovian non-symmetry random evolution to a diffusion process with drift is proved using problems of singular perturbation for the generators of evolutions. Relative compactness in $\mathbf{D}_{\mathbf{R}^n\timesΘ}[0,\infty)$ of the families of Markovian random evolutions is also shown.
We prove a new rearrangement inequality for multiple integrals, which partly generalizes a result of Friedberg and Luttinger (1976) and can be interpreted as involving symmetric rearrangements of domains around infinity. As applications, we prove two comparison results for general Levy processes and their symmetric rearrangements. The first application concerns the survival probability of a point particle in a Poisson field of moving traps following independent Levy motions. We show that the survival probability can only increase if the point particle does not move, and the traps and the Levy …
Given a two-dimensional correlated diffusion process, we determine the joint density of the first passage times of the process to some constant boundaries. This quantity depends on the joint density of the first passage time of the first crossing component and of the position of the second crossing component before its crossing time. First we show that these densities are solutions of a system of Volterra-Fredholm first kind integral equations. Then we propose a numerical algorithm to solve it and we describe how to use the algorithm to approximate the joint density of the first passage times.…
The celebrated Nualart-Peccati criterion [Ann. Probab. 33 (2005) 177-193] ensures the convergence in distribution toward a standard Gaussian random variable $N$ of a given sequence $\{X_n\}_{n\ge1}$ of multiple Wiener-Itô integrals of fixed order, if $\mathbb {E}[X_n^2]\to1$ and $\mathbb {E}[X_n^4]\to \mathbb {E}[N^4]=3$. Since its appearance in 2005, the natural question of ascertaining which other moments can replace the fourth moment in the above criterion has remained entirely open. Based on the technique recently introduced in [J. Funct. Anal. 266 (2014) 2341-2359], we settle this problem…
An Euler scheme for BSDEs via the Wiener chaos decomposition
提出基于Wiener混沌分解的Euler格式实现BSDE数值求解，无需前向-后向马尔可夫结构，适用于任意平方可积终端条件，并给出收敛性分析与数值验证。
The discrete-time approximation for nonlinear filtering problems is related to both of strong and weak approximations of stochastic differential equations. In this paper, we propose a new method of proof for the convergence of approximate nonlinear filter analyzed by Jean Picard (1984), and show a more general result than the original one. For the proof, we develop an analysis of Hilbert space valued functionals on Wiener space.
We study convergence to the invariant measure for a class of semilinear stochastic evolution equations driven by Lévy noise, including the case of cylindrical noise. For a certain class of equations we prove the exponential rate of convergence in the norm of total variation. Our general result is applied to a number of specific equations driven by cylindrical symmetric $α$-stable noise and/or cylindrical Wiener noise. We also consider the case of a "singular" Wiener process with unbounded covariance operator. In particular, in the equation with diagonal pure $α$-stable cylindrical noise introd…
The Fourth-Moment Theorem on Hilbert Spaces
本文在希尔伯特空间中建立了第四矩定理的条件，扩展了经典实值定理，并修正了前人工作中关于距离度量的不足。
In this paper, we show an approximation in law, in the space of the continuous functions on $[0,1]^2$, of two-parameter Gaussian processes that can be represented as a Wiener type integral by processes constructed from processes that converge to the Brownian sheet. As an application, we obtain a sequence of processes constructed from a Lévy sheet that converges in law towards the fractional Brownian sheet.
In the paper $k$-multiple self-intersection local time for planar Gaussian integrators generated by linear operator with nontrivial kernel is studied. In this case additional singularities arise in its formal Fourier--Wiener transform. In case $k=2$ the set of singularities is the finite number of points. In case $k>2$ it contains intervals and hyperplanes. In the first and the second cases using two different approaches related on structure of set of singularities we show that "new" singularities do not imply on the convergence of integral corresponding to the formal Fourier--Wiener transform…
Chen [Ann. Appl. Probab. {\bf 11} (2001), 1242--1262] derived exact convergence rates in a central limit theorem and a local limit theorem for a supercritical branching Wiener process.We extend Chen's results to a branching random walk under weaker moment conditions. For the branching Wiener process, our results sharpen Chen's by relaxing the second moment condition used by Chen to a moment condition of the form $ \E X (\ln^+X )^{1+λ}< \infty$. In the rate functions that we find for a branching random walk, we figure out some new terms which didn't appear in Chen's work.The results are establi…
New Constructions of Cubature Formulas on Wiener Space
本研究基于Lyons和Victoir的技术，首次构造出$\n\bbr^3$上维纳空间的7次显式求积公式，对比了随机泰勒展开与Log-ODE两种求积近似方法，通过实验验证了求积次数对收敛阶的影响及求积法在SDE弱近似中的效用，为后续研究提供参考。
Hölder estimates and weak convergences of certain weighted sum processes
本文针对固定阶Wiener混沌中的加权和过程，利用Malliavin分部积分、四阶矩定理和多维Young积分估计，证明了其Hölder连续性和泛函极限定理。
We use integration by parts formulas to give estimates for the $L^p$ norm of the Riesz transform. This is motivated by the representation formula for conditional expectations of functionals on the Wiener space already given in Malliavin and Thalmaier. As a consequence, we obtain regularity and estimates for the density of non degenerated functionals on the Wiener space. We also give a semi-distance which characterizes the convergence to the boundary of the set of the strict positivity points for the density.
Approximation convergence in the inverse first-passage time problem
针对逆首达时问题，提出一种修正的分段线性逼近方法，同时逼近边界起始值，并在绝对连续及导数一致有界条件下证明其子序列一致收敛，适用于反射维纳过程。
Generalized specific entropy on Wiener space with application to Martingale Optimal Transport
本文提出了一种基于泊松跳跃近似的特定熵框架，用于连续时间鞅传输问题，解决了传统熵正则化在连续时间中的局限性，并引入了广义特定熵函数在维纳空间上的显式表达，提出了连续时间特定熵鞅最优传输问题SEMOT。
扫码添加微信好友，提出您的宝贵建议 👇
💡 备注请填写：网站反馈
发表机构由 AI 从论文文本抽取，如发现错误，欢迎反馈修正建议。