S. Foss, T. Konstantopoulos, Bastien Mallein, Sanjay Ramassamy
Our object of study is the asymptotic growth of heaviest paths in a charged (weighted with signed weights) complete directed acyclic graph. Edge charges are i.i.d. random variables with common distribution $F$ supported on $[-infty,1]$ with essential supremum equal to $1$ (a charge of $-infty$ is understood as the absence of an edge). The asymptotic growth rate is a constant that we denote by $C(F)$. Even in the simplest case where $F=pdelta_1 + (1-p)delta_{-infty}$, corresponding to the longest path in the Barak-ErdH{o}s random graph, there is no closed-form expression for this function, but good bounds do exist. In this paper we construct a Markovian particle system that we call"Max Growth System"(MGS), and show how it is related to the charged random graph. The MGS is a generalization of the Infinite Bin Model that has been the object of study of a number of papers. We then identify a random functional of the process that admits a stationary version and whose expectation equals the unknown constant $C(F)$. Furthermore, we construct an effective perfect simulation algorithm for this functional which produces samples from the random functional.
{"title":"Estimation of the last passage percolation constant in a charged complete directed acyclic graph via perfect simulation","authors":"S. Foss, T. Konstantopoulos, Bastien Mallein, Sanjay Ramassamy","doi":"10.30757/ALEA.v20-19","DOIUrl":"https://doi.org/10.30757/ALEA.v20-19","url":null,"abstract":"Our object of study is the asymptotic growth of heaviest paths in a charged (weighted with signed weights) complete directed acyclic graph. Edge charges are i.i.d. random variables with common distribution $F$ supported on $[-infty,1]$ with essential supremum equal to $1$ (a charge of $-infty$ is understood as the absence of an edge). The asymptotic growth rate is a constant that we denote by $C(F)$. Even in the simplest case where $F=pdelta_1 + (1-p)delta_{-infty}$, corresponding to the longest path in the Barak-ErdH{o}s random graph, there is no closed-form expression for this function, but good bounds do exist. In this paper we construct a Markovian particle system that we call\"Max Growth System\"(MGS), and show how it is related to the charged random graph. The MGS is a generalization of the Infinite Bin Model that has been the object of study of a number of papers. We then identify a random functional of the process that admits a stationary version and whose expectation equals the unknown constant $C(F)$. Furthermore, we construct an effective perfect simulation algorithm for this functional which produces samples from the random functional.","PeriodicalId":49244,"journal":{"name":"Alea-Latin American Journal of Probability and Mathematical Statistics","volume":" ","pages":""},"PeriodicalIF":0.7,"publicationDate":"2021-10-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46072346","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
. We study the Matsumoto-Yor property in free probability. We prove three characterizations of free-GIG and free Poisson distributions by freeness properties together with some assumptions about conditional moments. Our main tools are subordination and Boolean cumulants. In particular, we establish a new connection between the additive subordination function and Boolean cumulants.
{"title":"The Matsumoto-Yor property in free probability via subordination and Boolean cumulants","authors":"Marcin Świeca","doi":"10.30757/alea.v19-55","DOIUrl":"https://doi.org/10.30757/alea.v19-55","url":null,"abstract":". We study the Matsumoto-Yor property in free probability. We prove three characterizations of free-GIG and free Poisson distributions by freeness properties together with some assumptions about conditional moments. Our main tools are subordination and Boolean cumulants. In particular, we establish a new connection between the additive subordination function and Boolean cumulants.","PeriodicalId":49244,"journal":{"name":"Alea-Latin American Journal of Probability and Mathematical Statistics","volume":" ","pages":""},"PeriodicalIF":0.7,"publicationDate":"2021-09-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42470611","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this article, we study a $d$-dimensional stochastic nonlinear heat equation (SNLH) with a quadratic nonlinearity, forced by a fractional space-time white noise: begin{equation*} left{begin{array}{l} partial_t u-Delta u= rho^2 u^2 + dot B , , quad tin [0,T] , , , xin mathbb{R}^d , , u_0=phi, . end{array} right. end{equation*} Two types of regimes are exhibited, depending on the ranges of the Hurst index $H=(H_0,...,H_d)$ $in (0,1)^{d+1}$. In particular, we show that the local well-posedness of (SNLH) resulting from the Da Prato-Debussche trick, is easily obtained when $2 H_0+sum_{i=1}^{d}H_i>d$. On the contrary, (SNLH) is much more difficult to handle when $2H_0+sum_{i=1}^{d}H_i leq d$. In this case, the model has to be interpreted in the Wick sense, thanks to a time-dependent renormalization. Helped with the regularising effect of the heat semigroup, we establish local well-posedness results for (SNLH) for all dimension $dgeq1.$
