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- [1] arXiv:2506.04424 [中文pdf, pdf, html, 其他]
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标题： Bakry-Émery梯度估计在Dyson布朗运动中的相变标题： A phase transition in the Bakry-Émery gradient estimate for Dyson Brownian motion评论： 欢迎评论主题： 概率 (math.PR) ; 数学物理 (math-ph) ; 泛函分析 (math.FA)
本文中，我们在与有限粒子 Dyson 布朗运动 (DBM) 对应的空间中发现了一个间隙，即 Bakry-Émery$N$-Ricci 张量的下界与逆温为$0<\beta<1$的 Bakry-Émery 梯度估计${\rm Ric}_N$和${\sf BE}$之间的差距。 也就是说，我们证明了，对于加权空间$(\mathbb R^n, w_\beta)$（其中$w_\beta=\prod_{i<j}^n |x_i-x_j|^\beta$和任意$N\in[n+\frac{\beta}{2}n(n-1),+\infty]$）， $\beta \ge 1 \implies {\rm Ric}_N \ge 0 \ \& \ {\sf BE}(0,N)$成立； $0 < \beta < 1 \implies {\rm Ric}_N \ge 0$成立而${\sf BE}(0,N)$不成立，这表明在小反温度参数下，Dyson布朗运动关于Bakry-Émery曲率下界的性质发生了相变。
In this paper, we find a gap between the lower bound of the Bakry-\'Emery $N$-Ricci tensor ${\rm Ric}_N$ and the Bakry-\'Emery gradient estimate ${\sf BE}$ in the space associated with the finite-particle Dyson Brownian motion (DBM) with inverse temperature $0<\beta<1$. Namely, we prove that, for the weighted space $(\mathbb R^n, w_\beta)$ with $w_\beta=\prod_{i<j}^n |x_i-x_j|^\beta$ and any $N\in[n+\frac{\beta}{2}n(n-1),+\infty]$, $\beta \ge 1 \implies {\rm Ric}_N \ge 0 \ \& \ {\sf BE}(0,N)$ hold; $0 < \beta < 1 \implies {\rm Ric}_N \ge 0$ holds while ${\sf BE}(0,N)$ does not, which shows a phase transition of the Dyson Brownian motion regarding the Bakry-\'Emery curvature bound in the small inverse temperature regime.
- [2] arXiv:2506.04560 [中文pdf, pdf, html, 其他]
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标题： 复独立同分布随机矩阵的最右特征值的收敛率的普适性标题： Universality of convergence rate of rightmost eigenvalue of complex IID random matrices主题： 概率 (math.PR)
设 $X$ 是一个 $n\times n$ 阶的矩阵，其独立同分布（i.i.d.）的元素为 $x_{ij} \stackrel{\text { d }}{=} n^{-1 / 2} \xi$，其中 $\xi$ 是一个均值为零、方差为一的复随机变量。 令 $\{\sigma_i\}_{1\le i\le n}$ 为 $X,$ 的特征值，$R_n:=\max_i \Re \sigma_i$ 和 $Z_n$ 是 $R_n.$ 的某个尺度变换版本。已经证明了在 $\xi.$ 满足某些矩条件时，$Z_n$ 弱收敛于 Gumbel 分布 $\Lambda$。我们进一步证明，对于具有独立同分布的复随机矩阵，... 条目 $$\sup_{x\in \mathbb{R}}|\mathbb{P}(Z_n \leq x)-e^{-e^{-x}}|=\frac{25\log \log n}{4e \log n}(1+o(1))$$ 和 $$ W_1\left(\mathcal{L}(Z_n), \Lambda\right)=\frac{25\log \log n}{4\log n}(1+o(1))$$ 对于足够大的 $n$，其中 $\mathcal{L}(Z_n)$ 是 $Z_n$的分布。
Let $X$ be an $n\times n$ matrix with independent and identically distributed (i.i.d.) entries $x_{ij} \stackrel{\text { d }}{=} n^{-1 / 2} \xi$ with $\xi$ being a complex random variable of mean zero and variance one. Let $\{\sigma_i\}_{1\le i\le n}$ be the eigenvalues of $X,$ and $R_n:=\max_i \Re \sigma_i$ and $Z_n$ be some scaled version of $R_n.$ It was proved that $Z_n$ converges weakly to the Gumbel distribution $\Lambda$ under certain moment conditions on $\xi.$ We further prove that for a complex random matrix with i.i.d. entries $$\sup_{x\in \mathbb{R}}|\mathbb{P}(Z_n \leq x)-e^{-e^{-x}}|=\frac{25\log \log n}{4e \log n}(1+o(1))$$ and $$ W_1\left(\mathcal{L}(Z_n), \Lambda\right)=\frac{25\log \log n}{4\log n}(1+o(1))$$ for sufficiently large $n$, where $\mathcal{L}(Z_n)$ is the distribution of $Z_n$.
- [3] arXiv:2506.04783 [中文pdf, pdf, 其他]
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标题： 具有吸收的谱负分支 L{é}vy 过程的总群体规模标题： Total progeny for spectrally negative branching L{é}vy processes with absorptionChristophe Profeta (LaMME)主题： 概率 (math.PR)
我们考虑一个光谱上为负的分支 Lévy 过程，在该过程中粒子会在越过零以下时被杀死。已知如果向 -$\infty$的漂移足够强以抵消繁殖率，则此类过程几乎肯定会灭绝。本文档研究了在过程寿命期间吸收于边界的粒子数的尾部分布渐近性，涵盖了次临界和临界情形。
We consider a spectrally negative branching L{\'e}vy process in which particles are killed upon crossing below zero. It is known that such a process becomes extinct almost surely if the drift toward -$\infty$ is sufficiently strong to counterbalance the reproduction rate. In this note, we study the tail asymptotics of the number of particles absorbed at the boundary during the lifetime of the process, in both the subcritical and critical regimes.
- [4] arXiv:2506.04797 [中文pdf, pdf, html, 其他]
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标题： 有限码与泊松可表示过程的随机支配标题： Finitary codings and stochastic domination for Poisson representable processes评论： 20页主题： 概率 (math.PR)
构造一个随机集，通过独立地以某种依赖于集合的概率选择整数的每个有限子集，并考虑这些选定集合的并集（直到平移）。 我们证明，当只有成对的集合以正概率被选中时，这种随机集是独立同分布（IID）过程的一个有限因子，从而回答了 Forsström、Gantert 和 Steif 提出的问题。 更一般地，我们证明只要由选定集合大小诱导的分布具有足够的指数矩，则这种情况总是成立，并且存在某些指数矩是必要的。 我们进一步证明，这种随机集被非平凡的伯努利渗流随机支配当且仅当存在有限指数矩，从而部分回答了 Forsström 等人提出的另一个问题。 我们还部分回答了第三个关于相变形式的问题。 这些结果在 $\mathbb{Z}^d$ 上同样成立，伴随 $d \ge 2$。 在一维情况下，如果由选定集合直径诱导的分布具有指数矩，我们进一步证明这种随机集是与 IID 过程有限同构的。
Construct a random set by independently selecting each finite subset of the integers with some probability depending on the set up to translations and taking the union of the selected sets. We show that when the only sets selected with positive probability are pairs, such a random set is a finitary factor of an IID process, answering a question of Forsstr\"om, Gantert and Steif. More generally, we show that this is the case whenever the distribution induced by the size of the selected sets has sufficient exponential moments, and that the existence of some exponential moment is necessary. We further show that such a random set is stochastically dominated by a non-trivial Bernoulli percolation if and only if there is a finite exponential moment, thereby partially answering another question of Forsstr\"om et al. We also give a partial answer to a third question regarding a form of phase transition. These results also hold on $\mathbb{Z}^d$ with $d \ge 2$. In the one-dimensional case, under the condition that the distribution induced by the diameter of the selected sets has an exponential moment, we further show that such a random set is finitarily isomorphic to an IID process.
