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Quantitative propagation of chaos for non-exchangeable diffusions via first-passage percolation 通过第一通道渗滤实现不可交换扩散的混沌定量传播
Pub Date : 2024-09-13 DOI: arxiv-2409.08882
Daniel Lacker, Lane Chun Yeung, Fuzhong Zhou
This paper develops a non-asymptotic approach to mean field approximationsfor systems of $n$ diffusive particles interacting pairwise. The interactionstrengths are not identical, making the particle system non-exchangeable. Themarginal law of any subset of particles is compared to a suitably chosenproduct measure, and we find sharp relative entropy estimates between the two.Building upon prior work of the first author in the exchangeable setting, weuse a generalized form of the BBGKY hierarchy to derive a hierarchy ofdifferential inequalities for the relative entropies. Our analysis of thiscomplicated hierarchy exploits an unexpected but crucial connection withfirst-passage percolation, which lets us bound the marginal entropies in termsof expectations of functionals of this percolation process.
本文针对成对相互作用的 $n$ 扩散粒子系统,提出了一种非渐近的平均场近似方法。相互作用的强度并不相同,因此粒子系统是不可交换的。我们将任何粒子子集的边际定律与适当选择的积度量进行比较,发现两者之间存在尖锐的相对熵估计值。在第一作者先前在可交换背景下所做工作的基础上,我们利用 BBGKY 层次的广义形式,推导出相对熵的差分不等式层次。我们对这一复杂层次结构的分析利用了与第一通道渗滤之间意想不到但却至关重要的联系,这让我们可以用这一渗滤过程的函数期望来约束边际熵。
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引用次数: 0
Integration by parts and invariant measure for KPZ KPZ 的分部积分和不变度量
Pub Date : 2024-09-13 DOI: arxiv-2409.08465
Yu Gu, Jeremy Quastel
Using Stein's method and a Gaussian integration by parts, we provide a directproof of the known fact that drifted Brownian motions are invariant measures(modulo height) for the KPZ equation.
利用斯坦因方法和高斯分部积分,我们直接证明了漂移布朗运动是 KPZ 方程的不变度量(高度模)这一已知事实。
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引用次数: 0
Integral formulas for two-layer Schur and Whittaker processes 两层舒尔和惠特克过程的积分公式
Pub Date : 2024-09-13 DOI: arxiv-2409.08927
Guillaume Barraquand
Stationary measures of last passage percolation with geometric weights andthe log-gamma polymer in a strip of the $mathbb Z^2$ lattice are characterizedin arXiv:2306.05983 using variants of Schur and Whittaker processes, calledtwo-layer Gibbs measures. In this article, we prove contour integral formulascharacterizing the multipoint joint distribution of two-layer Schur andWhittaker processes. We also express them as Doob transformed Markov processeswith explicit transition kernels. As an example of application of our formulas,we compute the growth rate of the KPZ equation on $[0,L]$ with arbitraryboundary parameters.
arXiv:2306.05983使用舒尔和维特克过程的变体(称为两层吉布斯量)描述了在$mathbb Z^2$晶格的条带中具有几何权重和对数伽马聚合物的最后通道渗流的静态量。在本文中,我们证明了描述两层舒尔和惠特克过程的多点联合分布的等高线积分公式。我们还将它们表示为具有明确过渡核的 Doob 变换马尔可夫过程。作为公式应用的一个例子,我们计算了具有任意边界参数的 $[0,L]$ 上 KPZ 方程的增长率。
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引用次数: 0
Tilted Solid-On-Solid is liquid: scaling limit of SOS with a potential on a slope 倾斜固一固是液体:具有斜坡势能的 SOS 的缩放极限
Pub Date : 2024-09-13 DOI: arxiv-2409.08745
Benoît Laslier, Eyal Lubetzky
The $(2+1)$D Solid-On-Solid (SOS) model famously exhibits a rougheningtransition: on an $Ntimes N$ torus with the height at the origin rooted at$0$, the variance of $h(x)$, the height at $x$, is $O(1)$ at largeinverse-temperature $beta$, vs. $asymp log |x|$ at small $beta$ (as in theGaussian free field (GFF)). The former--rigidity at large $beta$--is known fora wide class of $|nablaphi|^p$ models ($p=1$ being SOS) yet is believed tofail once the surface is on a slope (tilted boundary conditions). It isconjectured that the slope would destabilize the rigidity and induce theGFF-type behavior of the