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Variational closures for composite homogenised fluid flows 复合均质流体流动的变量闭包
arXiv - MATH - Probability Pub Date : 2024-09-16 DOI: arxiv-2409.10408
Theo Diamantakis, Ruiao Hu
{"title":"Variational closures for composite homogenised fluid flows","authors":"Theo Diamantakis, Ruiao Hu","doi":"arxiv-2409.10408","DOIUrl":"https://doi.org/arxiv-2409.10408","url":null,"abstract":"The Stochastic Advection by Lie Transport is a variational formulation of\u0000stochastic fluid dynamics introduced to model the effects of unresolved scales,\u0000whilst preserving the geometric structure of ideal fluid flows. In this work,\u0000we show that the SALT equations can arise from the decomposition of the fluid\u0000flow map into its mean and fluctuating components. The fluctuating component is\u0000realised as a prescribed stochastic diffeomorphism that introduces stochastic\u0000transport into the system and we construct it using homogenisation theory. The\u0000dynamics of the mean component are derived from a variational principle\u0000utilising particular forms of variations that preserve the composite structure\u0000of the flow. Using a new variational principle, we show that SALT equations can\u0000arise from random Lagrangians and are equivalent to random coefficient PDEs. We\u0000also demonstrate how to modify the composite flow and the associated\u0000variational principle to derive models inspired by the Lagrangian Averaged\u0000Euler-Poincare (LAEP) theory.","PeriodicalId":501245,"journal":{"name":"arXiv - MATH - Probability","volume":"27 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142262507","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
Network evolution with mesoscopic delay 具有中观延迟的网络演化
arXiv - MATH - Probability Pub Date : 2024-09-16 DOI: arxiv-2409.10307
Sayan Banerjee, Shankar Bhamidi, Partha Dey, Akshay Sakanaveeti
{"title":"Network evolution with mesoscopic delay","authors":"Sayan Banerjee, Shankar Bhamidi, Partha Dey, Akshay Sakanaveeti","doi":"arxiv-2409.10307","DOIUrl":"https://doi.org/arxiv-2409.10307","url":null,"abstract":"Fueled by the influence of real-world networks both in science and society,\u0000numerous mathematical models have been developed to understand the structure\u0000and evolution of these systems, particularly in a temporal context. Recent\u0000advancements in fields like distributed cyber-security and social networks have\u0000spurred the creation of probabilistic models of evolution, where individuals\u0000make decisions based on only partial information about the network's current\u0000state. This paper seeks to explore models that incorporate emph{network\u0000delay}, where new participants receive information from a time-lagged snapshot\u0000of the system. In the context of mesoscopic network delays, we develop\u0000probabilistic tools built on stochastic approximation to understand asymptotics\u0000of both local functionals, such as local neighborhoods and degree\u0000distributions, as well as global properties, such as the evolution of the\u0000degree of the network's initial founder. A companion paper explores the regime\u0000of macroscopic delays in the evolution of the network.","PeriodicalId":501245,"journal":{"name":"arXiv - MATH - Probability","volume":"19 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142262495","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
Approximating the signature of Brownian motion for high order SDE simulation 近似布朗运动特征，用于高阶 SDE 仿真
arXiv - MATH - Probability Pub Date : 2024-09-16 DOI: arxiv-2409.10118
James Foster
{"title":"Approximating the signature of Brownian motion for high order SDE simulation","authors":"James Foster","doi":"arxiv-2409.10118","DOIUrl":"https://doi.org/arxiv-2409.10118","url":null,"abstract":"The signature is a collection of iterated integrals describing the \"shape\" of\u0000a path. It appears naturally in the Taylor expansions of controlled\u0000differential equations and, as a consequence, is arguably the central object\u0000within rough path theory. In this paper, we will consider the signature of\u0000Brownian motion with time, and present both new and recently developed\u0000approximations for some of its integrals. Since these integrals (or equivalent\u0000L'{e}vy areas) are nonlinear functions of the Brownian path, they are not\u0000Gaussian and known to be challenging to simulate. To conclude the paper, we\u0000will present some applications of these approximations to the high order\u0000numerical simulation of stochastic differential equations (SDEs).","PeriodicalId":501245,"journal":{"name":"arXiv - MATH - Probability","volume":"65 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142262508","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
Notas sobre Teoría de colas y algunas aplicaciones 排队理论及一些应用说明
arXiv - MATH - Probability Pub Date : 2024-09-16 DOI: arxiv-2409.10735
Carlos E. Martínez-Rodríguez