- [5] arXiv:2506.04801 [中文pdf, pdf, html, 其他]
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标题： 二维和三维庞加莱域上一类非牛顿型微分型流体的随机动力学与不变测度标题： Random dynamics and invariant measures for a class of non-Newtonian fluids of differential type on 2D and 3D Poincaré domains主题： 概率 (math.PR) ; 偏微分方程分析 (math.AP)
本文中，我们研究二维和三维庞加莱区域 $\mathcal{O}$ （可能是有界或无界的）上一类不可压缩随机三阶流体（非牛顿流体）方程的适定性和渐近分析问题。首先，我们证明定义在 $\mathcal{O}$ 上的系统在齐次狄利克雷边界条件下存在唯一的弱解（解析意义下），并且它生成了一个随机动力系统 $\Psi$。其次，我们研究有界区域上的系统，利用紧致Sobolev嵌入 $\mathbb{H}^1(\mathcal{O}) \hookrightarrow\mathbb{L}^2(\mathcal{O})$，证明了在外部力属于 $\mathbb{H}^{-1}(\mathcal{O})+\mathbb{W}^{-1,\frac{4}{3}}(\mathcal{O})$ 的情况下，该系统在有界域上存在唯一的随机吸引子。最后，我们在外部力属于 $\mathbb{L}^{2}(\mathcal{O})$ 的无界庞加莱域上考虑该系统，并证明了存在唯一的随机吸引子。 为了在非有界域上得到唯一随机吸引子的存在性，由于缺乏紧的Sobolev嵌入$\mathbb{H}^1(\mathcal{O}) \hookrightarrow\mathbb{H}^2(\mathcal{O})$，我们采用了均匀尾部估计方法来证明$\Psi$的渐近紧性。 需要注意的是，由于所研究系统中存在多个非线性项，我们无法使用能量等式方法来获得$\Psi$在非有界域上的渐近紧性，这使得本工作在非有界域上的分析更加困难和有趣。 最后，作为随机吸引子存在性的结果，我们探讨了所研究系统的不变测度的存在性。
In this article, we consider a class of incompressible stochastic third-grade fluids (non-Newtonian fluids) equations on two- as well as three-dimensional Poincar\'e domains $\mathcal{O}$ (which may be bounded or unbounded). Our aims are to study the well-posedness and asymptotic analysis for the solutions of the underlying system. Firstly, we prove that the underlying system defined on $\mathcal{O}$ has a unique weak solution (in the analytic sense) under Dirichlet boundary condition and it also generates random dynamical system $\Psi$. Secondly, we consider the underlying system on bounded domains. Using the compact Sobolev embedding $\mathbb{H}^1(\mathcal{O}) \hookrightarrow\mathbb{L}^2(\mathcal{O})$, we prove the existence of a unique random attractor for the underlying system on bounded domains with external forcing in $\mathbb{H}^{-1}(\mathcal{O})+\mathbb{W}^{-1,\frac{4}{3}}(\mathcal{O})$. Thirdly, we consider the underlying system on unbounded Poincar\'e domains with external forcing in $\mathbb{L}^{2}(\mathcal{O})$ and show the existence of a unique random attractor. In order to obtain the existence of a unique random attractor on unbounded domains, due to the lack of compact Sobolev embedding $\mathbb{H}^1(\mathcal{O}) \hookrightarrow\mathbb{H}^2(\mathcal{O})$, we use the uniform-tail estimates method which helps us to demonstrate the asymptotic compactness of $\Psi$. Note that due to the presence of several nonlinear terms in the underlying system, we are not able to use the energy equality method to obtain the asymptotic compactness of $\Psi$ in unbounded domains, which makes the analysis of this work in unbounded domains more difficult and interesting. Finally, as a consequence of the existence of random attractors, we address the existence of invariant measures for underlying system.
- [6] arXiv:2506.04896 [中文pdf, pdf, html, 其他]
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标题： 关于阿尔伯特·希利亚耶夫在马尔可夫决策过程方面的开创性工作及后续发展标题： On Pioneering Works of Albert Shiryaev on Markov Decision Processes and Some Later Developments主题： 概率 (math.PR) ; 优化与控制 (math.OC)
本文献给阿尔贝特·希利亚耶夫大约六十年前发表的关于马尔可夫决策过程和不完全观测控制的三篇基础性论文。 其中一篇论文与O.V. 维斯科夫共同撰写。 我们讨论了这些论文中提出的一些结果以及许多丰富的思想，并概述了一些后续的发展。 最后，我们提到阿尔贝特·希利亚耶夫最近对跳跃型马尔可夫过程的柯尔莫哥洛夫方程以及对连续时间跳跃型马尔可夫过程的控制的一些研究。
This article is dedicated to three fundamental papers on Markov Decision Processes and on control with incomplete observations published by Albert Shiryaev approximately sixty years ago. One of these papers was coauthored with O.V. Viskov. We discuss some of the results and some of many rich ideas presented in these papers and survey some later developments. At the end we mention some recent studies of Albert Shiryaev on Kolmogorov's equations for jump Markov processes and on control of continuous-time jump Markov processes.
- [7] arXiv:2506.04911 [中文pdf, pdf, html, 其他]
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标题： 随机Volterra方程在具有广义核的凸域上的弱解标题： Weak solutions of Stochastic Volterra Equations in convex domains with general kernels主题： 概率 (math.PR)
我们为具有连续系数且可能具有一维非卷积奇异核的$d$维随机 Volterra 方程（SVEs）建立了新的弱存在性结果。通过引入一种逼近方案并证明其收敛性获得了这些结果。特别强调了解在闭凸集中的随机不变性。为此，我们将\cite{Alfonsi23}中引入的保持非负性的核概念扩展到非卷积核，并证明，在由相应的随机微分方程确定的闭凸集具有适当的随机不变性性质时，存在一个保持在该凸集内的 SVE 弱解。我们给出了一类满足假设的非卷积核，包括著名的分数核的非卷积扩展。我们将结果应用于具有平方根扩散系数和非卷积核的 SVE，证明了解在非负正交象限内弱存在且唯一。我们推导出 Laplace 变换的一个表示形式，即非卷积Riccati方程，并对此建立了存在性结果。
We establish new weak existence results for $d$-dimensional Stochastic Volterra Equations (SVEs) with continuous coefficients and possibly singular one-dimensional non-convolution kernels. These results are obtained by introducing an approximation scheme and showing its convergence. A particular emphasis is made on the stochastic invariance of the solution in a closed convex set. To do so, we extend the notion of kernels that preserve nonnegativity introduced in \cite{Alfonsi23} to non-convolution kernels and show that, under suitable stochastic invariance property of a closed convex set by the corresponding Stochastic Differential Equation, there exists a weak solution of the SVE that stays in this convex set. We present a family of non-convolution kernels that satisfy our assumptions, including a non-convolution extension of the well-known fractional kernel. We apply our results to SVEs with square-root diffusion coefficients and non-convolution kernels, for which we prove the weak existence and uniqueness of a solution that stays within the nonnegative orthant. We derive a representation of the Laplace transform in terms of a non-convolution Riccati equation, for which we establish an existence result.
- [8] arXiv:2506.05139 [中文pdf, pdf, html, 其他]
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标题： 无穷小自由性对于正交不变随机矩阵的应用标题： Infinitesimal freeness for orthogonally invariant random matricesGuillaume Cébron (Toulouse), James A Mingo (Queen's)评论： 44页主题： 概率 (math.PR) ; 算子代数 (math.OA)
我们引入了一种新的自由独立性概念，称为实无穷小自由性。 我们证明了独立的正交不变且具有无穷小律的元素是渐近实无穷小自由的。 我们引入了新的累积量，称为实无穷小累积量，并证明了实无穷小自由性等价于混合累积量的消失。 我们证明了累积量与乘积作为条目的公式。
We introduce a new kind of free independence, called real infinitesimal freeness. We show that independent orthogonally invariant with infinitesimal laws are asymptotically real infinitesimally free. We introduce new cumulants, called real infinitesimal cumulants and show that real infinitesimal freeness is equivalent to vanishing of mixed cumulants. We prove the formula for cumulants with products as entries.
