This contribution establishes exact tail asymptotics of sup(s,t)∈E X(s,t) for a large class of nonhomogeneous Gaussian random fields X on a bounded convex set E⊂R2, with variance function that attains its maximum on a segment on E. These findings extend the classical results for homogeneous Gaussian random fields and Gaussian random fields with unique maximum point of the variance. Applications of our result include the derivation of the exact tail asymptotics of the Shepp statistics for stationary Gaussian processes, Brownian bridge and fractional Brownian motion as well as the exact tail asymptotic expansion for the maximum loss and span of stationary Gaussian processes.
{"title":"Extremes of a class of nonhomogeneous Gaussian random fields","authors":"Krzysztof Dcebicki, E. Hashorva, L. Ji","doi":"10.1214/14-AOP994","DOIUrl":"https://doi.org/10.1214/14-AOP994","url":null,"abstract":"This contribution establishes exact tail asymptotics of sup(s,t)∈E X(s,t) for a large class of nonhomogeneous Gaussian random fields X on a bounded convex set E⊂R2, with variance function that attains its maximum on a segment on E. These findings extend the classical results for homogeneous Gaussian random fields and Gaussian random fields with unique maximum point of the variance. Applications of our result include the derivation of the exact tail asymptotics of the Shepp statistics for stationary Gaussian processes, Brownian bridge and fractional Brownian motion as well as the exact tail asymptotic expansion for the maximum loss and span of stationary Gaussian processes.","PeriodicalId":50763,"journal":{"name":"Annals of Probability","volume":"44 1","pages":"984-1012"},"PeriodicalIF":2.3,"publicationDate":"2014-05-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1214/14-AOP994","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"66010589","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Let M be a von Neumann algebra equipped with a faithful semifinite normal weight ϕ and N be a von Neumann subalgebra of M such that the restriction of ϕ to N is semifinite and such that N is invariant by the modular group of ϕ. Let E be the weight preserving conditional expectation from M onto N. As an application we show that there exists e0>0 such that for any free group Fn and any q≥4−e0, ∥Pt∥2→q≤1⇔t≥logq−1−−−−√, where (Pt) is the Poisson semigroup defined by the natural length function of Fn.
{"title":"A noncommutative martingale convexity inequality","authors":"'Eric Ricard, Quanhua Xu","doi":"10.1214/14-AOP990","DOIUrl":"https://doi.org/10.1214/14-AOP990","url":null,"abstract":"Let M be a von Neumann algebra equipped with a faithful semifinite normal weight ϕ and N be a von Neumann subalgebra of M such that the restriction of ϕ to N is semifinite and such that N is invariant by the modular group of ϕ. Let E be the weight preserving conditional expectation from M onto N. As an application we show that there exists e0>0 such that for any free group Fn and any q≥4−e0, \u0000∥Pt∥2→q≤1⇔t≥logq−1−−−−√, where (Pt) is the Poisson semigroup defined by the natural length function of Fn.","PeriodicalId":50763,"journal":{"name":"Annals of Probability","volume":"44 1","pages":"867-882"},"PeriodicalIF":2.3,"publicationDate":"2014-05-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1214/14-AOP990","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"66010725","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We consider a class of kinetically constrained interacting particle systems on Zd which play a key role in several heuristic qualitative and quantitative approaches to describe the complex behavior of glassy dynamics. With rate one and independently among the vertices of Zd, to each occupation variable ηx∈{0,1} a new value is proposed by tossing a (1−q)-coin. If a certain local constraint is satisfied by the current configuration the proposed move is accepted, otherwise it is rejected. For d=1, the constraint requires that there is a vacancy at the vertex to the left of the updating vertex. In this case, the process is the well-known East process. On Z2, the West or the South neighbor of the updating vertex must contain a vacancy, similarly, in higher dimensions. Despite of their apparent simplicity, in the limit q↘0 of low vacancy density, corresponding to a low temperature physical setting, these processes feature a rather complicated dynamic behavior with hierarchical relaxation time scales, heterogeneity and universality. Using renormalization group ideas, we first show that the relaxation time on Zd scales as the 1/d-root of the relaxation time of the East process, confirming indications coming from massive numerical simulations. Next, we compute the relaxation time in finite boxes by carefully analyzing the subtle energy-entropy competition, using a multiscale analysis, capacity methods and an algorithmic construction. Our results establish dynamic heterogeneity and a dramatic dependence on the boundary conditions. Finally, we prove a rather strong anisotropy property of these processes: the creation of a new vacancy at a vertex x out of an isolated one at the origin (a seed) may occur on (logarithmically) different time scales which heavily depend not only on the l1-norm of x but also on its direction.