{"title":"Notas sobre Teoría de colas y algunas aplicaciones","authors":"Carlos E. Martínez-Rodríguez","doi":"arxiv-2409.10735","DOIUrl":"https://doi.org/arxiv-2409.10735","url":null,"abstract":"This paper presents a comprehensive review of stochastic processes, with a\u0000particular focus on Markov chains and jump processes. The main results related\u0000to queuing systems are analyzed. Additionally, conditions that ensure the\u0000stability, or ergodicity, of such systems are presented. The paper also\u0000discusses stability results for queuing networks and their extension to\u0000visiting systems. Finally, key contributions concerning the Probability\u0000Generating Function, an essential tool in the analysis of the aforementioned\u0000processes, are introduced. The review is conducted from the perspective of\u0000queuing theory, grounded in the Kendall-Lee notation, emphasizing stability\u0000results and the computation of performance measures based on the specific\u0000characteristics of each process.","PeriodicalId":501245,"journal":{"name":"arXiv - MATH - Probability","volume":"3 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142262319","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
A Spectral Representation of a Weighted Random Vectorial Field: Potential Applications to Turbulence and the Problem of Anomalous Dissipation in the Inviscid Limit 加权随机矢量场的谱表示：湍流和不粘性极限中的异常耗散问题的潜在应用
arXiv - MATH - Probability Pub Date : 2024-09-16 DOI: arxiv-2409.10636
Steven D Miller
{"title":"A Spectral Representation of a Weighted Random Vectorial Field: Potential Applications to Turbulence and the Problem of Anomalous Dissipation in the Inviscid Limit","authors":"Steven D Miller","doi":"arxiv-2409.10636","DOIUrl":"https://doi.org/arxiv-2409.10636","url":null,"abstract":"Let ${mathfrak{G}}subsetmathbb{R}^{3}$ with $vol(mathfrak{G})sim L^{3}$.\u0000Let ${mathscr{T}}(x)$ be a Gaussian random field $forall~xinmathfrak{G}$\u0000with expectation $mathbf{E}[{mathscr{T}}(x)]=0$ and correlation\u0000$mathbf{E}[{mathscr{T}}(x)otimes{mathscr{T}}(y)]=K(x,y;lambda)$, an\u0000isotropic and regulated kernel with correlation length $lambda$. The field has\u0000a Karhunen-Loeve spectral representation\u0000${mathscr{T}}(x)=sum_{I=1}^{infty}mathrm{Z}^{1/2}_{I}f_{I}(x)otimesmathscr{Z}_{I}$,\u0000with eigenvalues $lbracemathrm{Z}_{I}rbrace$, eigenfunctions $lbrace\u0000f_{I}(x)rbrace $ and Gaussian random variables $mathscr{Z}_{I}$ with\u0000$mathbf{E}[mathscr{Z}_{I}]=0$ and\u0000$mathbf{E}[mathscr{Z}_{I}otimesmathscr{Z}_{J}]=delta_{IJ}$. If\u0000$mathfrak{G}$ contains incompressible fluid of viscosity $nu$ with velocity\u0000$u_{a}(x,t)$ that evolves via the Navier-Stokes equations with a high 'Reynolds\u0000function' $mathsf{RE}(x,t)=tfrac{|u_{a}(x,t)|L}{nu} $ then aspects of a\u0000turbulent flow with $mathsf{RE}(x,t)gg mathsf{RE}_{*}$, a critical Reynolds\u0000number, might be represented by the 'weighted' random field\u0000$mathscr{U}_{a}(x,t)=\u0000u_{a}(x,t)+mathrm{A}u_{a}(x,t)big(mathsf{RE}(x,t)-mathsf{RE}_{*}big)^{beta}sum_{I=1}^{infty}\u0000mathrm{Z}^{1/2}_{I}f_{I}(x)otimesmathscr{Z}_{I}$ where random fluctuations\u0000and amplitude scale nonlinearly with $mathsf{RE}(x,t)$, with mean\u0000$mathbf{E}[{mathscr{U}}_{a}(x,t)] =u_{a}(x,t)$. In the inviscid limit one can\u0000prove an anomalous dissipation-type law begin{align} lim_{nurightarrow\u00000}bigg(lim_{u_{a}(x,t)rightarrow {u}_{a}}sup~nu\u0000int_{mathfrak{G}}int_{0}^{T}{mathbf{E}}bigg[bigg|{nabla}_{a}{mathscr{U}}_{a}(x,s)bigg|^{2}bigg]dmathcal{V}(x)\u0000dsbigg)>0 end{align} iff $beta=tfrac{1}{2}$ and\u0000$sum_{I=1}^{infty}mathrm{Z}_{I}int_{{mathfrak{G}}}{nabla}_{a}f_{I}(x){nabla}^{a}f_{I}(x)dmathcal{V}(x)>0$.","PeriodicalId":501245,"journal":{"name":"arXiv - MATH - Probability","volume":"103 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142262318","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
KPZ equation from ASEP plus general speed-change drift 来自 ASEP 的 KPZ 公式加上一般速度变化漂移
arXiv - MATH - Probability Pub Date : 2024-09-16 DOI: arxiv-2409.10513
Kevin Yang
{"title":"KPZ equation from ASEP plus general speed-change drift","authors":"Kevin Yang","doi":"arxiv-2409.10513","DOIUrl":"https://doi.org/arxiv-2409.10513","url":null,"abstract":"We derive the KPZ equation as a continuum limit of height functions in\u0000asymmetric simple exclusion processes with a hyperbolic-scale drift that\u0000depends on the local particle configuration. To our knowledge, it is a first\u0000such result for a general class of particle systems with neither duality nor\u0000explicit invariant measures. The new tools to handle the lack of an invariant\u0000measure are estimates for Kolmogorov equations that produce a more robust proof\u0000of the Kipnis-Varadhan inequality. These tools are not exclusive to KPZ.","PeriodicalId":501245,"journal":{"name":"arXiv - MATH - Probability","volume":"16 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142262322","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