- [9] arXiv:2506.05246 [中文pdf, pdf, html, 其他]
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标题： 近视非交集在一个周期势中标题： Myopic non-intersection in a periodic potential评论： 26页，3个图主题： 概率 (math.PR) ; 数学物理 (math-ph)
我们引入了一类马尔可夫过程，这些过程被限制在一个长度为 T > 0 的移动时间窗口内避免相交，我们将这种设置称为近视非相交。特别地，我们研究了一组受到周期势约束的近视非相交布朗运动。我们的重点在于理解势场的约束效应与由非相交约束引起的排斥作用之间的相互作用。我们证明，在长时间极限下，当 T 和势场强度都变得很大时，该模型收敛到一组近视非相交随机游走，其动态在标准非相交动力学和排斥行为之间切换。本文的主要技术贡献是一种算法的提出，该算法基于接受-拒绝抽样方案的修改，提供了近视约束系统的显式构造。
We introduce a class of Markov processes conditioned to avoid intersection over a moving time window of length T>0, a setting we refer to as myopic non-intersection. In particular, we study a system of myopic non-intersecting Brownian motions subject to a periodic potential. Our focus lies in understanding the interplay between the confining effect of the potential and the repulsion induced by the non-intersection constraint. We show that, in the long time limit, and as both T and the strength of the potential become large, the model converges to a system of myopic non-intersecting random walks, which transitions between standard non-intersection dynamics and exclusion behavior. The main technical contribution of the paper is the introduction of an algorithm, based on a modification of the acceptance-rejection sampling scheme, that provides an explicit construction of myopically constrained systems.
新提交 (展示 9 之 9 条目 )
- [10] arXiv:2506.04386 (交叉列表自 cs.DS) [中文pdf, pdf, html, 其他]
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标题： 关于通过平稳时间的演化图的谣言传播标题： Rumors on evolving graphs through stationary times评论： 11页主题： 数据结构与算法 (cs.DS) ; 离散数学 (cs.DM) ; 概率 (math.PR)
我们研究了动态随机图中的谣言传播。从一个知情顶点开始，信息按照以下过程流动，直到到达图中的所有顶点（完成）。 在每一步 $k$，信息被传播到知情顶点的邻居，在第 $k$ 次生成的随机图中。 每一步中信息从一个顶点传播到另一个顶点的方式取决于“协议”。 当图是马尔可夫时间相关的时，我们提供了一种基于强平稳时间的方法来研究完成时间，并利用独立图的已知文献结果。 然后通过来自过去的耦合算法将强平稳时间的概念扩展到非马尔可夫动力学。 这允许将非马尔可夫动力学下的完成时间结果扩展到更广泛的情况。
We study rumor spreading in dynamic random graphs. Starting with a single informed vertex, the information flows until it reaches all the vertices of the graph (completion), according to the following process. At each step $k$, the information is propagated to neighbors of the informed vertices, in the $k$-th generated random graph. The way this information propagates from vertex to vertex at each step will depend on the ``protocol". We provide a method based on strong stationary times to study the completion time when the graphs are Markovian time dependent, using known results of the literature for independent graphs. The concept of strong stationary times is then extended to non-Markovian Dynamics using coupling from the past algorithms. This allows to extend results on completion times for non-Markov dynamics
- [11] arXiv:2506.04436 (交叉列表自 cs.SC) [中文pdf, pdf, html, 其他]
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标题： 超越最坏情况分析的符号计算：根隔离算法标题： Beyond Worst-Case Analysis for Symbolic Computation: Root Isolation Algorithms评论： 27页。 arXiv:2202.06428的扩展期刊版本。主题： 符号计算 (cs.SC) ; 计算复杂性 (cs.CC) ; 代数几何 (math.AG) ; 概率 (math.PR)
我们将超越最坏情况的分析引入到符号计算中。 这是一个几乎完全依赖于最坏情况位复杂度的广泛领域，我们从该领域的基本问题开始：隔离单变量多项式的实根。 这是符号计算中的一个基础问题，并且可以说是计算数学中最基本的问题之一。 这个问题有着悠久的历史，伴随着众多巧妙的算法，并构成了一个活跃的研究领域。 然而，文献中大多数现有的结果要么专注于位复杂度模型中的最坏情况分析，要么只是提供实验基准测试，而没有对观察到的结果进行任何理论上的解释。 我们的目标是解决根隔离算法的实际性能与最坏情况复杂性理论预测之间的差异：我们为具有整数系数的多项式开发了一个平滑分析框架以弥合这一差距。 我们展示了（准）线性的 （期望和平滑）复杂度界，针对的是笛卡尔算法，这是用于隔离具有整数系数的单变量多项式实根的最著名的符号算法之一。 我们的结果解释了笛卡尔求解器相比那些具有更优越最坏情况复杂性的复杂算法为何表现出令人惊讶的高效性。 我们还分析了斯图姆求解器，ANewDsc——一种结合了笛卡尔方法和牛顿算子的符号-数值算法，以及一种针对稀疏多项式的符号算法。
We introduce beyond-worst-case analysis into symbolic computation. This is an extensive field which almost entirely relies on worst-case bit complexity, and we start from a basic problem in the field: isolating the real roots of univariate polynomials. This is a fundamental problem in symbolic computation and it is arguably one of the most basic problems in computational mathematics. The problem has a long history decorated with numerous ingenious algorithms and furnishes an active area of research. However, most available results in literature either focus on worst-case analysis in the bit complexity model or simply provide experimental benchmarking without any theoretical justifications of the observed results. We aim to address the discrepancy between practical performance of root isolation algorithms and prescriptions of worst-case complexity theory: We develop a smoothed analysis framework for polynomials with integer coefficients to bridge this gap. We demonstrate (quasi-)linear (expected and smoothed) complexity bounds for Descartes algorithm, that is one most well know symbolic algorithms for isolating the real roots of univariate polynomials with integer coefficients. Our results explain the surprising efficiency of Descartes solver in comparison to sophisticated algorithms that have superior worst-case complexity. We also analyse the Sturm solver, ANewDsc a symbolic-numeric algorithm that combines Descartes with Newton operator, and a symbolic algorithm for sparse polynomials.
- [12] arXiv:2506.04700 (交叉列表自 cs.LG) [中文pdf, pdf, html, 其他]
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标题： 基于伯恩斯坦凸散度的神经隐式采样器的显式密度逼近标题： Explicit Density Approximation for Neural Implicit Samplers Using a Bernstein-Based Convex Divergence主题： 机器学习 (cs.LG) ; 人工智能 (cs.AI) ; 概率 (math.PR) ; 机器学习 (stat.ML)
基于排名的统计度量，例如不变统计损失（ISL），最近成为训练隐式生成模型的强大且实用有效的工具。 在这项工作中，我们引入了双-ISL，这是一种新颖的无显式似然目标函数，用于训练隐式生成模型，在ISL框架中互换了目标分布和模型分布的角色，从而在模型密度空间上产生一个凸优化问题。 我们证明，由此产生的基于排名的差异$d_K$具有以下性质：i）在弱收敛下以及与$L^1$范数相关时是连续的，并且 ii）在其第一个参数上是凸的——这些性质经典散度（如KL或Wasserstein距离）所不具备。 在此基础上，我们开发了一个理论框架，将$d_K$解释为密度比$q = p/\tilde p$在伯恩斯坦多项式基上的$L^2$投影，从中我们推导出截断误差的确切界限、精确的收敛速率以及截断密度近似的闭合形式表达。 我们进一步通过随机一维投影扩展了这一分析到多元设置，定义了一种切片双-ISL散度，保留了凸性和连续性。 我们实证表明，这些理论优势转化为实际优势。 具体而言，在多个基准测试中，双-ISL收敛更快，提供明显更平滑且更稳定的训练过程，并且比经典ISL和其他领先的隐式生成方法更有效地防止模式崩溃，同时还能提供明确的密度近似。
Rank-based statistical metrics, such as the invariant statistical loss (ISL), have recently emerged as robust and practically effective tools for training implicit generative models. In this work, we introduce dual-ISL, a novel likelihood-free objective for training implicit generative models that interchanges the roles of the target and model distributions in the ISL framework, yielding a convex optimization problem in the space of model densities. We prove that the resulting rank-based discrepancy $d_K$ is i) continuous under weak convergence and with respect to the $L^1$ norm, and ii) convex in its first argument-properties not shared by classical divergences such as KL or Wasserstein distances. Building on this, we develop a theoretical framework that interprets $d_K$ as an $L^2$-projection of the density ratio $q = p/\tilde p$ onto a Bernstein polynomial basis, from which we derive exact bounds on the truncation error, precise convergence rates, and a closed-form expression for the truncated density approximation. We further extend our analysis to the multivariate setting via random one-dimensional projections, defining a sliced dual-ISL divergence that retains both convexity and continuity. We empirically show that these theoretical advantages translate into practical ones. Specifically, across several benchmarks dual-ISL converges more rapidly, delivers markedly smoother and more stable training, and more effectively prevents mode collapse than classical ISL and other leading implicit generative methods-while also providing an explicit density approximation.