surface at small $beta$. The only rigorous result onthis is by Sheffield (2005): for these models of integer height functions, ifthe slope $theta$ is irrational, then Var$(h(x))toinfty$ with $|x|$ (with noknown quantitative bound). We study a family of SOS surfaces at a large enough fixed $beta$, on an$Ntimes N$ torus with a nonzero boundary condition slope $theta$, perturbedby a potential $V$ of strength $epsilon_beta$ per site (arbitrarily small).Our main result is (a) the measure on the height gradients $nabla h$ has aweak limit $mu_infty$ as $Ntoinfty$; and (b) the scaling limit of a samplefrom $mu_infty$ converges to a full plane GFF. In particular, we recover theasymptotics Var$(h(x))sim clog|x|$. To our knowledge, this is the firstexample of a tilted $|nablaphi|^p$ model, or a perturbation thereof, wherethe limit is recovered at large $beta$. The proof looks at random monotonesurfaces that approximate the SOS surface, and shows that (i) these form aweakly interacting dimer model, and (ii) the renormalization framework ofGiuliani, Mastropietro and Toninelli (2017) leads to the GFF limit. Newingredients are needed in both parts, including a nontrivial extension of[GMT17] from finite interactions to any long range summable interactions.
$(2+1)$D固-固(SOS)模型表现出著名的粗糙化转变:在一个原点高度为$0的$Ntimes N$环上,$h(x)$的方差,即$x$处的高度,在大逆温$beta$时为$O(1)$,而在小$beta$时为$asymp log ||x|$(如在高斯自由场(GFF)中)。前者--大$beta$时的刚性--对于一大类$|nablaphi|^p$模型($p=1$为SOS)是已知的,然而一旦表面处于斜坡上(倾斜边界条件),就会失效。据推测,斜坡会破坏刚度的稳定性,并在较小的 $beta$ 时诱发表面的 GFF 型行为。关于这一点的唯一严格结果是 Sheffield(2005)提出的:对于这些整数高度函数模型,如果斜率 $theta$ 是无理的,那么 Var$(h(x))toinfty$ 与 $|x|$ 有关(没有已知的定量约束)。我们研究了在一个足够大的固定$beta$下的SOS曲面族,它位于一个具有非零边界条件斜率$theta$的$Ntimes N$环上,并受到一个每个位点强度为$epsilon_beta$(任意小)的势$V$的扰动。我们的主要结果是:(a)高度梯度的度量 $nabla h$ 随着 $Ntoinfty$ 的增大而具有敬畏极限 $mu_infty$;(b)从 $mu_infty$ 开始的采样的缩放极限收敛到全平面 GFF。特别是,我们恢复了 Var$(h(x))sim clog|x|$ 的渐近线。据我们所知,这是倾斜$|nablaphi|^p$模型或其扰动的第一个实例,它在大beta$时恢复了极限。证明着眼于近似 SOS 曲面的随机单调曲面,并表明:(i) 这些曲面构成了一个弱相互作用的二聚体模型;(ii) Giuliani、Mastropietro 和 Toninelli(2017)的重正化框架导致了 GFF 极限。这两部分都需要新的成分,包括将[GMT17]从有限相互作用非难扩展到任何长程可求和相互作用。
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引用次数: 0
The exponential turnpike phenomenon for mean field game systems: weakly monotone drifts and small interactions 均场博弈系统的指数转弯现象:弱单调漂移和小相互作用
Pub Date : 2024-09-13 DOI: arxiv-2409.09193
Alekos Cecchin, Giovanni Conforti, Alain Durmus, Katharina Eichinger
This article aims at quantifying the long time behavior of solutions of meanfield PDE systems arising in the theory of Mean Field Games and McKean-Vlasovcontrol. Our main contribution is to show well-posedness of the ergodic problemand the exponential turnpike property of dynamic optimizers, which impliesexponential convergence to equilibrium for both optimal states and controls totheir ergodic counterparts. In contrast with previous works that require someversion of the Lasry-Lions monotonicity condition, our main assumption is aweak form of asymptotic monotonicity on the drift of the controlled dynamicsand some basic regularity and smallness conditions on the interaction terms.Our proof strategy is probabilistic and based on the construction ofcontractive couplings between controlled processes and forward-backwardstochastic differential equations. The flexibility of the coupling approachallows us to cover several interesting situations. For example, we do not needto restrict ourselves to compact domains and can work on the whole space$mathbb{R}^d$, we can cover the case of non-constant diffusion coefficientsand we can sometimes show turnpike estimates for the hessians of solutions tothe backward equation.