我们考虑了Zd上一类动力学约束的相互作用粒子系统，它们在描述玻璃动力学复杂行为的几种启发式定性和定量方法中起着关键作用。对于每个职业变量ηx∈{0,1}，在速率为1且独立于Zd的顶点之间，通过投掷(1−q)-硬币提出一个新值。如果当前配置满足某个局部约束，则建议的移动被接受，否则被拒绝。对于d=1，约束要求在更新顶点的左边顶点有一个空位。在本例中，该流程就是众所周知的East流程。在Z2上，更新顶点的西部或南部邻居必须包含一个空位，类似地，在更高的维度上。尽管它们表面上很简单，但在低空位密度的极限q d d 0下，对应于低温物理环境，这些过程具有相当复杂的动态行为，具有分层松弛时间尺度、异质性和普遍性。利用重整化群的思想，我们首先证明了Zd尺度上的弛豫时间是East过程弛豫时间的1/d-根，证实了大量数值模拟的结果。接下来，我们通过多尺度分析、容量方法和算法构建，仔细分析了微妙的能量熵竞争，计算了有限盒子中的松弛时间。我们的结果建立了动态异质性和对边界条件的戏剧性依赖。最后，我们证明了这些过程的一个相当强的各向异性性质:在原点(种子)孤立的顶点x上产生一个新的空位可能发生在(对数上)不同的时间尺度上，这不仅严重依赖于x的11范数，而且还依赖于它的方向。
{"title":"Relaxation to equilibrium of generalized East processes on $mathbb{Z}^{d}$: Renormalization group analysis and energy-entropy competition","authors":"P. Chleboun, A. Faggionato, F. Martinelli","doi":"10.1214/15-AOP1011","DOIUrl":"https://doi.org/10.1214/15-AOP1011","url":null,"abstract":"We consider a class of kinetically constrained interacting particle systems on Zd which play a key role in several heuristic qualitative and quantitative approaches to describe the complex behavior of glassy dynamics. With rate one and independently among the vertices of Zd, to each occupation variable ηx∈{0,1} a new value is proposed by tossing a (1−q)-coin. If a certain local constraint is satisfied by the current configuration the proposed move is accepted, otherwise it is rejected. For d=1, the constraint requires that there is a vacancy at the vertex to the left of the updating vertex. In this case, the process is the well-known East process. On Z2, the West or the South neighbor of the updating vertex must contain a vacancy, similarly, in higher dimensions. Despite of their apparent simplicity, in the limit q↘0 of low vacancy density, corresponding to a low temperature physical setting, these processes feature a rather complicated dynamic behavior with hierarchical relaxation time scales, heterogeneity and universality. Using renormalization group ideas, we first show that the relaxation time on Zd scales as the 1/d-root of the relaxation time of the East process, confirming indications coming from massive numerical simulations. Next, we compute the relaxation time in finite boxes by carefully analyzing the subtle energy-entropy competition, using a multiscale analysis, capacity methods and an algorithmic construction. Our results establish dynamic heterogeneity and a dramatic dependence on the boundary conditions. Finally, we prove a rather strong anisotropy property of these processes: the creation of a new vacancy at a vertex x out of an isolated one at the origin (a seed) may occur on (logarithmically) different time scales which heavily depend not only on the l1-norm of x but also on its direction.","PeriodicalId":50763,"journal":{"name":"Annals of Probability","volume":"44 1","pages":"1817-1863"},"PeriodicalIF":2.3,"publicationDate":"2014-04-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"66031627","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We consider a system of interacting diffusions on the integer lattice. By letting the mesh size go to zero and by using a suitable scaling, we show that the system converges (in a strong sense) to a solution of the stochastic heat equation on the real line. As a consequence, we obtain comparison inequalities for product moments of the stochastic heat equation with different nonlinearities.
