We study the fixation probability for two versions of the Moran process on the random graph $G_{n,p}$ at the threshold for connectivity. The Moran process models the spread of a mutant population in a network. Throughtout the process there are vertices of two types, mutants and non-mutants. Mutants have fitness $s$ and non-mutants have fitness 1. The process starts with a unique individual mutant located at the vertex $v_0$. In the Birth-Death version of the process a random vertex is chosen proportional to its fitness and then changes the type of a random neighbor to its own. The process continues until the set of mutants $X$ is empty or $[n]$. In the Death-Birth version a uniform random vertex is chosen and then takes the type of a random neighbor, chosen according to fitness. The process again continues until the set of mutants $X$ is empty or $[n]$. The {em fixation probability} is the probability that the process ends with $X=emptyset$. We give asymptotically correct estimates of the fixation probability that depend on degree of $v_0$ and its neighbors.,
{"title":"The Moran process on a random graph","authors":"Alan Frieze, Wesley Pegden","doi":"arxiv-2409.11615","DOIUrl":"https://doi.org/arxiv-2409.11615","url":null,"abstract":"We study the fixation probability for two versions of the Moran process on\u0000the random graph $G_{n,p}$ at the threshold for connectivity. The Moran process\u0000models the spread of a mutant population in a network. Throughtout the process\u0000there are vertices of two types, mutants and non-mutants. Mutants have fitness\u0000$s$ and non-mutants have fitness 1. The process starts with a unique individual\u0000mutant located at the vertex $v_0$. In the Birth-Death version of the process a\u0000random vertex is chosen proportional to its fitness and then changes the type\u0000of a random neighbor to its own. The process continues until the set of mutants\u0000$X$ is empty or $[n]$. In the Death-Birth version a uniform random vertex is\u0000chosen and then takes the type of a random neighbor, chosen according to\u0000fitness. The process again continues until the set of mutants $X$ is empty or\u0000$[n]$. The {em fixation probability} is the probability that the process ends\u0000with $X=emptyset$. We give asymptotically correct estimates of the fixation probability that\u0000depend on degree of $v_0$ and its neighbors.,","PeriodicalId":501245,"journal":{"name":"arXiv - MATH - Probability","volume":"11 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142262309","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}
In the sandpile model, vertices of a graph are allocated grains of sand. At each unit of time, a grain is added to a randomly chosen vertex. If that causes its number of grains to exceed its degree, that vertex is called unstable, and topples. In the Abelian sandpile model (ASM), topplings are deterministic, whereas in the stochastic sandpile model (SSM) they are random. We study the ASM and SSM on complete bipartite graphs. For the SSM, we provide a stochastic version of Dhar's burning algorithm to check if a given (stable) configuration is recurrent or not, with linear complexity. We also exhibit a bijection between sorted recurrent configurations and pairs of compatible Ferrers diagrams. We then provide a similar bijection for the ASM, and also interpret its recurrent configurations in terms of labelled Motzkin paths.
