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Spatial asymptotics for the parabolic Anderson models with generalized time–space Gaussian noise 具有广义时空高斯噪声的抛物型Anderson模型的空间渐近性
IF 2.3 1区 数学 Q1 STATISTICS & PROBABILITY Pub Date : 2016-03-01 DOI: 10.1214/15-AOP1006
Xia Chen
Partially motivated by the recent papers of Conus, Joseph and Khoshnevisan [Ann. Probab. 41 (2013) 2225–2260] and Conus et al. [Probab. Theory Related Fields 156 (2013) 483–533], this work is concerned with the precise spatial asymptotic behavior for the parabolic Anderson equation {∂u∂t(t,x)=12Δu(t,x)+V(t,x)u(t,x),u(0,x)=u0(x), where the homogeneous generalized Gaussian noise V(t,x) is, among other forms, white or fractional white in time and space. Associated with the Cole–Hopf solution to the KPZ equation, in particular, the precise asymptotic form limR→∞(logR)−2/3logmax|x|≤Ru(t,x)=342t3−−−√3a.s. is obtained for the parabolic Anderson model ∂tu=12∂2xxu+W˙u with the (1+1)-white noise W˙(t,x). In addition, some links between time and space asymptotics for the parabolic Anderson equation are also pursued.
部分原因是Conus, Joseph和Khoshnevisan最近的论文[Ann]。[Probab. 41(2013) 2225-2260]和Conus等人。理论相关领域156(2013)483-533],本文研究了抛物型Anderson方程{∂u∂t(t,x)=12Δu(t,x)+V(t,x)u(t,x),u(0,x)=u0(x)的精确空间渐近行为,其中齐次广义高斯噪声V(t,x)在时间和空间上为白色或分数白色。结合KPZ方程的Cole-Hopf解,得到了精确的渐近形式limR→∞(logR)−2/3logmax|x|≤Ru(t,x)=342t3−−−√3a。为抛物线型安德森模型∂tu=12∂2xxu+W˙u,具有(1+1)-白噪声W˙(t,x)。此外,还讨论了抛物型安德森方程的时间渐近性与空间渐近性之间的联系。
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引用次数: 44
Compensated fragmentation processes and limits of dilated fragmentations 补偿破碎过程和扩展破碎的极限
IF 2.3 1区 数学 Q1 STATISTICS & PROBABILITY Pub Date : 2016-03-01 DOI: 10.1214/14-AOP1000
J. Bertoin
A new class of fragmentation-type random processes is introduced, in which, roughly speaking, the accumulation of small dislocations which would instantaneously shatter the mass into dust, is compensated by an adequate dilation of the components. An important feature of these compensated fragmentations is that the dislocation measure $nu$ which governs their evolutions has only to fulfill the integral condition $int_{mathit{p}}$ (1-$mathit{p}_{1}$)$^{2}nu$(d$mathbf{p}$ < $infty$, where $mathbf{p}$ = ($mathit{p}_{1}$,…) denotes a generic mass-partition. This is weaker than the necessary and sufficient condition $int_{mathit{p}}$ (1-$mathit{p}_{1}$)$^{2}nu$(d$mathbf{p}$ < $infty$ for $nu$ to be the dislocation measure of a homogeneous fragmentation. Our main results show that such compensated fragmentations naturally arise as limits of homogeneous dilated fragmentations, and bear close connections to spectrally negative Levy processes.
本文介绍了一类新的碎裂型随机过程,在这种随机过程中,粗略地说,小的位错的积累会使质量瞬间粉碎成尘埃,而这种累积会被部件的适当膨胀所补偿。这些补偿碎片的一个重要特征是,控制它们演化的位错测度$nu$只需要满足积分条件$int_{mathit{p}}$ (1- $mathit{p}_{1}$) $^{2}nu$ (d $mathbf{p}$ < $infty$),其中$mathbf{p}$ = ($mathit{p}_{1}$,…)表示一般质量分配。这比$nu$作为均质破碎位错测度的充分必要条件$int_{mathit{p}}$ (1- $mathit{p}_{1}$) $^{2}nu$ (d $mathbf{p}$ < $infty$)弱。我们的主要结果表明,这种补偿碎片作为均匀膨胀碎片的极限自然出现,并且与谱负Levy过程密切相关。
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引用次数: 20
Central limit theorem for linear groups 线性群的中心极限定理
IF 2.3 1区 数学 Q1 STATISTICS & PROBABILITY Pub Date : 2016-03-01 DOI: 10.1214/15-AOP1002
Y. Benoist, Jean-François Quint
We prove a central limit theorem for random walks with finite variance on linear groups.