{"title":"Strong invariance and noise-comparison principles for some parabolic stochastic PDEs","authors":"Mathew Joseph, D. Khoshnevisan, C. Mueller","doi":"10.1214/15-AOP1009","DOIUrl":"https://doi.org/10.1214/15-AOP1009","url":null,"abstract":"We consider a system of interacting diffusions on the integer lattice. By letting the mesh size go to zero and by using a suitable scaling, we show that the system converges (in a strong sense) to a solution of the stochastic heat equation on the real line. As a consequence, we obtain comparison inequalities for product moments of the stochastic heat equation with different nonlinearities.","PeriodicalId":50763,"journal":{"name":"Annals of Probability","volume":"45 1","pages":"377-403"},"PeriodicalIF":2.3,"publicationDate":"2014-04-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1214/15-AOP1009","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"66031583","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The frog model is a growing system of random walks where a particle is added whenever a new site is visited. A longstanding open question is how often the root is visited on the infinite dd-ary tree. We prove the model undergoes a phase transition, finding it recurrent for d=2d=2 and transient for d≥5d≥5. Simulations suggest strong recurrence for d=2d=2, weak recurrence for d=3d=3, and transience for d≥4d≥4. Additionally, we prove a 0–1 law for all dd-ary trees, and we exhibit a graph on which a 0–1 law does not hold. To prove recurrence when d=2d=2, we construct a recursive distributional equation for the number of visits to the root in a smaller process and show the unique solution must be infinity a.s. The proof of transience when d=5d=5 relies on computer calculations for the transition probabilities of a large Markov chain. We also include the proof for d≥6d≥6, which uses similar techniques but does not require computer assistance.
{"title":"Recurrence and transience for the frog model on trees","authors":"C. Hoffman, Tobias Johnson, M. Junge","doi":"10.1214/16-AOP1125","DOIUrl":"https://doi.org/10.1214/16-AOP1125","url":null,"abstract":"The frog model is a growing system of random walks where a particle is added whenever a new site is visited. A longstanding open question is how often the root is visited on the infinite dd-ary tree. We prove the model undergoes a phase transition, finding it recurrent for d=2d=2 and transient for d≥5d≥5. Simulations suggest strong recurrence for d=2d=2, weak recurrence for d=3d=3, and transience for d≥4d≥4. Additionally, we prove a 0–1 law for all dd-ary trees, and we exhibit a graph on which a 0–1 law does not hold. To prove recurrence when d=2d=2, we construct a recursive distributional equation for the number of visits to the root in a smaller process and show the unique solution must be infinity a.s. The proof of transience when d=5d=5 relies on computer calculations for the transition probabilities of a large Markov chain. We also include the proof for d≥6d≥6, which uses similar techniques but does not require computer assistance.","PeriodicalId":50763,"journal":{"name":"Annals of Probability","volume":"45 1","pages":"2826-2854"},"PeriodicalIF":2.3,"publicationDate":"2014-04-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1214/16-AOP1125","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"66047774","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We consider stochastic processes on complete, locally compact tree-like metric spaces (T,r)(T,r) on their “natural scale” with boundedly finite speed measure νν. Given a triple (T,r,ν)(T,r,ν) such a speed-νν motion on (T,r)(T,r) can be characterized as the unique strong Markov process which if restricted to compact subtrees satisfies for all x,y∈Tx,y∈T and all positive, bounded measurable ff, Ex[∫τy0dsf(Xs)]=2∫Tν(dz)r(y,c(x,y,z))f(z)<∞, Ex[∫0τydsf(Xs)]=2∫Tν(dz)r(y,c(x,y,z))f(z)<∞, where c(x,y,z)c(x,y,z) denotes the branch point generated by x,y,zx,y,z. If (T,r)(T,r) is a discrete tree, XX is a continuous time nearest neighbor random walk which jumps from vv to v′∼vv′∼v at rate 12⋅(ν({v})⋅r(v,v′))−112⋅(ν({v})⋅r(v,v′))−1. If (T,r)(T,r) is path-connected, XX has continuous paths and equals the νν-Brownian motion which was recently constructed in [Trans. Amer. Math. Soc. 365 (2013) 3115–3150]. In this paper, we show that speed-νnνn motions on (Tn,rn)(Tn,rn) converge weakly in path space to the speed-νν motion on (T,r)(T,r) provided that the underlying triples of metric measure spaces converge in the Gromov–Hausdorff-vague topology introduced in [Stochastic Process. Appl. 126 (2016) 2527–2553].