Optimality Gap of Decentralized Submodular Maximization under Probabilistic Communication 概率通信下分散式次模态最大化的最优性差距
arXiv - MATH - Probability Pub Date : 2024-09-16 DOI: arxiv-2409.09979
Joan Vendrell, Solmaz Kia
{"title":"Optimality Gap of Decentralized Submodular Maximization under Probabilistic Communication","authors":"Joan Vendrell, Solmaz Kia","doi":"arxiv-2409.09979","DOIUrl":"https://doi.org/arxiv-2409.09979","url":null,"abstract":"This paper considers the problem of decentralized submodular maximization\u0000subject to partition matroid constraint using a sequential greedy algorithm\u0000with probabilistic inter-agent message-passing. We propose a\u0000communication-aware framework where the probability of successful communication\u0000between connected devices is considered. Our analysis introduces the notion of\u0000the probabilistic optimality gap, highlighting its potential influence on\u0000determining the message-passing sequence based on the agent's broadcast\u0000reliability and strategic decisions regarding agents that can broadcast their\u0000messages multiple times in a resource-limited environment. This work not only\u0000contributes theoretical insights but also has practical implications for\u0000designing and analyzing decentralized systems in uncertain communication\u0000environments. A numerical example demonstrates the impact of our results.","PeriodicalId":501245,"journal":{"name":"arXiv - MATH - Probability","volume":"122 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142262509","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
Singularity of solutions to singular SPDEs 奇异 SPDEs 解的奇异性
arXiv - MATH - Probability Pub Date : 2024-09-16 DOI: arxiv-2409.10037
Martin Hairer, Seiichiro Kusuoka, Hirotatsu Nagoji
{"title":"Singularity of solutions to singular SPDEs","authors":"Martin Hairer, Seiichiro Kusuoka, Hirotatsu Nagoji","doi":"arxiv-2409.10037","DOIUrl":"https://doi.org/arxiv-2409.10037","url":null,"abstract":"Building on the notes [Hai17], we give a sufficient condition for the\u0000marginal distribution of the solution of singular SPDEs on the $d$-dimensional\u0000torus to be singular with respect to the law of the Gaussian measure induced by\u0000the linearised equation. As applications we obtain the singularity of the\u0000$Phi^4_3$-measure with respect to the Gaussian free field measure and the\u0000border of parameters for the fractional $Phi^4$-measure to be singular with\u0000respect to the Gaussian free field measure. Our approach is applicable to quite\u0000a large class of singular SPDEs.","PeriodicalId":501245,"journal":{"name":"arXiv - MATH - Probability","volume":"27 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142262496","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
Mimicking and Conditional Control with Hard Killing 硬杀伤的模仿和条件控制
arXiv - MATH - Probability Pub Date : 2024-09-16 DOI: arxiv-2409.10650
Rene Carmona, Daniel Lacker
{"title":"Mimicking and Conditional Control with Hard Killing","authors":"Rene Carmona, Daniel Lacker","doi":"arxiv-2409.10650","DOIUrl":"https://doi.org/arxiv-2409.10650","url":null,"abstract":"We first prove a mimicking theorem (also known as a Markovian projection\u0000theorem) for the marginal distributions of an Ito process conditioned to not\u0000have exited a given domain. We then apply this new result to the proof of a\u0000conjecture of P.L. Lions for the optimal control of conditioned processes.","PeriodicalId":501245,"journal":{"name":"arXiv - MATH - Probability","volume":"32 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142262317","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
Mesoscopic Universality for Circular Orthogonal Polynomial Ensembles 圆正交多项式集合的中观普遍性
arXiv - MATH - Probability Pub Date : 2024-09-15 DOI: arxiv-2409.09803
Jonathan Breuer, Daniel Ofner
{"title":"Mesoscopic Universality for Circular Orthogonal Polynomial Ensembles","authors":"Jonathan Breuer, Daniel Ofner","doi":"arxiv-2409.09803","DOIUrl":"https://doi.org/arxiv-2409.09803","url":null,"abstract":"We study mesoscopic fluctuations of orthogonal polynomial ensembles on the\u0000unit circle. We show that asymptotics of such fluctuations are stable under\u0000decaying perturbations of the recurrence coefficients, where the appropriate\u0000decay rate depends on the scale considered. By directly proving Gaussian limits\u0000for certain constant coefficient ensembles, we obtain mesoscopic scale Gaussian\u0000limits for a large class of orthogonal polynomial ensembles on the unit circle. As a corollary we prove mesocopic central limit theorems (for all mesoscopic\u0000scales) for the $beta=2$ circular Jacobi ensembles with real parameter\u0000$delta>-1/2$.","PeriodicalId":501245,"journal":{"name":"arXiv - MATH - Probability","volume":"30 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142262512","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
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