- [13] arXiv:2506.04878 (交叉列表自 math.ST) [中文pdf, pdf, html, 其他]
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标题： kTULA：一种具有改进的超线性对数梯度KL界的Langevin抽样算法标题： kTULA: A Langevin sampling algorithm with improved KL bounds under super-linear log-gradients主题： 统计理论 (math.ST) ; 机器学习 (cs.LG) ; 概率 (math.PR) ; 机器学习 (stat.ML)
受深度学习应用的启发，其中全局Lipschitz连续性条件通常不满足，我们研究了从具有超线性增长的对数梯度的分布中采样的问题。我们提出了一种基于改进型 Langevin 动力学的新算法，称为kTULA，以解决上述采样问题，并为其性能提供了理论保证。更具体地说，我们在 Kullback-Leibler (KL) 散度中建立了非渐近收敛界，其最佳收敛率等于 $2-\overline{\epsilon}$, $\overline{\epsilon}>0$，这显著改进了现有文献中的相关结果。这使我们能够在 Wasserstein-2 距离中获得改进的非渐近误差界，该界可以进一步用于推导出kTULA解决相关优化问题的非渐近保证。为了展示kTULA的适用性，我们将所提出的算法应用于从高维双井势分布中采样的问题以及涉及神经网络的优化问题。我们表明，我们的主要结果可以用于提供kTULA性能的理论保证。
Motivated by applications in deep learning, where the global Lipschitz continuity condition is often not satisfied, we examine the problem of sampling from distributions with super-linearly growing log-gradients. We propose a novel tamed Langevin dynamics-based algorithm, called kTULA, to solve the aforementioned sampling problem, and provide a theoretical guarantee for its performance. More precisely, we establish a non-asymptotic convergence bound in Kullback-Leibler (KL) divergence with the best-known rate of convergence equal to $2-\overline{\epsilon}$, $\overline{\epsilon}>0$, which significantly improves relevant results in existing literature. This enables us to obtain an improved non-asymptotic error bound in Wasserstein-2 distance, which can be used to further derive a non-asymptotic guarantee for kTULA to solve the associated optimization problems. To illustrate the applicability of kTULA, we apply the proposed algorithm to the problem of sampling from a high-dimensional double-well potential distribution and to an optimization problem involving a neural network. We show that our main results can be used to provide theoretical guarantees for the performance of kTULA.
- [14] arXiv:2506.05194 (交叉列表自 math.CO) [中文pdf, pdf, html, 其他]
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标题： 星图分解与定向标题： Star decompositions via orientations主题： 组合数学 (math.CO) ; 概率 (math.PR)
一个图的$k$-星分解是指将其边划分为若干个$k$-星（即以某一公共顶点为中心的$k$条边）。本文研究了如下问题：给定$k \leq d/2$，随机$d$-正则图是否具有一个$k$-星分解（渐近几乎必然成立，前提是边的数量能被$k$整除）？Delcourt、Greenhill、Isaev、Lidický和Postle证明了渐近几乎必然成立的结果。 对于每个奇数$k$，利用之前关于满足某些模$k$度条件的定向图的结果，存在性已知。 本文给出了一个直接且自包含的证明，适用于每个$d$和每个$k<d/2-1$。 事实上，我们证明了更强的结果。 令$s\geq 1$表示$d/(2k)$的整数部分。 我们证明随机$d$-正则图几乎处处成立。 具有一个$k$-星分解，使得每个顶点为中心的星的数量要么是$s$要么是$s+1$。 此外，如果$k < d/3$或$k \leq d/2 - 2.6 \log d$，我们可以甚至指定带有$s$颗星的顶点集合，只要它的大小合适。
A $k$-star decomposition of a graph is a partition of its edges into $k$-stars (i.e., $k$ edges with a common vertex). The paper studies the following problem: given $k \leq d/2$, does the random $d$-regular graph have a $k$-star decomposition (asymptotically almost surely, provided that the number of edges is divisible by $k$)? Delcourt, Greenhill, Isaev, Lidick\'y, and Postle proved the a.a.s. existence for every odd $k$ using earlier results regarding orientations satisfying certain degree conditions modulo $k$. In this paper we give a direct, self-contained proof that works for every $d$ and every $k<d/2-1$. In fact, we prove stronger results. Let $s\geq 1$ denote the integer part of $d/(2k)$. We show that the random $d$-regular graph a.a.s. has a $k$-star decomposition such that the number of stars centered at each vertex is either $s$ or $s+1$. Moreover, if $k < d/3$ or $k \leq d/2 - 2.6 \log d$, we can even prescribe the set of vertices with $s$ stars, as long as it is of the appropriate size.
- [15] arXiv:2506.05269 (交叉列表自 quant-ph) [中文pdf, pdf, html, 其他]
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标题： 量子 dynamical 半群的状态空间分解标题： State Space Decomposition of Quantum Dynamical Semigroups评论： 这将在IEEE qCCL 2025上发表。主题： 量子物理 (quant-ph) ; 数学物理 (math-ph) ; 优化与控制 (math.OC) ; 概率 (math.PR)
开放量子系统的连续时间平均演化由时间连续的量子信道（完全正且迹保持的线性映射）半群描述。 Baumgartner 和 Narnhofer 提出了一个关于基本 Hilbert 空间的通用分解，这些子空间也称为禁闭态。 我们提出对该结果的一种新解读，受 Carbone 和 Pautrat 工作的启发。 此外，我们将该分解应用于一类开放量子随机游走和量子轨迹，并研究其唯一性。
The mean evolution of an open quantum system in continuous time is described by a time continuous semigroup of quantum channels (completely positive and trace-preserving linear maps). Baumgartner and Narnhofer presented a general decomposition of the underlying Hilbert space into a sum of invariant subspaces, also called enclosures. We propose a new reading of this result, inspired by the work of Carbone and Pautrat. In addition, we apply this decomposition to a class of open quantum random walks and to quantum trajectories, where we study its uniqueness.
交叉提交 (展示 6 之 6 条目 )
- [16] arXiv:2004.12815 (替换) [中文pdf, pdf, 其他]
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标题： 洛伦兹系统的噪声诱导过渡标题： A noise-induced transition in the Lorenz system评论： 修正了第5节中的数学印刷错误主题： 概率 (math.PR) ; 经典分析与常微分方程 (math.CA) ; 动力系统 (math.DS)
我们研究了经典Lorenz系统的随机扰动，参数范围为原点是全局吸引子的情况。 我们证明，在最后一个分量中加入噪声会导致唯一遍历不变测度到恰好两个遍历不变测度的转变。 分歧阈值依赖于噪声的强度：如果噪声较弱，则唯一的不变测度是高斯测度，而足够强的噪声会引起第二个遍历不变测度的出现。
We consider a stochastic perturbation of the classical Lorenz system in the range of parameters for which the origin is the global attractor. We show that adding noise in the last component causes a transition from a unique to exactly two ergodic invariant measures. The bifurcation threshold depends on the strength of the noise: if the noise is weak, the only invariant measure is Gaussian, while strong enough noise causes the appearance of a second ergodic invariant measure.