{"title":"Abelian and stochastic sandpile models on complete bipartite graphs","authors":"Thomas Selig, Haoyue Zhu","doi":"arxiv-2409.11811","DOIUrl":"https://doi.org/arxiv-2409.11811","url":null,"abstract":"In the sandpile model, vertices of a graph are allocated grains of sand. At\u0000each unit of time, a grain is added to a randomly chosen vertex. If that causes\u0000its number of grains to exceed its degree, that vertex is called unstable, and\u0000topples. In the Abelian sandpile model (ASM), topplings are deterministic,\u0000whereas in the stochastic sandpile model (SSM) they are random. We study the\u0000ASM and SSM on complete bipartite graphs. For the SSM, we provide a stochastic\u0000version of Dhar's burning algorithm to check if a given (stable) configuration\u0000is recurrent or not, with linear complexity. We also exhibit a bijection\u0000between sorted recurrent configurations and pairs of compatible Ferrers\u0000diagrams. We then provide a similar bijection for the ASM, and also interpret\u0000its recurrent configurations in terms of labelled Motzkin paths.","PeriodicalId":501245,"journal":{"name":"arXiv - MATH - Probability","volume":"46 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142262310","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}
Piotr Dyszewski, Samuel G. G. Johnston, Sandra Palau, Joscha Prochno
Take a self-similar fragmentation process with dislocation measure $nu$ and index of self-similarity $alpha > 0$. Let $e^{-m_t}$ denote the size of the largest fragment in the system at time $tgeq 0$. We prove fine results for the asymptotics of the stochastic process $(s_{t geq 0}$ for a broad class of dislocation measures. In the case where the process has finite activity (i.e. $nu$ is a finite measure with total mass $lambda>0$), we show that setting begin{equation*} g(t) :=frac{1}{alpha}left(log t - log log t + log(alpha lambda)right), qquad tgeq 0, end{equation*} we have $lim_{t to infty} (m_t - g(t)) = 0$ almost-surely. In the case where the process has infinite activity, we impose the mild regularity condition that the dislocation measure satisfies begin{equation*} nu(1-s_1 > delta ) = delta^{-theta} ell(1/delta), end{equation*} for some $theta in (0,1)$ and $ell:(0,infty) to (0,infty)$ slowly varying at infinity. Under this regularity condition, we find that if begin{equation*} g(t) :=frac{1}{alpha}left( log t - (1-theta) log log t - log ell left( log t ~ellleft( log t right)^{frac{1}{1-theta}} right) + c(alpha,theta) right), qquad tgeq 0, end{equation*} then $lim_{t to infty} (m_t - g(t)) = 0$ almost-surely. Here $c(alpha,theta) := log alpha -(1-theta)log(1-theta) - log Gamma(1-theta)$. Our results sharpen significantly the best prior result on general self-similar fragmentation processes, due to Bertoin, which states that $m_t = (1+o(1)) frac{1}{alpha} log t$.
取一个自相似分裂过程，其错位度量为 $nu$，自相似度指数为 $alpha > 0$。让 $e^{-m_t}$ 表示时间 $tgeq 0$ 时系统中最大碎片的大小。我们证明了随机过程$(s_{t geq 0}$对于一类广泛的位移度量的渐近性的精细结果。在过程具有有限活动的情况下（即nu$是总质量为$lambda>0$的有限度量）的情况下，我们证明 settingbegin{equation*} g(t) ：=frac{1}{alpha}left(log t -log log t +log(alpha lambda)right), qquad tgeq 0, end{equation*} 我们几乎可以肯定 $lim_{tto infty} (m_t - g(t)) = 0$。在过程具有无限活动的情况下，我们施加了一个温和的规则性条件，即位错度量满足 begin{equation*}nu(1-s_1 > delta ) = delta^{-theta}ell(1/delta), end{equation*} for some $theta in (0,1)$ and$ell:(0,infty) to (0,infty)$ slowly varying at infinity.在这个正则条件下，我们会发现如果g(t):=frac{1}{alpha}left(（log t - (1-theta) log log t - log ell left(log t ~ellleft( （log t （right）^{frac{1}{1-theta}})right) +c(alpha,theta) right), qquad tgeq 0, end{equation*} 那么 $lim_{t toinfty} (m_t - g(t)) = 0$ 几乎是肯定的。这里 $c(alpha,theta) := log alpha-(1-theta)log(1-theta) - log Gamma(1-theta)$.我们的结果极大地改进了贝托因提出的关于一般自相似分裂过程的最佳先验结果，即 $m_t = (1+o(1)) frac{1}{alpha}log t$。
{"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":"3 1","pages":""},"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}
Charles BertucciCMAP, Pierre Louis LionsCdF, CEREMADE
We provide a simple $C^{1,1}$ approximation of the squared Wasserstein distance on R^d when one of the two measures is fixed. This approximation converges locally uniformly. More importantly, at points where the differential of the squared Wasserstein distance exists, it attracts the differentials of the approximations at nearby points. Our method relies on the Hilbertian lifting of PL Lions and on the regularization in Hilbert spaces of Lasry and Lions. We then provide an application of this result by using it to establish a comparison principle for an Hamilton-Jacobi equation on the set of probability measures.