{"title":"Invariance principle for variable speed random walks on trees","authors":"S. Athreya, Wolfgang Lohr, A. Winter","doi":"10.1214/15-AOP1071","DOIUrl":"https://doi.org/10.1214/15-AOP1071","url":null,"abstract":"We consider stochastic processes on complete, locally compact tree-like metric spaces (T,r)(T,r) on their “natural scale” with boundedly finite speed measure νν. Given a triple (T,r,ν)(T,r,ν) such a speed-νν motion on (T,r)(T,r) can be characterized as the unique strong Markov process which if restricted to compact subtrees satisfies for all x,y∈Tx,y∈T and all positive, bounded measurable ff, \u0000 \u0000Ex[∫τy0dsf(Xs)]=2∫Tν(dz)r(y,c(x,y,z))f(z)<∞, \u0000Ex[∫0τydsf(Xs)]=2∫Tν(dz)r(y,c(x,y,z))f(z)<∞, \u0000 \u0000where c(x,y,z)c(x,y,z) denotes the branch point generated by x,y,zx,y,z. If (T,r)(T,r) is a discrete tree, XX is a continuous time nearest neighbor random walk which jumps from vv to v′∼vv′∼v at rate 12⋅(ν({v})⋅r(v,v′))−112⋅(ν({v})⋅r(v,v′))−1. If (T,r)(T,r) is path-connected, XX has continuous paths and equals the νν-Brownian motion which was recently constructed in [Trans. Amer. Math. Soc. 365 (2013) 3115–3150]. In this paper, we show that speed-νnνn motions on (Tn,rn)(Tn,rn) converge weakly in path space to the speed-νν motion on (T,r)(T,r) provided that the underlying triples of metric measure spaces converge in the Gromov–Hausdorff-vague topology introduced in [Stochastic Process. Appl. 126 (2016) 2527–2553].","PeriodicalId":50763,"journal":{"name":"Annals of Probability","volume":"109 1","pages":"625-667"},"PeriodicalIF":2.3,"publicationDate":"2014-04-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1214/15-AOP1071","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"66033248","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
G. Prato, Franco Flandoli, M. Röckner, A. Veretennikov
We prove pathwise uniqueness for a class of stochastic dierential equations (SDE) on a Hilbert space with cylindrical Wiener noise, whose non-linear drift parts are sums of the subdierential of a convex function and a bounded part. This generalizes a classical result by one of the authors to innite dimensions. Our results also generalize
{"title":"Strong uniqueness for SDEs in Hilbert spaces with nonregular drift","authors":"G. Prato, Franco Flandoli, M. Röckner, A. Veretennikov","doi":"10.1214/15-AOP1016","DOIUrl":"https://doi.org/10.1214/15-AOP1016","url":null,"abstract":"We prove pathwise uniqueness for a class of stochastic dierential equations (SDE) on a Hilbert space with cylindrical Wiener noise, whose non-linear drift parts are sums of the subdierential of a convex function and a bounded part. This generalizes a classical result by one of the authors to innite dimensions. Our results also generalize","PeriodicalId":50763,"journal":{"name":"Annals of Probability","volume":"44 1","pages":"1985-2023"},"PeriodicalIF":2.3,"publicationDate":"2014-04-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1214/15-AOP1016","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"66031703","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Consider an Ito process X satisfying the stochastic differential equation dX=a(X)dt+b(X)dW where a,b are smooth and W is a multidimensional Brownian motion. Suppose that Wn has smooth sample paths and that Wn converges weakly to W. A central question in stochastic analysis is to understand the limiting behavior of solutions Xn to the ordinary differential equation dXn=a(Xn)dt+b(Xn)dWn. The classical Wong–Zakai theorem gives sufficient conditions under which Xn converges weakly to X provided that the stochastic integral ∫b(X)dW is given the Stratonovich interpretation. The sufficient conditions are automatic in one dimension, but in higher dimensions the correct interpretation of ∫b(X)dW depends sensitively on how the smooth approximation Wnis chosen. In applications, a natural class of smooth approximations arise by setting Wn(t)=n−1/2∫nt0v∘ϕsds where ϕt is a flow (generated, e.g., by an ordinary differential equation) and v is a mean zero observable. Under mild conditions on ϕt, we give a definitive answer to the interpretation question for the stochastic integral ∫b(X)dW. Our theory applies to Anosov or Axiom A flows ϕt, as well as to a large class of nonuniformly hyperbolic flows (including the one defined by the well-known Lorenz equations) and our main results do not require any mixing assumptions on ϕt. The methods used in this paper are a combination of rough path theory and smooth ergodic theory.