- [17] arXiv:2312.02965 (替换) [中文pdf, pdf, html, 其他]
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标题： 概率分布的条件约束和非约束量化标题： Conditional constrained and unconstrained quantization for probability distributions主题： 概率 (math.PR)
本文中，我们引入并在$\mathbb{R}^k,$上的波尔概率测度的条件量化概念方面进行了发展，同时考虑了受限和不受限框架。 对于每种设定，我们都定义了相关的量化误差、维数和系数，并对特定类别的概率分布提供了明确的计算。 不受限情况下的一个关键结果是，所有$ n$-均值最优集的并集在测度的支持集中是稠密的。 此外，我们证明了在条件受限量化中，如果条件集包含在约束族的并集中，则对于任何波尔概率测度，下界和上界的量化维数以及相应的系数不会受到条件集的影响。 相比之下，如果条件集不包含在这个并集中，这些性质可能不再成立，这一点通过各种例子加以说明。
In this paper, we introduce and develop the concept of conditional quantization for Borel probability measures on $\mathbb{R}^k,$ considering both constrained and unconstrained frameworks. For each setting, we define the associated quantization errors, dimensions, and coefficients, and provide explicit computations for specific classes of probability distributions. A key result in the unconstrained case is that the union of all optimal sets of $ n$-means is dense in the support of the measure. Furthermore, we demonstrate that in conditional constrained quantization, if the conditional set is contained within the union of the constraint family, then the lower and upper quantization dimensions, as well as the corresponding coefficients, remain unaffected by the conditional set for any Borel probability measure. In contrast, if the conditional set is not contained within this union, these properties may no longer hold, as illustrated through various examples.
- [18] arXiv:2403.04324 (替换) [中文pdf, pdf, html, 其他]
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标题： 可数状态空间下的次线性期望结构标题： Sublinear expectation structure under countable state space主题： 概率 (math.PR)
在这项研究中，我们提出了可数状态空间下的次线性期望结构。 为了描述一个有趣的“非线性随机化”试验，基于凸紧致域，我们在可数状态空间下引入了一组概率测度。 对应于彭实戈提出的次线性期望算子，我们考虑了可数状态空间下的相关概念。 在可数状态框架下，次线性期望可以通过一个新的重复求和公式显式计算，并给出了一些有趣的例子。 此外，我们建立了次线性期望的单调收敛定理、Fatou引理和控制收敛定理。 随后，我们研究了每个概率测度下的独立性，并在此基础上建立了次线性期望下的大数定律，得到了次线性期望下的最大分布。
In this study, we propose the sublinear expectation structure under countable state space. To describe an interesting "nonlinear randomized" trial, based on a convex compact domain, we introduce a family of probability measures under countable state space. Corresponding the sublinear expectation operator introduced by S. Peng, we consider the related notation under countable state space. Within the countable state framework, the sublinear expectation can be explicitly calculated by a novel repeated summation formula, and some interesting examples are given. Furthermore, we establish Monotone convergence theorem, Fatou's lemma and Dominated convergence theorem of sublinear expectation. Afterwards, we consider the independence under each probability measure, upon which we establish the sublinear law of large numbers and obtain the maximal distribution under sublinear expectation.
- [19] arXiv:2408.03218 (替换) [中文pdf, pdf, html, 其他]
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标题： 关于随机几何图投影中的穿越次数和应力的极限定理标题： Limit theorems for the number of crossings and stress in projections of the random geometric graph主题： 概率 (math.PR)
我们研究了在由投影随机几何图到某个紧凸集 $W\subset \mathbb{R}^d$, $d\geq 3$ 上的平面时产生的随机图绘制中的边交叉数。 这些交叉的位置形成了一个点过程的支持集。 我们证明了如果期望的交叉数收敛到一个正且有限的值，则该点过程以 Kantorovich-Rubinstein 距离收敛到一个泊松点过程。 我们进一步证明了一个多元中心极限定理，涉及交叉数和另一个称为应力的变量，在随机几何图中期望顶点度数收敛到一个正有限值的情况下成立。
We consider the number of edge crossings in a random graph drawing generated by projecting a random geometric graph on some compact convex set $W\subset \mathbb{R}^d$, $d\geq 3$, onto a plane. The positions of these crossings form the support of a point process. We show that if the expected number of crossings converges to a positive but finite value, this point process converges to a Poisson point process in the Kantorovich-Rubinstein distance. We further show a multivariate central limit theorem between the number of crossings and a second variable called the stress that holds when the expected vertex degree in the random geometric graph converges to a positive finite value.
- [20] arXiv:2409.11330 (替换) [中文pdf, pdf, 其他]
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标题： 依赖参数的粗糙随机微分方程及其在粗糙偏微分方程中的应用标题： Parameter dependent rough SDEs with applications to rough PDEs评论： 引言重写，并添加了更多参考文献。证明大幅简化。结果保持不变。主题： 概率 (math.PR)
随机微分方程（粗糙SDEs）由Friz、Hocquet和Lê最近在arXiv:2106.10340中引入，已成为研究“双重”SDEs（在部分条件下的情形下），其动机来自路径滤波与控制、金融中的波动率建模以及具有共同噪声的平均场随机动力学等领域的一个通用工具。 尽管整体的动力学可能高度非马尔可夫，但条件下的动力学通常可以是马尔可夫的。在自然（甚至线性）情况下，由此产生的随机偏微分方程可能超出了现有技术的能力范围。 本文解决了这一背景下的一个关键问题，即粗糙Kolmogorov向后方程的正则解的适定性问题。 为此，我们研究了依赖参数的粗糙SDEs在$\mathscr{L}$-可微性意义下的情况（类似于Krylov，2008年的工作）。在相关工作中，我们将展示如何通过这种方法消除Zakai方程、Kushner-Stratonovich方程和非线性Fokker-Planck随机方程适定性中的维度相关的正则性假设。
Rough stochastic differential equations (rough SDEs), recently introduced by Friz, Hocquet and L\^e in arXiv:2106.10340, have emerged as a versatile tool to study "doubly" SDEs under partial conditioning (with motivation from pathwise filtering and control, volatility modelling in finance and mean-field stochastic dynamics with common noise ...). While the full dynamics may be highly non-Markovian, the conditional dynamics often are. In natural (and even linear) situations, the resulting stochastic PDEs can be beyond existing technology. The present work then tackles a key problem in this context, which is the well-posedness of regular solution to the rough Kolmogorov backward equation. To this end, we study parameter dependent rough SDEs in sense of $\mathscr{L}$-differentiability (as in Krylov, 2008). In companion works, we will show how this removes dimension-dependent regularity assumptions for well-posedness of the Zakai, Kushner-Stratonovich and nonlinear Fokker-Planck stochastic equations.
- [21] arXiv:2409.14734 (替换) [中文pdf, pdf, html, 其他]
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标题： 准分数驱动波动模型的连续时间极限标题： The continuous-time limit of quasi score-driven volatility models评论： 在线发表于《时间序列分析期刊》主题： 概率 (math.PR) ; 计量经济学 (econ.EM)
本文探讨了一类表征波动性的准得分驱动（QSD）模型的连续时间极限。随着采样频率增加且时间间隔趋于零，该模型弱收敛于一个连续时间随机波动率模型，其中两个布朗运动相关，从而捕捉市场中的杠杆效应。随后，我们指出非退化相关性的一个必要条件是驱动创新分布与计算得分分布不同，并且至少有一个是对称的。接着，我们通过两个典型例子加以说明。作为应用，QSD 模型被用作相关随机波动率扩散的近似，并进行了准最大似然估计。仿真结果验证了该方法的有效性，特别是在估计相关系数方面。
This paper explores the continuous-time limit of a class of Quasi Score-Driven (QSD) models that characterize volatility. As the sampling frequency increases and the time interval tends to zero, the model weakly converges to a continuous-time stochastic volatility model where the two Brownian motions are correlated, thereby capturing the leverage effect in the market. Subsequently, we identify that a necessary condition for non-degenerate correlation is that the distribution of driving innovations differs from that of computing score, and at least one being asymmetric. We then illustrate this with two typical examples. As an application, the QSD model is used as an approximation for correlated stochastic volatility diffusions and quasi maximum likelihood estimation is performed. Simulation results confirm the method's effectiveness, particularly in estimating the correlation coefficient.
