arXiv - MATH - Probability最新文献_第5页

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

arXiv - MATH - Probability Pub Date : 2024-09-13 DOI: arxiv-2409.08882

Daniel Lacker, Lane Chun Yeung, Fuzhong Zhou

引用次数: 0

Integration by parts and invariant measure for KPZ KPZ 的分部积分和不变度量

arXiv - MATH - Probability Pub Date : 2024-09-13 DOI: arxiv-2409.08465

Yu Gu, Jeremy Quastel

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

arXiv - MATH - Probability Pub Date : 2024-09-13 DOI: arxiv-2409.08927

Guillaume Barraquand

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

arXiv - MATH - Probability Pub Date : 2024-09-13 DOI: arxiv-2409.08745

Benoît Laslier, Eyal Lubetzky

{"title":"Tilted Solid-On-Solid is liquid: scaling limit of SOS with a potential on a slope","authors":"Benoît Laslier, Eyal Lubetzky","doi":"arxiv-2409.08745","DOIUrl":"https://doi.org/arxiv-2409.08745","url":null,"abstract":"The $(2+1)$D Solid-On-Solid (SOS) model famously exhibits a roughening\u0000transition: on an $Ntimes N$ torus with the height at the origin rooted at\u0000$0$, the variance of $h(x)$, the height at $x$, is $O(1)$ at large\u0000inverse-temperature $beta$, vs. $asymp log |x|$ at small $beta$ (as in the\u0000Gaussian free field (GFF)). The former--rigidity at large $beta$--is known for\u0000a wide class of $|nablaphi|^p$ models ($p=1$ being SOS) yet is believed to\u0000fail once the surface is on a slope (tilted boundary conditions). It is\u0000conjectured that the slope would destabilize the rigidity and induce the\u0000GFF-type behavior of the surface at small $beta$. The only rigorous result on\u0000this is by Sheffield (2005): for these models of integer height functions, if\u0000the slope $theta$ is irrational, then Var$(h(x))toinfty$ with $|x|$ (with no\u0000known quantitative bound). We study a family of SOS surfaces at a large enough fixed $beta$, on an\u0000$Ntimes N$ torus with a nonzero boundary condition slope $theta$, perturbed\u0000by a potential $V$ of strength $epsilon_beta$ per site (arbitrarily small).\u0000Our main result is (a) the measure on the height gradients $nabla h$ has a\u0000weak limit $mu_infty$ as $Ntoinfty$; and (b) the scaling limit of a sample\u0000from $mu_infty$ converges to a full plane GFF. In particular, we recover the\u0000asymptotics Var$(h(x))sim clog|x|$. To our knowledge, this is the first\u0000example of a tilted $|nablaphi|^p$ model, or a perturbation thereof, where\u0000the limit is recovered at large $beta$. The proof looks at random monotone\u0000surfaces that approximate the SOS surface, and shows that (i) these form a\u0000weakly interacting dimer model, and (ii) the renormalization framework of\u0000Giuliani, Mastropietro and Toninelli (2017) leads to the GFF limit. New\u0000ingredients are needed in both parts, including a nontrivial extension of\u0000[GMT17] from finite interactions to any long range summable interactions.","PeriodicalId":501245,"journal":{"name":"arXiv - MATH - Probability","volume":"2 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142262516","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

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

arXiv - MATH - Probability Pub Date : 2024-09-13 DOI: arxiv-2409.09193

Alekos Cecchin, Giovanni Conforti, Alain Durmus, Katharina Eichinger

{"title":"The exponential turnpike phenomenon for mean field game systems: weakly monotone drifts and small interactions","authors":"Alekos Cecchin, Giovanni Conforti, Alain Durmus, Katharina Eichinger","doi":"arxiv-2409.09193","DOIUrl":"https://doi.org/arxiv-2409.09193","url":null,"abstract":"This article aims at quantifying the long time behavior of solutions of mean\u0000field PDE systems arising in the theory of Mean Field Games and McKean-Vlasov\u0000control. Our main contribution is to show well-posedness of the ergodic problem\u0000and the exponential turnpike property of dynamic optimizers, which implies\u0000exponential convergence to equilibrium for both optimal states and controls to\u0000their ergodic counterparts. In contrast with previous works that require some\u0000version of the Lasry-Lions monotonicity condition, our main assumption is a\u0000weak form of asymptotic monotonicity on the drift of the controlled dynamics\u0000and some basic regularity and smallness conditions on the interaction terms.\u0000Our proof strategy is probabilistic and based on the construction of\u0000contractive couplings between controlled processes and forward-backward\u0000stochastic differential equations. The flexibility of the coupling approach\u0000allows us to cover several interesting situations. For example, we do not need\u0000to restrict ourselves to compact domains and can work on the whole space\u0000$mathbb{R}^d$, we can cover the case of non-constant diffusion coefficients\u0000and we can sometimes show turnpike estimates for the hessians of solutions to\u0000the backward equation.","PeriodicalId":501245,"journal":{"name":"arXiv - MATH - Probability","volume":"63 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142262506","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