证明了线性群上有限方差随机漫步的一个中心极限定理。
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引用次数: 87
On the perimeter of excursion sets of shot noise random fields 散粒噪声随机场偏移集的周长
IF 2.3 1区 数学 Q1 STATISTICS & PROBABILITY Pub Date : 2016-01-01 DOI: 10.1214/14-AOP980
H. Biermé, A. Desolneux
In this paper, we use the framework of functions of bounded variation and the coarea formula to give an explicit computation for the expectation of the perimeter of excursion sets of shot noise random fields in dimension n≥1. This will then allow us to derive the asymptotic behavior of these mean perimeters as the intensity of the underlying homogeneous Poisson point process goes to infinity. In particular, we show that two cases occur: we have a Gaussian asymptotic behavior when the kernel function of the shot noise has no jump part, whereas the asymptotic is non-Gaussian when there are jumps.
本文利用有界变分函数框架和共面积公式,给出了n≥1维的散粒噪声随机场偏移集周长期望的显式计算。这样我们就可以推导出当齐次泊松点过程的强度趋于无穷时这些平均周长的渐近行为。特别地,我们证明了两种情况:当散粒噪声的核函数没有跳跃部分时,我们有高斯渐近行为,而当有跳跃时,渐近行为是非高斯的。
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引用次数: 24
Interlacements and the wired uniform spanning forest 交错和有线均匀跨越森林
IF 2.3 1区 数学 Q1 STATISTICS & PROBABILITY Pub Date : 2015-12-28 DOI: 10.1214/17-AOP1203
Tom Hutchcroft
We extend the Aldous-Broder algorithm to generate the wired uniform spanning forests (WUSFs) of infinite, transient graphs. We do this by replacing the simple random walk in the classical algorithm with Sznitman's random interlacement process. We then apply this algorithm to study the WUSF, showing that every component of the WUSF is one-ended almost surely in any graph satisfying a certain weak anchored isoperimetric condition, that the number of `excessive ends' in the WUSF is non-random in any graph, and also that every component of the WUSF is one-ended almost surely in any transient unimodular random rooted graph. The first two of these results answer positively two questions of Lyons, Morris and Schramm, while the third extends a recent result of the author. Finally, we construct a counterexample showing that almost sure one-endedness of WUSF components is not preserved by rough isometries of the underlying graph, answering negatively a further question of Lyons, Morris and Schramm.
我们扩展了Aldous-Broder算法来生成无限暂态图的有线均匀生成森林(WUSFs)。我们将经典算法中的简单随机游走替换为Sznitman随机交错过程。应用该算法对WUSF进行了研究,结果表明,在满足一定弱锚定等周条件的任意图中,WUSF的每个分量几乎肯定是一端的,在任意图中,WUSF的“过端”数都是非随机的,在任意暂态单模随机根图中,WUSF的每个分量几乎肯定是一端的。前两个结果肯定地回答了里昂、莫里斯和施拉姆的两个问题,而第三个结果扩展了作者最近的一个结果。最后,我们构造了一个反例,表明底层图的粗糙等边不保留WUSF组件的几乎肯定的一端性,否定地回答了Lyons, Morris和Schramm的进一步问题。
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引用次数: 29
Size biased couplings and the spectral gap for random regular graphs 随机正则图的尺寸偏置耦合和谱隙
IF 2.3 1区 数学 Q1 STATISTICS & PROBABILITY Pub Date : 2015-10-20 DOI: 10.1214/17-AOP1180
Nicholas A. Cook, L. Goldstein, Tobias Johnson
Let λλ be the second largest eigenvalue in absolute value of a uniform random dd-regular graph on nn vertices. It was famously conjectured by Alon and proved by Friedman that if dd is fixed independent of nn, then λ=2d−1−−−−√+o(1)λ=2d−1+o(1) with high probability. In the present work, we show that λ=O(d−−√)λ=O(d) continues to hold with high probability as long as d=O(n2/3)d=O(n2/3), making progress toward a conjecture of Vu that the bound holds for all 1≤d≤n/21≤d≤n/2. Prior to this work the best result was obtained by Broder, Frieze, Suen and Upfal (1999) using the configuration model, which hits a barrier at d=o(n1/2)d=o(n1/2). We are able to go beyond this barrier by proving concentration of measure results directly for the uniform distribution on dd-regular graphs. These come as consequences of advances we make in the theory of concentration by size biased couplings. Specifically, we obtain Bennett-type tail estimates for random variables admitting certain unbounded size biased couplings.