- [22] arXiv:2410.22038 (替换) [中文pdf, pdf, html, 其他]
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标题： 一个关于混合物的Cramér-Wold定理标题： A Cramér-Wold theorem for mixtures评论： 12页主题： 概率 (math.PR)
我们展示了如何将一族多元概率分布的Cramér-Wold定理用于生成针对来自同一族分布的混合分布（凸组合）的类似定理。 利用这一抽象结果，我们建立了多元高斯分布混合的Cramér-Wold定理。 根据该定理，可以通过将这两种混合投影到某些预先确定的有限数量的直线上来区分它们，直线的数量仅取决于涉及的高斯分布总数和环境维度。 对于多元 $t$-分布的混合，也得到了类似的结论。
We show how a Cram\'er-Wold theorem for a family of multivariate probability distributions can be used to generate a similar theorem for mixtures (convex combinations) of distributions drawn from the same family. Using this abstract result, we establish a Cram\'er-Wold theorem for mixtures of multivariate Gaussian distributions. According to this theorem, two such mixtures can be distinguished by projecting them onto a certain predetermined finite set of lines, the number of lines depending only on the total number Gaussian distributions involved and on the ambient dimension. A similar result is also obtained for mixtures of multivariate $t$-distributions.
- [23] arXiv:2411.07103 (替换) [中文pdf, pdf, html, 其他]
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标题： 关于全正性和伯努利停止问题之间的联系标题： On a connection between total positivity and Bernoulli stopping problems评论： 13页，1个图。修正了一些错误，并删除了第3.4节和第4.3节。主题： 概率 (math.PR)
考虑最大化序列独立伯努利试验成功时停止所获得的非负奖励的期望收益的最佳停止问题。 这些伯努利停止问题由连接停止收益和继续收益的递推关系刻画。 该递推关系用于在单峰继续收益下证明近视策略的最优性。 此外，通过将试验的成功时刻嵌入马尔可夫链，链的转移矩阵将停止收益映射到继续收益。 一个全正性论证表明，链的单峰函数的期望在初始状态下单峰。 此性质建立了停止收益的单峰性作为近视规则最优性的充分条件。 这一结果的说明性应用包括第$m$个最后成功问题及其推广。
Consider the optimal stopping problem of maximising the expected payoff in a game where a nonnegative reward is granted upon stopping on a success in a sequence of independent Bernoulli trials. These Bernoulli stopping problems are characterised by a recurrence relation connecting the stopping and continuation payoffs. This recurrence is used to establish optimality of the myopic strategy under unimodal continuation payoffs. Further, by embedding the success epochs of the trials into a Markov chain, the transition matrix of the chain maps the stopping payoffs into continuation payoffs. A total positivity argument shows that the expectation of a unimodal function of the chain is shown to be unimodal in the initial state. This property establishes the unimodality of the stopping payoffs as a sufficient condition for the optimality of the myopic rule. Illustrative applications of this result include the $m$th last-success problem and its generalisation.
- [24] arXiv:2502.21015 (替换) [中文pdf, pdf, html, 其他]
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标题： 不变子空间和布朗位移的$C_{00}$-性质标题： Invariant subspaces and the $C_{00}$-property of Brownian Shifts评论： 26页。进行了彻底的修订，现在结果已在向量值设定中得到发展。主题： 概率 (math.PR) ; 复变量 (math.CV) ; 泛函分析 (math.FA) ; 算子代数 (math.OA)
我们引入向量值Hardy空间上的Brown运动移位，并描述它们的不变子空间。然后，我们将Brown运动移位限制到它们的不变子空间上，并分类何时它们是么正等价的。此外，我们证明了一个渐近性质，指出归一化的Brown运动移位属于经典的$C_{00}$-类。
We introduce Brownian shifts on vector-valued Hardy spaces and describe their invariant subspaces. We then consider the restriction of Brownian shifts to their invariant subspaces and classify when they are unitarily equivalent. Additionally, we prove an asymptotic property stating that normalized Brownian shifts belong to the classical $C_{00}$-class.
- [25] arXiv:2504.07552 (替换) [中文pdf, pdf, html, 其他]
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标题： 超临界高斯乘性混沌的唯一性标题： Uniqueness of supercritical Gaussian multiplicative chaos评论： 21页。v2： minor fixes（已修正）主题： 概率 (math.PR) ; 数学物理 (math-ph)
我们证明了，对于一类对数相关的高斯场的一般卷积近似，归一化的超临界高斯乘性混沌测度在稳定意义下收敛到一个非平凡的极限。 这个极限仅依赖于正则化的选择，通过一个乘法常数体现出来，并且可以被描述为具有随机强度的原子测度的积分，其中随机强度以临界高斯乘性混沌的形式表示。
We show that, for general convolution approximations to a large class of log-correlated Gaussian fields, the properly normalised supercritical Gaussian multiplicative chaos measures converge stably to a nontrivial limit. This limit depends on the choice of regularisation only through a multiplicative constant and can be characterised as an integrated atomic measure with a random intensity expressed in terms of the critical Gaussian multiplicative chaos.
- [26] arXiv:2505.09400 (替换) [中文pdf, pdf, html, 其他]
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标题： 结构化凝聚过程、凝聚方程和多类型分支过程标题： Structured coalescents, coagulation equations and multi-type branching processes评论： 26页，5幅图主题： 概率 (math.PR)
考虑一个由$d$个群体组成的结构化种群，其中迁移率与一个正参数$K$成正比。 我们从$d$个群体中均匀分布地采样$N_K$个个体，并追溯它们的祖先谱系。 在一个群体内，我们假设祖先谱系成对凝聚的速率是恒定的，如同Kingman凝聚模型。 我们将每个祖先谱系与样本中它的后代集合（或块）相对应，并用$d$维的经验测度向量来编码系统的状态；第$i$个分量记录群体$i$中存在的块以及构成每个块的谱系的初始位置。 我们感兴趣的是经验测度过程的渐近行为，当$K\to\infty$时。 我们考虑两种情形：临界抽样情形，其中$N_K \sim K$，以及大样本情形，其中$N_K \gg K$。 经过适当的时间-空间尺度变换后，我们证明经验测度过程收敛到一个$d$维凝聚方程的解。 在临界抽样情形下，该解可以用多类型分支过程表示。 在大样本情形下，该解可以用多类型连续状态分支过程的入口律表示。
Consider a structured population consisting of $d$ colonies, with migration rates that are proportional to a positive parameter $K$. We sample $N_K$ individuals distributed evenly across the $d$ colonies and trace their ancestral lineages back. Within a colony, we assume that pairs of ancestral lineages coalesce at a constant rate, as in a Kingman's coalescent. We identify each ancestral lineage with the set, or block, of its descendants in the sample and we encode the state of the system using a $d$-dimensional vector of empirical measures; the $i$-th component records the blocks present in colony $i$ and the initial location of the lineages composing each block. We are interested in the asymptotic behaviour of the process of empirical measures as $K\to\infty$. We consider two regimes: the critical sampling regime, where $N_K \sim K$ and the large sampling regime where $N_K \gg K$. After an appropriate time-space scaling, we show that the process of empirical measures converges to the solution of a $d$-dimensional coagulation equation. In the critical sampling regime, the solution can be represented in terms of a multi-type branching process. In the large sampling regime, the solution can be represented in terms of the entrance law of a multi-type continuous state branching process.