本文旨在量化平均场博弈和麦金-弗拉索夫控制理论中出现的平均场 PDE 系统解的长期行为。我们的主要贡献在于证明了遍历问题的良好提出性和动态优化器的指数岔道特性,这意味着最优状态和控制都指数收敛于其遍历对应的均衡状态。与以往需要对 Lasry-Lions 单调性条件进行某种反转的研究不同,我们的主要假设是受控动态漂移的渐近单调性形式,以及交互项的一些基本正则性和微小性条件。我们的证明策略是概率性的,基于受控过程与前向-后向随机微分方程之间契约耦合的构建。耦合方法的灵活性使我们能够涵盖几种有趣的情况。例如,我们不需要局限于紧凑域,可以在整个空间中工作;我们可以涵盖非恒定扩散系数的情况;有时我们还可以展示后向方程解的赫斯估计值。
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引用次数: 0
The Yaglom limit for branching Brownian motion with absorption and slightly subcritical drift 具有吸收和轻微次临界漂移的分支布朗运动的雅格洛姆极限
Pub Date : 2024-09-13 DOI: arxiv-2409.08789
Julien Berestycki, Jiaqi Liu, Bastien Mallein, Jason Schweinsberg
Consider branching Brownian motion with absorption in which particles moveindependently as one-dimensional Brownian motions with drift $-rho$, eachparticle splits into two particles at rate one, and particles are killed whenthey reach the origin. Kesten (1978) showed that this process dies out withprobability one if and only if $rho geq sqrt{2}$. We show that in thesubcritical case when $rho > sqrt{2}$, the law of the process conditioned onsurvival until time $t$ converges as $t rightarrow infty$ to aquasi-stationary distribution, which we call the Yaglom limit. We give aconstruction of this quasi-stationary distribution. We also study theasymptotic behavior as $rho downarrow sqrt{2}$ of this quasi-stationarydistribution. We show that the logarithm of the number of particles and thelocation of the highest particle are of order $epsilon^{-1/3}$, and we obtaina limit result for the empirical distribution of the particle locations.
考虑有吸收的分支布朗运动,其中粒子作为漂移为 $-rho$ 的一维布朗运动独立运动,每个粒子以 1 的速率分裂成两个粒子,当粒子到达原点时被杀死。Kesten(1978)证明,当且仅当 $rho geq sqrt{2}$ 时,这一过程会以 1 的概率消亡。我们证明,在$rho > sqrt{2}$的次临界情况下,以存活到时间$t$为条件的过程规律随着$t rightarrow infty$收敛到水稳态分布,我们称之为Yaglom极限。我们给出了这种准稳态分布的构造。我们还研究了这种准稳态分布的$rho downarrow sqrt{2}$ 的渐近行为。我们证明粒子数量的对数和最高粒子的位置都是 $epsilon^{-1/3}$ 的,并且得到了粒子位置经验分布的极限结果。
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引用次数: 0
Multiscaling limit theorems for stochastic FPDE with cyclic long-range dependence 具有循环长程依赖性的随机 FPDE 的多尺度极限定理
Pub Date : 2024-09-13 DOI: arxiv-2409.09215
Maha Mosaad A Alghamdi, Nikolai Leonenko, Andriy Olenko
The paper studies solutions of stochastic partial differential equations withrandom initial conditions. First, it overviews some of the known results onscaled solutions of such equations and provides several explicit motivatingexamples. Then, it proves multiscaling limit theorems for renormalizedsolutions for the case of initial conditions subordinated to the randomprocesses with cyclic long-range dependence. Two cases of stochastic partialdifferential equations are examined. The spectral and covariancerepresentations for the corresponding limit random fields are derived.Additionally, it is discussed why analogous results are not valid forsubordinated cases with Hermite ranks greater than 1. Numerical examples thatillustrate the obtained theoretical results are presented.