{"title":"Study of a fractional stochastic heat equation","authors":"N. Schaeffer","doi":"10.30757/alea.v20-15","DOIUrl":"https://doi.org/10.30757/alea.v20-15","url":null,"abstract":"In this article, we study a $d$-dimensional stochastic nonlinear heat equation (SNLH) with a quadratic nonlinearity, forced by a fractional space-time white noise: begin{equation*} left{begin{array}{l} partial_t u-Delta u= rho^2 u^2 + dot B , , quad tin [0,T] , , , xin mathbb{R}^d , , u_0=phi, . end{array} right. end{equation*} Two types of regimes are exhibited, depending on the ranges of the Hurst index $H=(H_0,...,H_d)$ $in (0,1)^{d+1}$. In particular, we show that the local well-posedness of (SNLH) resulting from the Da Prato-Debussche trick, is easily obtained when $2 H_0+sum_{i=1}^{d}H_i>d$. On the contrary, (SNLH) is much more difficult to handle when $2H_0+sum_{i=1}^{d}H_i leq d$. In this case, the model has to be interpreted in the Wick sense, thanks to a time-dependent renormalization. Helped with the regularising effect of the heat semigroup, we establish local well-posedness results for (SNLH) for all dimension $dgeq1.$","PeriodicalId":49244,"journal":{"name":"Alea-Latin American Journal of Probability and Mathematical Statistics","volume":" ","pages":""},"PeriodicalIF":0.7,"publicationDate":"2021-09-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47685706","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In extending Stein's method to new target distributions, the first step is to find a Stein operator that suitably characterises the target distribution. In this paper, we introduce a widely applicable technique for proving sufficiency of these Stein characterisations, which can be applied when the Stein operators are linear differential operators with polynomial coefficients. The approach involves performing an asymptotic analysis to prove that only one characteristic function satisfies a certain differential equation associated to the Stein characterisation. We use this approach to prove that all Stein operators with linear coefficients characterise their target distribution, and verify on a case-by-case basis that all polynomial Stein operators in the literature with coefficients of degree at most two are characterising. For $X$ denoting a standard Gaussian random variable and $H_p$ the $p$-th Hermite polynomial, we also prove, amongst other examples, that the Stein operators for $H_p(X)$, $p=3,4,ldots,8$, with coefficients of minimal possible degree characterise their target distribution, and that the Stein operators for the products of $p=3,4,ldots,8$ independent standard Gaussian random variables are characterising (in both settings the Stein operators for the cases $p=1,2$ are already known to be characterising). We leverage our Stein characterisations of $H_3(X)$ and $H_4(X)$ to derive characterisations of these target distributions in terms of iterated Gamma operators from Malliavin calculus, that are natural in the context of the Malliavin-Stein method.
{"title":"An asymptotic approach to proving sufficiency of Stein characterisations","authors":"E. Azmoodeh, Dario Gasbarra, Robert E. Gaunt","doi":"10.30757/alea.v20-06","DOIUrl":"https://doi.org/10.30757/alea.v20-06","url":null,"abstract":"In extending Stein's method to new target distributions, the first step is to find a Stein operator that suitably characterises the target distribution. In this paper, we introduce a widely applicable technique for proving sufficiency of these Stein characterisations, which can be applied when the Stein operators are linear differential operators with polynomial coefficients. The approach involves performing an asymptotic analysis to prove that only one characteristic function satisfies a certain differential equation associated to the Stein characterisation. We use this approach to prove that all Stein operators with linear coefficients characterise their target distribution, and verify on a case-by-case basis that all polynomial Stein operators in the literature with coefficients of degree at most two are characterising. For $X$ denoting a standard Gaussian random variable and $H_p$ the $p$-th Hermite polynomial, we also prove, amongst other examples, that the Stein operators for $H_p(X)$, $p=3,4,ldots,8$, with coefficients of minimal possible degree characterise their target distribution, and that the Stein operators for the products of $p=3,4,ldots,8$ independent standard Gaussian random variables are characterising (in both settings the Stein operators for the cases $p=1,2$ are already known to be characterising). We leverage our Stein characterisations of $H_3(X)$ and $H_4(X)$ to derive characterisations of these target distributions in terms of iterated Gamma operators from Malliavin calculus, that are natural in the context of the Malliavin-Stein method.","PeriodicalId":49244,"journal":{"name":"Alea-Latin American Journal of Probability and Mathematical Statistics","volume":" ","pages":""},"PeriodicalIF":0.7,"publicationDate":"2021-09-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42323526","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
. We consider a branching stable process with positive jumps, i.e. a continuous-time branching process in which the particles evolve independently as stable Lévy processes with positive jumps. Assuming the branching mechanism is critical or subcritical, we compute the asymptotics of the maximum location ever reached by a particle of the process.