设λ为nn顶点上的一致随机dd-正则图的绝对值中的第二大特征值。著名的猜想是由Alon和Friedman证明,如果dd是独立于nn的固定的,那么λ=2d−1−−−√+o(1)λ=2d−1+o(1)具有高概率。在本工作中,我们证明了只要d=O(n2/3)d=O(n2/3), λ=O(d−−√)λ=O(d)继续以高概率成立,从而进一步证明了Vu的一个猜想,即对于所有1≤d≤n/21≤d≤n/2,界都成立。在此工作之前,Broder, Frieze, Suen和upfall(1999)使用配置模型获得了最佳结果,该模型在d=o(n1/2)d=o(n1/2)处遇到障壁。通过直接证明dd-正则图上均匀分布的测量结果的集中,我们能够超越这个障碍。这些都是我们在尺寸偏差耦合集中理论中取得进展的结果。具体地说,我们得到了具有无界尺寸偏差耦合的随机变量的bennett型尾估计。
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引用次数: 51
The chaotic representation property of compensated-covariation stable families of martingales 补偿协变鞅稳定族的混沌表示性质
IF 2.3 1区 数学 Q1 STATISTICS & PROBABILITY Pub Date : 2015-09-29 DOI: 10.1214/15-AOP1066
P. D. Tella, H. Engelbert
In the present paper, we study the chaotic representation property for certain families XX of square integrable martingales on a finite time interval [0,T][0,T]. For this purpose, we introduce the notion of compensated-covariation stability of such families. The chaotic representation property will be defined using iterated integrals with respect to a given family XX of square integrable martingales having deterministic mutual predictable covariation ⟨X,Y⟩⟨X,Y⟩ for all X,Y∈XX,Y∈X. The main result of the present paper is stated in Theorem 5.8 below: If XX is a compensated-covariation stable family of square integrable martingales such that ⟨X,Y⟩⟨X,Y⟩ is deterministic for all X,Y∈XX,Y∈X and, furthermore, the system of monomials generated by XX is total in L2(Ω,FXT,P)L2(Ω,FTX,P), then XX possesses the chaotic representation property with respect to the σσ-field FXTFTX. We shall apply this result to the case of Levy processes. Relative to the filtration FLFL generated by a Levy process LL, we construct families of martingales which possess the chaotic representation property. As an illustration of the general results, we will also discuss applications to continuous Gaussian families of martingales and independent families of compensated Poisson processes. We conclude the paper by giving, for the case of Levy processes, several examples of concrete families XX of martingales including Teugels martingales.
本文研究了有限时间区间[0,T][0,T]上若干平方可积鞅族XX的混沌表示性质。为此,我们引入了这类族的补偿共变稳定性的概念。混沌表示性质将使用对给定族XX的平方可积鞅的迭代积分来定义,该族XX对所有X,Y∈XX,Y∈X具有确定的相互可预测的协变⟨X,Y⟩⟨X,Y⟩。本文的主要结果在下面的定理5.8中陈述:如果XX是一个补偿协变稳定的平方可积鞅族,使得⟨X,Y⟩⟨X,Y⟩对所有X,Y∈XX,Y∈X是确定的,并且,XX生成的单项式系统在L2(Ω,FXT,P)L2(Ω,FTX,P)中是全的,那么XX对σσ-域FXTFTX具有混沌表示性质。我们将把这个结果应用于列维过程的情况。相对于由Levy过程生成的滤波FLFL,我们构造了具有混沌表示性质的鞅族。为了说明一般结果,我们还将讨论在连续高斯鞅族和独立补偿泊松过程族中的应用。对于列维过程,我们给出了包括Teugels鞅在内的鞅的几个具体族XX的例子,以此来结束本文。
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引用次数: 9
Polarity of points for Gaussian random fields 高斯随机场点的极性
IF 2.3 1区 数学 Q1 STATISTICS & PROBABILITY Pub Date : 2015-05-20 DOI: 10.1214/17-AOP1176
R. Dalang, C. Mueller, Yimin Xiao
We show that for a wide class of Gaussian random fields, points are polar in the critical dimension. Examples of such random fields include solutions of systems of linear stochastic partial differential equations with deterministic coefficients, such as the stochastic heat equation or wave equation with space–time white noise, or colored noise in spatial dimensions k≥1k≥1. Our approach builds on a delicate covering argument developed by M. Talagrand [Ann. Probab. 23 (1995) 767–775; Probab. Theory Related Fields 112 (1998) 545–563] for the study of fractional Brownian motion, and uses a harmonizable representation of the solutions of these stochastic PDEs.