{"title":"An approximation of the squared Wasserstein distance and an application to Hamilton-Jacobi equations","authors":"Charles BertucciCMAP, Pierre Louis LionsCdF, CEREMADE","doi":"arxiv-2409.11793","DOIUrl":"https://doi.org/arxiv-2409.11793","url":null,"abstract":"We provide a simple $C^{1,1}$ approximation of the squared Wasserstein\u0000distance on R^d when one of the two measures is fixed. This approximation\u0000converges locally uniformly. More importantly, at points where the differential\u0000of the squared Wasserstein distance exists, it attracts the differentials of\u0000the approximations at nearby points. Our method relies on the Hilbertian\u0000lifting of PL Lions and on the regularization in Hilbert spaces of Lasry and\u0000Lions. We then provide an application of this result by using it to establish a\u0000comparison principle for an Hamilton-Jacobi equation on the set of probability\u0000measures.","PeriodicalId":501245,"journal":{"name":"arXiv - MATH - Probability","volume":"30 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142262320","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}
Super-tree random measure's (STRM's) were introduced by Allouba, Durrett, Hawkes and Perkins as a simple stochastic model which emulates a superprocess at a fixed time. A STRM $nu$ arises as the a.s. limit of a sequence of empirical measures for a discrete time particle system which undergoes independent supercritical branching and independent random displacement (spatial motion) of children from their parents. We study the connectedness properties of the closed support of a STRM ($mathrm{supp}(nu)$) for a particular choice of random displacement. Our main results are distinct sufficient conditions for the a.s. total disconnectedness (TD) of $mathrm{supp}(nu)$, and for percolation on $mathrm{supp}(nu)$ which will imply a.s. existence of a non-trivial connected component in $mathrm{supp}(nu)$. We illustrate a close connection between a subclass of these STRM's and super-Brownian motion (SBM). For these particular STRM's the above results give a.s. TD of the support in three and higher dimensions and the existence of a non-trivial connected component in two dimensions, with the three-dimensional case being critical. The latter two-dimensional result assumes that $p_c(mathbb{Z}^2)$, the critical probability for site percolation on $mathbb{Z}^2$, is less than $1-e^{-1}$. (There is strong numerical evidence supporting this condition although the known rigorous bounds fall just short.) This gives evidence that the same connectedness properties should hold for SBM. The latter remains an interesting open problem in dimensions $2$ and $3$ ever since it was first posed by Don Dawson over $30$ years ago.
{"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":"40 1","pages":""},"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}
In this paper we show that the random degree constrained process (a time-evolving random graph model with degree constraints) has a local weak limit, provided that the underlying host graphs are high degree almost regular. We, moreover, identify the limit object as a multi-type branching process, by combining coupling arguments with the analysis of a certain recursive tree process. Using a spectral characterization, we also give an asymptotic expansion of the critical time when the giant component emerges in the so-called random $d$-process, resolving a problem of Warnke and Wormald for large $d$.
{"title":"Local limit of the random degree constrained process","authors":"Balázs Ráth, Márton Szőke, Lutz Warnke","doi":"arxiv-2409.11747","DOIUrl":"https://doi.org/arxiv-2409.11747","url":null,"abstract":"In this paper we show that the random degree constrained process (a\u0000time-evolving random graph model with degree constraints) has a local weak\u0000limit, provided that the underlying host graphs are high degree almost regular.\u0000We, moreover, identify the limit object as a multi-type branching process, by\u0000combining coupling arguments with the analysis of a certain recursive tree\u0000process. Using a spectral characterization, we also give an asymptotic\u0000expansion of the critical time when the giant component emerges in the\u0000so-called random $d$-process, resolving a problem of Warnke and Wormald for\u0000large $d$.","PeriodicalId":501245,"journal":{"name":"arXiv - MATH - Probability","volume":"12 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142262308","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}
We study the large deviation principle (LDP) for locally damped nonlinear wave equations perturbed by a bounded noise. When the noise is sufficiently non-degenerate, we establish the LDP for empirical distributions with lower bound of a local type. The primary challenge is the lack of compactness due to the absence of smoothing effect. This is overcome by exploiting the asymptotic compactness for the dynamics of waves, introducing the concept of asymptotic exponential tightness for random measures, and establishing a new LDP approach for random dynamical systems.