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

arXiv - MATH - Probability Pub Date : 2024-09-13 DOI: arxiv-2409.08789

Julien Berestycki, Jiaqi Liu, Bastien Mallein, Jason Schweinsberg

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

arXiv - MATH - Probability Pub Date : 2024-09-13 DOI: arxiv-2409.09215

Maha Mosaad A Alghamdi, Nikolai Leonenko, Andriy Olenko

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

arXiv - MATH - Probability Pub Date : 2024-09-13 DOI: arxiv-2409.09024

Nishant Chandgotia, Brian Marcus, Jacob Richey, Chengyu Wu

{"title":"Shifts of Finite Type Obtained by Forbidding a Single Pattern","authors":"Nishant Chandgotia, Brian Marcus, Jacob Richey, Chengyu Wu","doi":"arxiv-2409.09024","DOIUrl":"https://doi.org/arxiv-2409.09024","url":null,"abstract":"Given a finite word $w$, Guibas and Odlyzko (J. Combin. Theory Ser. A, 30,\u00001981, 183-208) showed that the autocorrelation polynomial $phi_w(t)$ of $w$,\u0000which records the set of self-overlaps of $w$, explicitly determines for each\u0000$n$, the number $|B_n(w)|$ of words of length $n$ that avoid $w$. We consider\u0000this and related problems from the viewpoint of symbolic dynamics, focusing on\u0000the setting of $X_{{w}}$, the space of all bi-infinite sequences that avoid\u0000$w$. We first summarize and elaborate upon (J. Combin. Theory Ser. A, 30, 1981,\u0000183-208) and other work to show that the sequence $|B_n(w)|$ is equivalent to\u0000several invariants of $X_{{w}}$. We then give a finite-state labeled\u0000graphical representation $L_w$ of $X_{{w}}$ and show that $w$ can be\u0000recovered from the graph isomorphism class of the unlabeled version of $L_w$.\u0000Using $L_w$, we apply ideas from probability and Perron-Frobenius theory to\u0000obtain results comparing features of $X_{{w}}$ for different $w$. Next, we\u0000give partial results on the problem of classifying the spaces $X_{{w}}$ up to\u0000conjugacy. Finally, we extend some of our results to spaces of\u0000multi-dimensional arrays that avoid a given finite pattern.","PeriodicalId":501245,"journal":{"name":"arXiv - MATH - Probability","volume":"19 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142262708","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

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

arXiv - MATH - Probability Pub Date : 2024-09-12 DOI: arxiv-2409.07689

Pietro Caputo, Zongchen Chen, Yuzhou Gu, Yury Polyanskiy

{"title":"Entropy Contractions in Markov Chains: Half-Step, Full-Step and Continuous-Time","authors":"Pietro Caputo, Zongchen Chen, Yuzhou Gu, Yury Polyanskiy","doi":"arxiv-2409.07689","DOIUrl":"https://doi.org/arxiv-2409.07689","url":null,"abstract":"This paper considers the speed of convergence (mixing) of a finite Markov\u0000kernel $P$ with respect to the Kullback-Leibler divergence (entropy). Given a\u0000Markov kernel one defines either a discrete-time Markov chain (with the\u0000$n$-step transition kernel given by the matrix power $P^n$) or a\u0000continuous-time Markov process (with the time-$t$ transition kernel given by\u0000$e^{t(P-mathrm{Id})}$). The contraction of entropy for $n=1$ or $t=0+$ are\u0000characterized by the famous functional inequalities, the strong data processing\u0000inequality (SDPI) and the modified log-Sobolev inequality (MLSI), respectively.\u0000When $P=KK^*$ is written as the product of a kernel and its adjoint, one could\u0000also consider the ``half-step'' contraction, which is the SDPI for $K$, while\u0000the ``full-step'' contraction refers to the SDPI for $P$. The work [DMLM03]\u0000claimed that these contraction coefficients (half-step, full-step, and\u0000continuous-time) are generally within a constant factor of each other. We\u0000disprove this and related conjectures by working out a number of different\u0000counterexamples. In particular, we construct (a) a continuous-time Markov\u0000process that contracts arbitrarily faster than its discrete-time counterpart;\u0000and (b) a kernel $P$ such that $P^{m+1}$ contracts arbitrarily better than\u0000$P^m$. Hence, our main conclusion is that the four standard inequalities\u0000comparing five common notions of entropy and variance contraction are generally\u0000not improvable. In the process of analyzing the counterexamples, we survey and sharpen the\u0000tools for bounding the contraction coefficients and characterize properties of\u0000extremizers of the respective functional inequalities. As our examples range\u0000from Bernoulli-Laplace model, random walks on graphs, to birth-death chains,\u0000the paper is also intended as a tutorial on computing MLSI, SDPI and other\u0000constants for these types of commonly occurring Markov chains.","PeriodicalId":501245,"journal":{"name":"arXiv - MATH - Probability","volume":"106 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142212024","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

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

arXiv - MATH - Probability Pub Date : 2024-09-12 DOI: arxiv-2409.08021

Sandra Cerrai, Mengzi Xie

引用次数: 0