arXiv - MATH - Probability最新文献

The Moran process on a random graph 随机图上的莫兰过程

arXiv - MATH - Probability Pub Date : 2024-09-18 DOI: arxiv-2409.11615

Alan Frieze, Wesley Pegden

引用次数: 0

Abelian and stochastic sandpile models on complete bipartite graphs 完整双向图上的非线性和随机沙堆模型

arXiv - MATH - Probability Pub Date : 2024-09-18 DOI: arxiv-2409.11811

Thomas Selig, Haoyue Zhu

引用次数: 0

The largest fragment in self-similar fragmentation processes of positive index 正指数自相似破碎过程中的最大碎片

arXiv - MATH - Probability Pub Date : 2024-09-18 DOI: arxiv-2409.11795

Piotr Dyszewski, Samuel G. G. Johnston, Sandra Palau, Joscha Prochno

{"title":"The largest fragment in self-similar fragmentation processes of positive index","authors":"Piotr Dyszewski, Samuel G. G. Johnston, Sandra Palau, Joscha Prochno","doi":"arxiv-2409.11795","DOIUrl":"https://doi.org/arxiv-2409.11795","url":null,"abstract":"Take a self-similar fragmentation process with dislocation measure $nu$ and\u0000index of self-similarity $alpha > 0$. Let $e^{-m_t}$ denote the size of the\u0000largest fragment in the system at time $tgeq 0$. We prove fine results for the\u0000asymptotics of the stochastic process $(s_{t geq 0}$ for a broad class of\u0000dislocation measures. In the case where the process has finite activity (i.e.\u0000$nu$ is a finite measure with total mass $lambda>0$), we show that setting\u0000begin{equation*} g(t) :=frac{1}{alpha}left(log t - log log t +\u0000log(alpha lambda)right), qquad tgeq 0, end{equation*} we have $lim_{t\u0000to infty} (m_t - g(t)) = 0$ almost-surely. In the case where the process has\u0000infinite activity, we impose the mild regularity condition that the dislocation\u0000measure satisfies begin{equation*} nu(1-s_1 > delta ) = delta^{-theta}\u0000ell(1/delta), end{equation*} for some $theta in (0,1)$ and\u0000$ell:(0,infty) to (0,infty)$ slowly varying at infinity. Under this\u0000regularity condition, we find that if begin{equation*} g(t)\u0000:=frac{1}{alpha}left( log t - (1-theta) log log t - log ell left(\u0000log t ~ellleft( log t right)^{frac{1}{1-theta}} right) +\u0000c(alpha,theta) right), qquad tgeq 0, end{equation*} then $lim_{t to\u0000infty} (m_t - g(t)) = 0$ almost-surely. Here $c(alpha,theta) := log alpha\u0000-(1-theta)log(1-theta) - log Gamma(1-theta)$. Our results sharpen\u0000significantly the best prior result on general self-similar fragmentation\u0000processes, due to Bertoin, which states that $m_t = (1+o(1)) frac{1}{alpha}\u0000log t$.","PeriodicalId":501245,"journal":{"name":"arXiv - MATH - Probability","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-09-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142262306","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

An approximation of the squared Wasserstein distance and an application to Hamilton-Jacobi equations 瓦瑟斯坦距离平方的近似值及其在汉密尔顿-雅可比方程中的应用

arXiv - MATH - Probability Pub Date : 2024-09-18 DOI: arxiv-2409.11793

Charles BertucciCMAP, Pierre Louis LionsCdF, CEREMADE

引用次数: 0

Total disconnectedness and percolation for the supports of super-tree random measures 超树随机测量支持的全断开性和渗流

arXiv - MATH - Probability Pub Date : 2024-09-18 DOI: arxiv-2409.11841

Edwin Perkins, Delphin Sénizergues

{"title":"Total disconnectedness and percolation for the supports of super-tree random measures","authors":"Edwin Perkins, Delphin Sénizergues","doi":"arxiv-2409.11841","DOIUrl":"https://doi.org/arxiv-2409.11841","url":null,"abstract":"Super-tree random measure's (STRM's) were introduced by Allouba, Durrett,\u0000Hawkes and Perkins as a simple stochastic model which emulates a superprocess\u0000at a fixed time. A STRM $nu$ arises as the a.s. limit of a sequence of\u0000empirical measures for a discrete time particle system which undergoes\u0000independent supercritical branching and independent random displacement\u0000(spatial motion) of children from their parents. We study the connectedness\u0000properties of the closed support of a STRM ($mathrm{supp}(nu)$) for a\u0000particular choice of random displacement. Our main results are distinct\u0000sufficient conditions for the a.s. total disconnectedness (TD) of\u0000$mathrm{supp}(nu)$, and for percolation on $mathrm{supp}(nu)$ which will\u0000imply a.s. existence of a non-trivial connected component in\u0000$mathrm{supp}(nu)$. We illustrate a close connection between a subclass of\u0000these STRM's and super-Brownian motion (SBM). For these particular STRM's the\u0000above results give a.s. TD of the support in three and higher dimensions and\u0000the existence of a non-trivial connected component in two dimensions, with the\u0000three-dimensional case being critical. The latter two-dimensional result\u0000assumes that $p_c(mathbb{Z}^2)$, the critical probability for site percolation\u0000on $mathbb{Z}^2$, is less than $1-e^{-1}$. (There is strong numerical evidence\u0000supporting this condition although the known rigorous bounds fall just short.)\u0000This gives evidence that the same connectedness properties should hold for SBM.\u0000The latter remains an interesting open problem in dimensions $2$ and $3$ ever\u0000since it was first posed by Don Dawson over $30$ years ago.","PeriodicalId":501245,"journal":{"name":"arXiv - MATH - Probability","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-09-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142262305","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

Local limit of the random degree constrained process 随机度受限过程的局部极限

arXiv - MATH - Probability Pub Date : 2024-09-18 DOI: arxiv-2409.11747

Balázs Ráth, Márton Szőke, Lutz Warnke

引用次数: 0

Local large deviations for randomly forced nonlinear wave equations with localized damping 具有局部阻尼的随机强迫非线性波方程的局部大偏差

arXiv - MATH - Probability Pub Date : 2024-09-18 DOI: arxiv-2409.11717

Yuxuan Chen, Ziyu Liu, Shengquan Xiang, Zhifei Zhang

引用次数: 0

On the number of elements beyond the ones actually observed 关于实际观测之外的元素数量

arXiv - MATH - Probability Pub Date : 2024-09-17 DOI: arxiv-2409.11364

Eugenio Regazzini