{"title":"Local large deviations for randomly forced nonlinear wave equations with localized damping","authors":"Yuxuan Chen, Ziyu Liu, Shengquan Xiang, Zhifei Zhang","doi":"arxiv-2409.11717","DOIUrl":"https://doi.org/arxiv-2409.11717","url":null,"abstract":"We study the large deviation principle (LDP) for locally damped nonlinear\u0000wave equations perturbed by a bounded noise. When the noise is sufficiently\u0000non-degenerate, we establish the LDP for empirical distributions with lower\u0000bound of a local type. The primary challenge is the lack of compactness due to\u0000the absence of smoothing effect. This is overcome by exploiting the asymptotic\u0000compactness for the dynamics of waves, introducing the concept of asymptotic\u0000exponential tightness for random measures, and establishing a new LDP approach\u0000for random dynamical systems.","PeriodicalId":501245,"journal":{"name":"arXiv - MATH - Probability","volume":"30 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142262311","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}
In this work, a variant of the birth and death chain with constant intensities, originally introduced by Bruno de Finetti way back in 1957, is revisited. This fact is also underlined by the choice of the title, which is clearly a literal translation of the original one. Characteristic of the variant is that it allows negative jumps of any magnitude. And this, as explained in the paper, might be useful in offering some insight into the issue, arising in numerous situations, of inferring the number of the undetected elements of a given population. One thinks, for example, of problems concerning abundance or richness of species. The author's purpose is twofold: to align the original de Finetti's construction with the modern, well-established theory of the continuous-time Markov chains with discrete state space and show how it could be used to make probabilistic previsions on the number of the unseen elements of a population. With the aim of enhancing the possible practical applications of the model, one discusses the statistical point estimation of the rates which characterize its infinitesimal description.
在这部作品中，我们重新审视了布鲁诺-德-菲内蒂（Bruno de Finetti）早在 1957 年就提出的具有恒定强度的生死链变体。标题的选择也凸显了这一事实，它显然是对原标题的直译。该变式的特点是允许任何大小的负跳跃。正如论文中解释的那样，这可能有助于深入了解在许多情况下出现的问题，即推断特定种群中未检测到的元素的数量。例如，我们会想到有关物种丰度或丰富度的问题。作者的目的有两个：将德菲内蒂的原始结构与现代成熟的离散空间连续时间马尔可夫链理论相结合，并说明如何利用它来对种群中未发现元素的数量进行概率预测。为了加强该模型可能的实际应用，我们讨论了该模型无穷小描述的特征率的统计点估计。
{"title":"On the number of elements beyond the ones actually observed","authors":"Eugenio Regazzini","doi":"arxiv-2409.11364","DOIUrl":"https://doi.org/arxiv-2409.11364","url":null,"abstract":"In this work, a variant of the birth and death chain with constant\u0000intensities, originally introduced by Bruno de Finetti way back in 1957, is revisited. This fact is also underlined\u0000by the choice of the title, which is clearly a literal translation of the original one. Characteristic of\u0000the variant is that it allows negative jumps of any magnitude. And this, as explained in the paper,\u0000might be useful in offering some insight into the issue, arising in numerous situations, of inferring the\u0000number of the undetected elements of a given population. One thinks, for example, of problems\u0000concerning abundance or richness of species. The author's purpose is twofold: to align the original de Finetti's\u0000construction with the modern, well-established theory of the continuous-time Markov chains with discrete\u0000state space and show how it could be used to make probabilistic previsions on the number of the unseen\u0000elements of a population. With the aim of enhancing the possible practical applications of the model,\u0000one discusses the statistical point estimation of the rates which characterize its infinitesimal\u0000description.","PeriodicalId":501245,"journal":{"name":"arXiv - MATH - Probability","volume":"12 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142262312","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}
We study notions of hyperuniformity for invariant locally square-integrable point processes in regular trees. We show that such point processes are never geometrically hyperuniform, and if the diffraction measure has support in the complementary series then the process is geometrically hyperfluctuating along all subsequences of radii. A definition of spectral hyperuniformity and stealth of a point process is given in terms of vanishing of the complementary series diffraction and sub-Poissonian decay of the principal series diffraction near the endpoints of the principal spectrum. Our main contribution is providing examples of stealthy invariant random lattice orbits in trees whose number variance grows strictly slower than the volume along some unbounded sequence of radii. These random lattice orbits are constructed from the fundamental groups of complete graphs and the Petersen graph.