我们证明了对于一类广泛的高斯随机场,点在临界维上是极的。此类随机场的例子包括具有确定性系数的线性随机偏微分方程系统的解,例如具有时空白噪声或空间维度k≥1k≥1的彩色噪声的随机热方程或波动方程。我们的方法建立在一个微妙的覆盖论点的基础上。约23 (1995)767-775;Probab。理论相关领域112(1998)545-563]对分数布朗运动的研究,并使用这些随机偏微分方程解的协调表示。
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引用次数: 26
Wired cycle-breaking dynamics for uniform spanning forests 均匀跨越森林的有线循环打破动力学
IF 2.3 1区 数学 Q1 STATISTICS & PROBABILITY Pub Date : 2015-04-15 DOI: 10.1214/15-AOP1063
Tom Hutchcroft
We prove that every component of the wired uniform spanning forest (WUSFWUSF) is one-ended almost surely in every transient reversible random graph, removing the bounded degree hypothesis required by earlier results. We deduce that every component of the WUSFWUSF is one-ended almost surely in every supercritical Galton–Watson tree, answering a question of Benjamini, Lyons, Peres and Schramm [Ann. Probab. 29 (2001) 1–65]. Our proof introduces and exploits a family of Markov chains under which the oriented WUSFWUSF is stationary, which we call the wired cycle-breaking dynamics.
我们证明了有线均匀生成森林(WUSFWUSF)的每一个分量在每一个瞬态可逆随机图中几乎肯定是一端的,从而消除了先前结果所要求的有界度假设。我们推断出,在每个超临界高尔顿-沃森树中,WUSFWUSF的每个组分几乎都是一端的,回答了Benjamini, Lyons, Peres和Schramm [Ann]的问题。可能。29(2001)1-65]。我们的证明引入并利用了一组马尔可夫链,在这些马尔可夫链下,导向的WUSFWUSF是平稳的,我们称之为有线破环动力学。
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引用次数: 15
Intermittency and multifractality: A case study via parabolic stochastic PDEs 间歇性和多重分形:一个基于抛物型随机偏微分方程的案例研究
IF 2.3 1区 数学 Q1 STATISTICS & PROBABILITY Pub Date : 2015-03-20 DOI: 10.1214/16-AOP1147
D. Khoshnevisan, Kunwoo Kim, Yimin Xiao
Let denote space-time white noise, and consider the following stochastic partial dierential equations: (i) _ u = 1 u 00 +u , started identically at one; and (ii) _ Z = 1 Z 00 + , started identically at zero. It is well known that the solution to (i) is intermittent, whereas the solution to (ii) is not. And the two equations are known to be in dierent universality classes. We prove that the tall peaks of both systems are multifractals in a natural large-scale sense. Some of this work is extended to also establish the multifractal behavior of the peaks of stochastic PDEs on R+ R d with d > 2. G. Lawler has asked us if intermittency is the same as multifractality. The present work gives a negative answer to this question. As a byproduct of our methods, we prove also that the peaks of the Brownian motion form a large-scale monofractal, whereas the peaks of the Ornstein{Uhlenbeck process on R are multifractal. Throughout, we make extensive use of the macroscopic fractal theory of M.T. Barlow and S.J. Taylor [3, 4]. We expand on aspects of the Barlow{Taylor theory, as well.
设为时空白噪声,并考虑以下随机偏微分方程:(i) _ u = 1 u 00 +u,在1相等开始;和(ii) _ Z = 1z00 +,相同地从零开始。众所周知,(i)的解决方案是间歇性的,而(ii)的解决方案则不是。已知这两个方程属于不同的通用性类。我们证明了这两个体系的高峰在自然的大尺度意义上是多重分形的。在此基础上进一步推广了随机偏微分方程在R+ R+ d上的多重分形行为。G. Lawler问我们间歇性是否等同于多重分形。目前的工作对这个问题给出了否定的答案。作为我们的方法的副产品,我们还证明了布朗运动的峰形成一个大尺度的单分形,而R上的Ornstein{Uhlenbeck过程的峰是多重分形。自始至终,我们广泛运用了M.T. Barlow和S.J. Taylor的宏观分形理论[3,4]。我们也扩展了巴洛{泰勒理论的各个方面。
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引用次数: 56
期刊
Annals of Probability
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