- [27] arXiv:2506.01306 (替换) [中文pdf, pdf, html, 其他]
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标题： Coulomb气体积分的渐近性，Temperley-Lieb型代数和纯划分函数标题： Asymptotic of Coulomb gas integral, Temperley-Lieb type algebras and pure partition functions评论： 34页，9幅图主题： 概率 (math.PR) ; 数学物理 (math-ph)
在这篇补充说明中，我们研究了几类库仑气体积分的渐近行为，并为多个径向系统$\mathrm{SLE}(\kappa)$和一般多重弦向系统$\mathrm{SLE}(\kappa)$构造了纯划分函数。 对于径向和弦向两种情况，我们都证明了基态解$J_{\alpha}^{(m,n)}(\boldsymbol{x})$对无矢量方程的线性无关性，其中$\kappa \in (0,8)$取无理数值。 特别地，我们证明了基态解 $J^{(m,n)}_\alpha \in B_{m,n}$，由具有 $m$ 筛选电荷的链接模式 $\alpha$ 标记，当 $\kappa$ 为无理数时是线性无关的。 这是通过为每个链接模式 $\beta$ 构造一个对偶泛函 $l_\beta \in B^{*}_{m,n}$ 实现的，该泛函使得对应于 Temperley-Lieb 型代数的蜿蜒矩阵由 $M_{\alpha\beta} = l_{\beta}(J^{(m,n)}_\alpha)$ 给出。 该矩阵的行列式具有显式表达式，并且对于无理数$\kappa$非零，从而证明了所需的线性无关性。 由此得出，我们通过乘以回文矩阵的逆矩阵，为每种连接模式$\alpha$构造多重$\mathrm{SLE}(\kappa)$系统的纯划分函数$Z_{\alpha}(\boldsymbol{x})$。 此方法还可以推广到径向和弦两种情况下激发态解$K_{\alpha}$的渐近分析中。
In this supplementary note, we study the asymptotic behavior of several types of Coulomb gas integrals and construct the pure partition functions for multiple radial $\mathrm{SLE}(\kappa)$ and general multiple chordal $\mathrm{SLE}(\kappa)$ systems. For both radial and chordal cases, we prove the linear independence of the ground state solutions $J_{\alpha}^{(m,n)}(\boldsymbol{x})$ to the null vector equations for irrational values of $\kappa \in (0,8)$. In particular, we show that the ground state solutions $J^{(m,n)}_\alpha \in B_{m,n}$, indexed by link patterns $\alpha$ with $m$ screening charges, are linearly independent when $\kappa$ is irrational. This is achieved by constructing, for each link pattern $\beta$, a dual functional $l_\beta \in B^{*}_{m,n}$ such that the meander matrix of the corresponding Temperley-Lieb type algebra is given by $M_{\alpha\beta} = l_{\beta}(J^{(m,n)}_\alpha)$. The determinant of this matrix admits an explicit expression and is nonzero for irrational $\kappa$, establishing the desired linear independence. As a consequence, we construct the pure partition functions $Z_{\alpha}(\boldsymbol{x})$ of the multiple $\mathrm{SLE}(\kappa)$ systems for each link pattern $\alpha$ by multiplying the inverse of the meander matrix. This method can also be extended to the asymptotic analysis of the excited state solutions $K_{\alpha}$ in both radial and chordal cases.
- [28] arXiv:2209.10166 (替换) [中文pdf, pdf, html, 其他]
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标题： 基于迭代积分和神经网络的混沌套期保值标题： Chaotic Hedging with Iterated Integrals and Neural Networks主题： 数学金融 (q-fin.MF) ; 机器学习 (cs.LG) ; 概率 (math.PR) ; 计算金融 (q-fin.CP) ; 机器学习 (stat.ML)
本文中，我们基于给定的指数可积连续半鞅，推导出了一种基于迭代Stratonovich积分的$L^p$-混沌展开。通过忽略展开的正交性，我们证明了每个$p$-可积泛函$p \in [1,\infty)$可以被有限项的迭代Stratonovich积分所逼近。利用（可能是随机的）神经网络作为被积函数，我们因此得到了$p$-可积金融衍生品在$L^p$-意义下的通用逼近结果。此外，我们还可以近似求解$L^p$-套期保值问题（对于$p = 2$与二次套期保值问题一致），其中近似的套期保值策略可以在短时间内以封闭形式计算出来。
In this paper, we derive an $L^p$-chaos expansion based on iterated Stratonovich integrals with respect to a given exponentially integrable continuous semimartingale. By omitting the orthogonality of the expansion, we show that every $p$-integrable functional, $p \in [1,\infty)$, can be approximated by a finite sum of iterated Stratonovich integrals. Using (possibly random) neural networks as integrands, we therefere obtain universal approximation results for $p$-integrable financial derivatives in the $L^p$-sense. Moreover, we can approximately solve the $L^p$-hedging problem (coinciding for $p = 2$ with the quadratic hedging problem), where the approximating hedging strategy can be computed in closed form within short runtime.
- [29] arXiv:2404.15483 (替换) [中文pdf, pdf, html, 其他]
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标题： Büchi目标在并发随机博弈中的策略复杂度标题： Strategy Complexity of Büchi Objectives in Concurrent Stochastic Games评论： 在EC '25上发表的论文完整版本主题： 计算机科学与博弈论 (cs.GT) ; 概率 (math.PR)
我们研究可数图上的双人零和并发（即同时移动）随机 Büchi 博弈和瞬时博弈。 两个博弈者 Max 和 Min 分别寻求最大化和最小化满足博弈目标的概率。 Büchi 的目标是无限次地访问给定的一组目标状态。 这可以看作是最大化每日奖励的预期 $\limsup$ 的一个特例，其中所有每日奖励都在 $\{0,1\}$ 范围内。 瞬时博弈的目标是无限次地访问任何状态，即每个有限子集最终都会永远存在。 瞬时博弈只能在无限博弈图中实现。 我们证明，在 Büchi 博弈中，始终存在仅使用计步器（离散时钟）加上 1 位公共内存的 $\varepsilon$ 最优 Max 策略。 这个上界适用于所有可数图，但即使对于有限图的特殊情况，这也是一个新的结果。 这个上界是严格的，因为即使在有限的博弈图上，仅使用步数计数器或有限内存的最大策略也是不够的。 这个上界是一个稍强的新结果的结果： 对于组合的Büchi和瞬时性目标，$\varepsilon$-最优最大策略只需要1比特的公共内存（但不能是无记忆的）。 我们的证明技巧也得到了一个密切相关的结果，即 对于单独的瞬时性目标，$\varepsilon$-最优最大策略可以选择为无记忆的。
We study 2-player zero-sum concurrent (i.e., simultaneous move) stochastic B\"uchi games and Transience games on countable graphs. Two players, Max and Min, seek respectively to maximize and minimize the probability of satisfying the game objective. The B\"uchi objective is to visit a given set of target states infinitely often. This can be seen as a special case of maximizing the expected $\limsup$ of the daily rewards, where all daily rewards are in $\{0,1\}$. The Transience objective is to visit no state infinitely often, i.e., every finite subset of the states is eventually left forever. Transience can only be met in infinite game graphs. We show that in B\"uchi games there always exist $\varepsilon$-optimal Max strategies that use just a step counter (discrete clock) plus 1 bit of public memory. This upper bound holds for all countable graphs, but it is a new result even for the special case of finite graphs. The upper bound is tight in the sense that Max strategies that use just a step counter, or just finite memory, are not sufficient even on finite game graphs. This upper bound is a consequence of a slightly stronger new result: $\varepsilon$-optimal Max strategies for the combined B\"uchi and Transience objective require just 1 bit of public memory (but cannot be memoryless). Our proof techniques also yield a closely related result, that $\varepsilon$-optimal Max strategies for the Transience objective alone can be chosen as memoryless.