{"title":"Hyperuniformity in regular trees","authors":"Mattias Byléhn","doi":"arxiv-2409.10998","DOIUrl":"https://doi.org/arxiv-2409.10998","url":null,"abstract":"We study notions of hyperuniformity for invariant locally square-integrable\u0000point processes in regular trees. We show that such point processes are never\u0000geometrically hyperuniform, and if the diffraction measure has support in the\u0000complementary series then the process is geometrically hyperfluctuating along\u0000all subsequences of radii. A definition of spectral hyperuniformity and stealth\u0000of a point process is given in terms of vanishing of the complementary series\u0000diffraction and sub-Poissonian decay of the principal series diffraction near\u0000the endpoints of the principal spectrum. Our main contribution is providing\u0000examples of stealthy invariant random lattice orbits in trees whose number\u0000variance grows strictly slower than the volume along some unbounded sequence of\u0000radii. These random lattice orbits are constructed from the fundamental groups\u0000of complete graphs and the Petersen graph.","PeriodicalId":501245,"journal":{"name":"arXiv - MATH - Probability","volume":"50 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142262316","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}
In this paper we generalize Krylov's theory on parameter-dependent stochastic differential equations to the framework of rough stochastic differential equations (rough SDEs), as initially introduced by Friz, Hocquet and L^e. We consider a stochastic equation of the form $$ dX_t^zeta = b_t(zeta,X_t^zeta) dt + sigma_t(zeta,X_t^zeta) dB_t + beta_t (zeta,X_t^zeta) dmathbf{W}_t,$$ where $zeta$ is a parameter, $B$ denotes a Brownian motion and $mathbf{W}$ is a deterministic H"older rough path. We investigate the conditions under which the solution $X$ exhibits continuity and/or differentiability with respect to the parameter $zeta$ in the $mathscr{L}$-sense, as defined by Krylov. As an application, we present an existence-and-uniqueness result for a class of rough partial differential equations (rough PDEs) of the form $$-du_t = L_t u_t dt + Gamma_t u_t dmathbf{W}_t, quad u_T =g.$$ We show that the solution admits a Feynman--Kac type representation in terms of the solution of an appropriate rough SDE, where the initial time and the initial state play the role of parameters.
{"title":"Parameter dependent rough SDEs with applications to rough PDEs","authors":"Fabio Bugini, Peter K. Friz, Wilhelm Stannat","doi":"arxiv-2409.11330","DOIUrl":"https://doi.org/arxiv-2409.11330","url":null,"abstract":"In this paper we generalize Krylov's theory on parameter-dependent stochastic\u0000differential equations to the framework of rough stochastic differential\u0000equations (rough SDEs), as initially introduced by Friz, Hocquet and L^e. We\u0000consider a stochastic equation of the form $$ dX_t^zeta = b_t(zeta,X_t^zeta)\u0000 dt + sigma_t(zeta,X_t^zeta) dB_t + beta_t (zeta,X_t^zeta)\u0000dmathbf{W}_t,$$ where $zeta$ is a parameter, $B$ denotes a Brownian motion\u0000and $mathbf{W}$ is a deterministic H\"older rough path. We investigate the\u0000conditions under which the solution $X$ exhibits continuity and/or\u0000differentiability with respect to the parameter $zeta$ in the\u0000$mathscr{L}$-sense, as defined by Krylov. As an application, we present an existence-and-uniqueness result for a class\u0000of rough partial differential equations (rough PDEs) of the form $$-du_t = L_t\u0000u_t dt + Gamma_t u_t dmathbf{W}_t, quad u_T =g.$$ We show that the solution\u0000admits a Feynman--Kac type representation in terms of the solution of an\u0000appropriate rough SDE, where the initial time and the initial state play the\u0000role of parameters.","PeriodicalId":501245,"journal":{"name":"arXiv - MATH - Probability","volume":"2 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142262314","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}
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