本文研究具有随机初始条件的随机偏微分方程的解。首先,它概述了关于此类方程缩放解的一些已知结果,并提供了几个明确的激励示例。然后,针对初始条件从属于具有循环长程依赖性的随机过程的情况,证明了重规范化解的多尺度极限定理。研究了随机偏微分方程的两种情况。此外,还讨论了为什么类似结果不适用于赫米特秩大于 1 的从属情况。
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引用次数: 0
Shifts of Finite Type Obtained by Forbidding a Single Pattern 通过禁止单一模式获得的有限类型转变
Pub Date : 2024-09-13 DOI: arxiv-2409.09024
Nishant Chandgotia, Brian Marcus, Jacob Richey, Chengyu Wu
Given a finite word $w$, Guibas and Odlyzko (J. Combin. Theory Ser. A, 30,1981, 183-208) showed that the autocorrelation polynomial $phi_w(t)$ of $w$,which records the set of self-overlaps of $w$, explicitly determines for each$n$, the number $|B_n(w)|$ of words of length $n$ that avoid $w$. We considerthis and related problems from the viewpoint of symbolic dynamics, focusing onthe setting of $X_{{w}}$, the space of all bi-infinite sequences that avoid$w$. We first summarize and elaborate upon (J. Combin. Theory Ser. A, 30, 1981,183-208) and other work to show that the sequence $|B_n(w)|$ is equivalent toseveral invariants of $X_{{w}}$. We then give a finite-state labeledgraphical representation $L_w$ of $X_{{w}}$ and show that $w$ can berecovered from the graph isomorphism class of the unlabeled version of $L_w$.Using $L_w$, we apply ideas from probability and Perron-Frobenius theory toobtain results comparing features of $X_{{w}}$ for different $w$. Next, wegive partial results on the problem of classifying the spaces $X_{{w}}$ up toconjugacy. Finally, we extend some of our results to spaces ofmulti-dimensional arrays that avoid a given finite pattern.
给定一个有限词 $w$,Guibas 和 Odlyzko (J. Combin. Theory Ser. A, 30,1981, 183-208) 发现,记录了 $w$ 的自重叠集合的 $w$ 的自相关多项式 $phi_w(t)$,明确地决定了每个 $n$ 的长度为 $n$ 的词中避开 $w$ 的词的数目 $|B_n(w)|$。我们从符号动力学的角度来考虑这个问题及相关问题,重点是 $X_{{w}}$,即所有避开 $w$ 的双无限序列的空间。我们首先总结并阐述了 (J. Combin. Theory Ser. A, 30, 1981,183-208)和其他工作,以证明序列 $|B_n(w)|$ 等价于 $X_{{w}}$ 的几个不变式。然后,我们给出了$X_{{w}}$的有限状态标注图表示$L_w$,并证明$w$可以从未标明版本的$L_w$的图同构类中得到。利用$L_w$,我们应用概率论和佩伦-弗罗贝尼斯理论的思想,得到了比较不同$w$下$X_{{w}}$特征的结果。接下来,我们给出了对直到共轭的空间 $X_{{w}$ 的分类问题的部分结果。最后,我们将部分结果扩展到避免给定有限模式的多维阵列空间。
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引用次数: 0
Entropy Contractions in Markov Chains: Half-Step, Full-Step and Continuous-Time 马尔可夫链中的熵收缩:半步、全步和连续时间
Pub Date : 2024-09-12 DOI: arxiv-2409.07689
Pietro Caputo, Zongchen Chen, Yuzhou Gu, Yury Polyanskiy
This paper considers the speed of convergence (mixing) of a finite Markovkernel $P$ with respect to the Kullback-Leibler divergence (entropy). Given aMarkov kernel one defines either a discrete-time Markov chain (with the$n$-step transition kernel given by the matrix power $P^n$) or acontinuous-time Markov process (with the time-$t$ transition kernel given by$e^{t(P-mathrm{Id})}$). The contraction of entropy for $n=1$ or $t=0+$ arecharacterized by the famous functional inequalities, the strong data processinginequality (SDPI) and the modified log-Sobolev inequality (MLSI), respectively.When $P=KK^*$ is written as the product of a kernel and its adjoint, one couldalso consider the ``half-step'' contraction, which is the SDPI for $K$, whilethe ``full-step'' contraction refers to the SDPI for $P$. The work [DMLM03]claimed that these contraction coefficients (half-step, full-step, andcontinuous-time) are generally within a constant factor of each other. Wedisprove this and related conjectures by working out a number of differentcounterexamples. In particular, we construct (a) a continuous-time Markovprocess that contracts arbitrarily faster than its discrete-time counterpart;and (b) a kernel $P$ such that $P^{m+1}$ contracts arbitrarily better than$P^m$. Hence, our main conclusion is that the four standard inequalitiescomparing five common notions of entropy and variance contraction are generallynot improvable. In the process of analyzing the counterexamples, we survey and sharpen thetools for bounding the contraction coefficients and characterize properties ofextremizers of the respective functional inequalities. As our examples rangefrom Bernoulli-Laplace model, random walks on graphs, to birth-death chains,the paper is also intended as a tutorial on computing MLSI, SDPI and otherconstants for these types of commonly occurring Markov chains.