考虑一个Ito过程X满足随机微分方程dX=a(X)dt+b(X)dW，其中a,b是光滑的，W是多维布朗运动。假设Wn具有光滑的样本路径，并且Wn弱收敛于w。随机分析中的一个中心问题是理解常微分方程dXn= A (Xn)dt+b(Xn)dWn的解Xn的极限行为。经典的Wong-Zakai定理给出了Xn弱收敛于X的充分条件，在此条件下，随机积分∫b(X)dW给出了Stratonovich解释。在一维中充分条件是自动的，但在高维中∫b(X)dW的正确解释敏感地取决于Wnis如何选择光滑近似。在应用中，通过设置Wn(t)=n−1/2∫nt0v°ϕsds，产生了一类自然的光滑近似，其中，ϕt是一个流(例如由常微分方程产生)，v是一个可观测的平均零。在温和条件下，我们给出了随机积分∫b(X)dW的解释问题的确定答案。我们的理论适用于阿诺索夫流或公理A流，以及一大类非均匀双曲流(包括由著名的洛伦兹方程定义的流)，我们的主要结果不需要任何混合假设。本文采用的方法是粗糙路径理论与光滑遍历理论的结合。
{"title":"Smooth approximation of stochastic differential equations","authors":"David Kelly, I. Melbourne","doi":"10.1214/14-AOP979","DOIUrl":"https://doi.org/10.1214/14-AOP979","url":null,"abstract":"Consider an Ito process X satisfying the stochastic differential equation dX=a(X)dt+b(X)dW where a,b are smooth and W is a multidimensional Brownian motion. Suppose that Wn has smooth sample paths and that Wn converges weakly to W. A central question in stochastic analysis is to understand the limiting behavior of solutions Xn to the ordinary differential equation dXn=a(Xn)dt+b(Xn)dWn. \u0000The classical Wong–Zakai theorem gives sufficient conditions under which Xn \u0000converges weakly to X provided that the stochastic integral ∫b(X)dW is given the Stratonovich interpretation. The sufficient conditions are automatic in one dimension, but in higher dimensions the correct interpretation of ∫b(X)dW depends sensitively on how the smooth approximation Wnis chosen. \u0000In applications, a natural class of smooth approximations arise by setting Wn(t)=n−1/2∫nt0v∘ϕsds where ϕt is a flow (generated, e.g., by an ordinary differential equation) and v is a mean zero observable. Under mild conditions on ϕt, we give a definitive answer to the interpretation question for the stochastic integral ∫b(X)dW. Our theory applies to Anosov or Axiom A flows ϕt, as well as to a large class of nonuniformly hyperbolic flows (including the one defined by the well-known Lorenz equations) and our main results do not require any mixing assumptions on ϕt. \u0000The methods used in this paper are a combination of rough path theory and smooth ergodic theory.","PeriodicalId":50763,"journal":{"name":"Annals of Probability","volume":"44 1","pages":"479-520"},"PeriodicalIF":2.3,"publicationDate":"2014-03-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1214/14-AOP979","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"66010097","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Let $X,X_1,X_2,ldots$ be i.i.d. mean zero random vectors with values in a separable Banach space $B$, $S_n=X_1+cdots+X_n$ for $nge1$, and assume ${c_n:nge1}$ is a suitably regular sequence of constants. Furthermore, let $S_{(n)}(t),0le tle1$ be the corresponding linearly interpolated partial sum processes. We study the cluster sets $A=C({S_n/c_n})$ and $mathcal{A}=C({S_{(n)}(cdot)/c_n})$. In particular, $A$ and $mathcal{A}$ are shown to be nonrandom, and we derive criteria when elements in $B$ and continuous functions $f:[0,1]to B$ belong to $A$ and $mathcal{A}$, respectively. When $B=mathbb{R}^d$ we refine our clustering criteria to show both $A$ and $mathcal{A}$ are compact, symmetric, and star-like, and also obtain both upper and lower bound sets for $mathcal{A}$. When the coordinates of $X$ in $mathbb{R}^d$ are independent random variables, we are able to represent $mathcal {A}$ in terms of $A$ and the classical Strassen set $mathcal{K}$, and, except for degenerate cases, show $mathcal{A}$ is strictly larger than the lower bound set whenever $dge2$. In addition, we show that for any compact, symmetric, star-like subset $A$ of $mathbb{R}^d$, there exists an $X$ such that the corresponding functional cluster set $mathcal{A}$ is always the lower bound subset. If $d=2$, then additional refinements identify $mathcal{A}$ as a subset of ${(x_1g_1,x_2g_2):(x_1,x_2)in A,g_1,g_2inmathcal{K}}$, which is the functional cluster set obtained when the coordinates are assumed to be independent.