- [30] arXiv:2405.19256 (替换) [中文pdf, pdf, html, 其他]
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标题： 弱生成采样器以有效采样随机微分方程的不变分布标题： Weak Generative Sampler to Efficiently Sample Invariant Distribution of Stochastic Differential Equation评论： 33页，19幅图主题： 机器学习 (cs.LG) ; 数学物理 (math-ph) ; 动力系统 (math.DS) ; 数值分析 (math.NA) ; 概率 (math.PR)
从伊藤扩散过程采样不变分布是随机模拟中的一个重大挑战。传统的随机微分方程数值求解器需要精细的时间步长和较长的仿真时间，这会导致有偏且相关的样本。当前基于深度学习的方法通过求解平稳Fokker-Planck方程来确定以深度神经网络形式表示的不变概率密度函数，但它们通常不直接解决从计算得到的密度函数中采样的问题。在这项工作中，我们引入了一个框架，使用弱生成采样器（WGS）直接生成由平稳Fokker-Planck方程导出的变换映射所诱导的独立同分布（iid）样本。我们提出的损失函数基于Fokker-Planck方程的弱形式，并结合归一化流来刻画不变分布并促进从基础分布中生成样本。我们的随机测试函数避免了传统弱形式中需要的min-max优化。我们的方法既不需要计算密集的雅可比行列式，也不需要变换映射的可逆性。我们框架的关键组成部分是在生成的数据样本中心相关的高斯核函数形式的自适应选择的测试函数族。在多个基准例子上的实验结果表明了我们方法的有效性和可扩展性，它提供了较低的计算成本和出色的探索多个亚稳态的能力。
Sampling invariant distributions from an It\^o diffusion process presents a significant challenge in stochastic simulation. Traditional numerical solvers for stochastic differential equations require both a fine step size and a lengthy simulation period, resulting in biased and correlated samples. The current deep learning-based method solves the stationary Fokker--Planck equation to determine the invariant probability density function in the form of deep neural networks, but they generally do not directly address the problem of sampling from the computed density function. In this work, we introduce a framework that employs a weak generative sampler (WGS) to directly generate independent and identically distributed (iid) samples induced by a transformation map derived from the stationary Fokker--Planck equation. Our proposed loss function is based on the weak form of the Fokker--Planck equation, integrating normalizing flows to characterize the invariant distribution and facilitate sample generation from a base distribution. Our randomized test function circumvents the need for min-max optimization in the traditional weak formulation. Our method necessitates neither the computationally intensive calculation of the Jacobian determinant nor the invertibility of the transformation map. A crucial component of our framework is the adaptively chosen family of test functions in the form of Gaussian kernel functions with centers related to the generated data samples. Experimental results on several benchmark examples demonstrate the effectiveness and scalability of our method, which offers both low computational costs and excellent capability in exploring multiple metastable states.
- [31] arXiv:2410.06307 (替换) [中文pdf, pdf, html, 其他]
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标题： 模型预测控制对于不安分的多臂老虎机几乎是最佳的标题： Model Predictive Control is Almost Optimal for Restless Bandit评论： 已被COLT 2025接受并审阅主题： 优化与控制 (math.OC) ; 机器学习 (cs.LG) ; 概率 (math.PR) ; 机器学习 (stat.ML)
我们研究了离散时间无限时域平均奖励的不安分马尔可夫多臂老虎机（RMAB）问题。 我们提出了一种基于\emph{模型预测控制}的非平稳策略，具有滚动计算时域$\tau$。 在每个时隙，该策略求解一个$\tau$时域线性规划，其第一个控制值被保留为 RMAB 的控制。 我们的解决方案假设最少，并以$\tau$和手臂数量$N$来量化最优性的损失。 我们证明了其次优性差距一般为$O(1/\sqrt{N})$，在局部稳定性条件下为$\exp(-\Omega(N))$。 我们的证明基于动态控制领域的一个框架，称为\emph{耗散性}。 与最先进的方法相比，我们的解决方案易于实现且在实践中表现出色。 此外，我们的解决方案和证明方法都可以轻松推广到更一般的约束MDP设置，因此，应该会引起新兴的RMAB社区的极大兴趣。
We consider the discrete time infinite horizon average reward restless markovian bandit (RMAB) problem. We propose a \emph{model predictive control} based non-stationary policy with a rolling computational horizon $\tau$. At each time-slot, this policy solves a $\tau$ horizon linear program whose first control value is kept as a control for the RMAB. Our solution requires minimal assumptions and quantifies the loss in optimality in terms of $\tau$ and the number of arms, $N$. We show that its sub-optimality gap is $O(1/\sqrt{N})$ in general, and $\exp(-\Omega(N))$ under a local-stability condition. Our proof is based on a framework from dynamic control known as \emph{dissipativity}. Our solution easy to implement and performs very well in practice when compared to the state of the art. Further, both our solution and our proof methodology can easily be generalized to more general constrained MDP settings and should thus, be of great interest to the burgeoning RMAB community.
- [32] arXiv:2502.06072 (替换) [中文pdf, pdf, html, 其他]
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标题： 基于投影的Lyapunov方法在完全异构弱耦合MDP中的应用标题： Projection-based Lyapunov method for fully heterogeneous weakly-coupled MDPs评论： 34页，更新了相关工作以包含一个遗漏的结果主题： 机器学习 (cs.LG) ; 优化与控制 (math.OC) ; 概率 (math.PR)
异质性对许多现实世界中的大规模决策问题构成了根本性的挑战，但这一领域仍然很大程度上未被研究。 本文研究了此类问题的一个显著类别——弱耦合马尔可夫决策过程（WCMDPs）的完全异质设定。 每个WCMDP由 $N$个臂（或子问题）组成，在完全异质设定下，这些臂具有不同的模型参数，当 $N$ 较大时会导致维度灾难。 我们证明了，在温和假设下，一个可以有效计算的策略在完全异质WCMDPs的每个臂的长期平均奖励中实现了 $O(1/\sqrt{N})$ 的最优性差距，当 $N$ 变得很大时。 这是首个关于完全异质平均奖励WCMDPs的渐近最优性结果。 我们的主要技术创新增强了基于投影的李雅普诺夫函数的构建，即使在完全异质的情况下也能验证奖励和成本收敛到最优区域。
Heterogeneity poses a fundamental challenge for many real-world large-scale decision-making problems but remains largely understudied. In this paper, we study the fully heterogeneous setting of a prominent class of such problems, known as weakly-coupled Markov decision processes (WCMDPs). Each WCMDP consists of $N$ arms (or subproblems), which have distinct model parameters in the fully heterogeneous setting, leading to the curse of dimensionality when $N$ is large. We show that, under mild assumptions, an efficiently computable policy achieves an $O(1/\sqrt{N})$ optimality gap in the long-run average reward per arm for fully heterogeneous WCMDPs as $N$ becomes large. This is the first asymptotic optimality result for fully heterogeneous average-reward WCMDPs. Our main technical innovation is the construction of projection-based Lyapunov functions that certify the convergence of rewards and costs to an optimal region, even under full heterogeneity.
- [33] arXiv:2505.03789 (替换) [中文pdf, pdf, html, 其他]
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标题： 一种学习鞅的高阶深度神经网络的新架构标题： A new architecture of high-order deep neural networks that learn martingales评论： 19页，3幅图主题： 机器学习 (cs.LG) ; 概率 (math.PR) ; 计算金融 (q-fin.CP)
提出了一种基于求解随机微分方程 (SDE) 高阶弱逼近算法的新深度学习神经网络架构。 该架构使深度学习模型能够高效地学习鞅。 还研究了基于此架构的深度神经网络在金融衍生品定价问题中的行为。 该新架构的核心在于显式龙格-库塔型高阶弱逼近算法，其中逼近仅通过目标 SDE 的向量场的迭代复合与线性组合来实现。
A new deep-learning neural network architecture based on high-order weak approximation algorithms for stochastic differential equations (SDEs) is proposed. The architecture enables the efficient learning of martingales by deep learning models. The behaviour of deep neural networks based on this architecture, when applied to the problem of pricing financial derivatives, is also examined. The core of this new architecture lies in the high-order weak approximation algorithms of the explicit Runge--Kutta type, wherein the approximation is realised solely through iterative compositions and linear combinations of vector fields of the target SDEs.