本文探讨了有限马尔可夫核 $P$ 与库尔贝克-莱布勒发散(熵)的收敛(混合)速度。给定马尔可夫核,可以定义离散时间马尔可夫链(其$n$步过渡核由矩阵幂$P^n$给出)或连续时间马尔可夫过程(其时间-$t$过渡核由$e^{t(P-mathrm{Id})}$给出)。当 $P=KK^*$ 被写成核及其矢量的乘积时,我们还可以考虑 "半步 "收缩,即 $K$ 的 SDPI,而 "全步 "收缩指的是 $P$ 的 SDPI。工作[DMLM03]声称,这些收缩系数(半步、全步和连续时间)通常在一个常数因子范围内。我们通过一系列不同的反例证明了这一猜想及相关猜想。特别是,我们构建了(a)一个连续时间马尔可夫过程,其收缩速度任意快于其离散时间对应过程;以及(b)一个核$P$,使得$P^{m+1}$的收缩速度任意优于$P^m$。因此,我们的主要结论是,比较熵和方差收缩的五个常见概念的四个标准不等式一般是无法改进的。在分析反例的过程中,我们考察并改进了约束收缩系数的工具,并描述了各自函数不等式的求极限者的性质。由于我们的例子涵盖了从伯努利-拉普拉斯模型、图上的随机漫步到出生-死亡链等各种类型,因此本文也可作为计算这些常见马尔可夫链的 MLSI、SDPI 和其他常数的教程。
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引用次数: 0
The small-mass limit for some constrained wave equations with nonlinear conservative noise 具有非线性保守噪声的某些约束波方程的小质量极限
Pub Date : 2024-09-12 DOI: arxiv-2409.08021
Sandra Cerrai, Mengzi Xie
We study the small-mass limit, also known as the Smoluchowski-Kramersdiffusion approximation (see cite{kra} and cite{smolu}), for a system ofstochastic damped wave equations, whose solution is constrained to live in theunitary sphere of the space of square-integrable functions on the interval$(0,L)$. The stochastic perturbation is given by a nonlinear multiplicativeGaussian noise, where the stochastic differential is understood in Stratonovichsense. Due to its particular structure, such noise not only conserves$mathbb{P}$-a.s. the constraint, but also preserves a suitable energyfunctional. In the limit, we derive a deterministic system, that remainsconfined to the unit sphere of $L^2$, but includes additional terms. Theseterms depend on the reproducing kernel of the noise and account for theinteraction between the constraint and the particular conservative noise wechoose.
我们研究了一个随机阻尼波方程系统的小质量极限,也称为斯莫卢霍夫斯基-克拉默扩散近似(见 cite{kra} 和 cite{smolu}),其解受限于区间$(0,L)$上平方可积分函数空间的单元球内。随机扰动由非线性乘法高斯噪声给出,其中的随机微分从斯特拉顿维奇意义上理解。由于其特殊的结构,这种噪声不仅保留了$mathbb{P}$-a.s. 约束,而且还保留了一个合适的能量函数。在极限中,我们推导出一个确定性系统,它仍然限定在 $L^2$ 的单位球内,但包含附加项。这些项取决于噪声的再现核,并考虑了约束与我们选择的特定保守噪声之间的相互作用。
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引用次数: 0
期刊
arXiv - MATH - Probability
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