{"title":"cluster sets for partial sums and partial sum processes","authors":"U. Einmahl, J. Kuelbs","doi":"10.1214/12-AOP827","DOIUrl":"https://doi.org/10.1214/12-AOP827","url":null,"abstract":"Let $X,X_1,X_2,ldots$ be i.i.d. mean zero random vectors with values in a separable Banach space $B$, $S_n=X_1+cdots+X_n$ for $nge1$, and assume ${c_n:nge1}$ is a suitably regular sequence of constants. Furthermore, let $S_{(n)}(t),0le tle1$ be the corresponding linearly interpolated partial sum processes. We study the cluster sets $A=C({S_n/c_n})$ and $mathcal{A}=C({S_{(n)}(cdot)/c_n})$. In particular, $A$ and $mathcal{A}$ are shown to be nonrandom, and we derive criteria when elements in $B$ and continuous functions $f:[0,1]to B$ belong to $A$ and $mathcal{A}$, respectively. When $B=mathbb{R}^d$ we refine our clustering criteria to show both $A$ and $mathcal{A}$ are compact, symmetric, and star-like, and also obtain both upper and lower bound sets for $mathcal{A}$. When the coordinates of $X$ in $mathbb{R}^d$ are independent random variables, we are able to represent $mathcal {A}$ in terms of $A$ and the classical Strassen set $mathcal{K}$, and, except for degenerate cases, show $mathcal{A}$ is strictly larger than the lower bound set whenever $dge2$. In addition, we show that for any compact, symmetric, star-like subset $A$ of $mathbb{R}^d$, there exists an $X$ such that the corresponding functional cluster set $mathcal{A}$ is always the lower bound subset. If $d=2$, then additional refinements identify $mathcal{A}$ as a subset of ${(x_1g_1,x_2g_2):(x_1,x_2)in A,g_1,g_2inmathcal{K}}$, which is the functional cluster set obtained when the coordinates are assumed to be independent.","PeriodicalId":50763,"journal":{"name":"Annals of Probability","volume":"42 1","pages":"1121-1160"},"PeriodicalIF":2.3,"publicationDate":"2014-03-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1214/12-AOP827","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"65971544","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We introduce an interacting particle system in which two families of reflected diffusions interact in a singular manner near a deterministic interface II. This system can be used to model the transport of positive and negative charges in a solar cell or the population dynamics of two segregated species under competition. A related interacting random walk model with discrete state spaces has recently been introduced and studied in Chen and Fan (2014). In this paper, we establish the functional law of large numbers for this new system, thereby extending the hydrodynamic limit in Chen and Fan (2014) to reflected diffusions in domains with mixed-type boundary conditions, which include absorption (harvest of electric charges). We employ a new and direct approach that avoids going through the delicate BBGKY hierarchy.
{"title":"Systems of interacting diffusions with partial annihilation through membranes","authors":"Zhen-Qing Chen, W. Fan","doi":"10.1214/15-AOP1047","DOIUrl":"https://doi.org/10.1214/15-AOP1047","url":null,"abstract":"We introduce an interacting particle system in which two families of reflected diffusions interact in a singular manner near a deterministic interface II. This system can be used to model the transport of positive and negative charges in a solar cell or the population dynamics of two segregated species under competition. A related interacting random walk model with discrete state spaces has recently been introduced and studied in Chen and Fan (2014). In this paper, we establish the functional law of large numbers for this new system, thereby extending the hydrodynamic limit in Chen and Fan (2014) to reflected diffusions in domains with mixed-type boundary conditions, which include absorption (harvest of electric charges). We employ a new and direct approach that avoids going through the delicate BBGKY hierarchy.","PeriodicalId":50763,"journal":{"name":"Annals of Probability","volume":"45 1","pages":"100-146"},"PeriodicalIF":2.3,"publicationDate":"2014-03-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1214/15-AOP1047","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"66032664","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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