{"title":"Extreme values of critical and subcritical branching stable processes with positive jumps","authors":"C. Profeta","doi":"10.30757/alea.v19-57","DOIUrl":"https://doi.org/10.30757/alea.v19-57","url":null,"abstract":". We consider a branching stable process with positive jumps, i.e. a continuous-time branching process in which the particles evolve independently as stable Lévy processes with positive jumps. Assuming the branching mechanism is critical or subcritical, we compute the asymptotics of the maximum location ever reached by a particle of the process.","PeriodicalId":49244,"journal":{"name":"Alea-Latin American Journal of Probability and Mathematical Statistics","volume":" ","pages":""},"PeriodicalIF":0.7,"publicationDate":"2021-09-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41987055","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
This paper builds upon the research of Corwin and Knizel who proved the existence of stationary measures for the KPZ equation on an interval and characterized them through a Laplace transform formula. Bryc, Kuznetsov, Wang and Wesolowski found a probabilistic description of the stationary measures in terms of a Doob transform of some Markov kernels; essentially at the same time, another description connecting the stationary measures to the exponential functionals of the Brownian motion appeared in work of Barraquand and Le Doussal. Our first main result clarifies and proves the equivalence of the two probabilistic description of these stationary measures. We then use the Markovian description to give rigorous proofs of some of the results claimed in Barraquand and Le Doussal. We analyze how the stationary measures of the KPZ equation on finite interval behave at large scale. We investigate which of the limits of the steady states of the KPZ equation obtained recently by G. Barraquand and P. Le Doussal can be represented by Markov processes in spatial variable under an additional restriction on the range of parameters.
本文以Corwin和Knizel的研究为基础，证明了区间上KPZ方程的平稳测度的存在性，并用拉普拉斯变换公式对其进行了刻画。Bryc、Kuznetsov、Wang和Wesolowski用一些马尔可夫核的Doob变换找到了平稳测度的概率描述;基本上在同一时间，另一种将平稳测度与布朗运动的指数函数联系起来的描述出现在Barraquand和Le dousal的著作中。我们的第一个主要结果澄清并证明了这些平稳测度的两个概率描述的等价性。然后，我们使用马尔可夫描述对Barraquand和Le Doussal提出的一些结果给出严格的证明。我们分析了有限区间上KPZ方程的平稳测度在大尺度上的表现。我们研究了G. Barraquand和P. Le Doussal最近得到的KPZ方程的稳态极限中，在参数范围的附加限制下，哪些可以用空间变量中的马尔可夫过程表示。
{"title":"Markov limits of steady states of the KPZ equation on an interval","authors":"W. Bryc, A. Kuznetsov","doi":"10.30757/ALEA.v19-53","DOIUrl":"https://doi.org/10.30757/ALEA.v19-53","url":null,"abstract":"This paper builds upon the research of Corwin and Knizel who proved the existence of stationary measures for the KPZ equation on an interval and characterized them through a Laplace transform formula. Bryc, Kuznetsov, Wang and Wesolowski found a probabilistic description of the stationary measures in terms of a Doob transform of some Markov kernels; essentially at the same time, another description connecting the stationary measures to the exponential functionals of the Brownian motion appeared in work of Barraquand and Le Doussal. Our first main result clarifies and proves the equivalence of the two probabilistic description of these stationary measures. We then use the Markovian description to give rigorous proofs of some of the results claimed in Barraquand and Le Doussal. We analyze how the stationary measures of the KPZ equation on finite interval behave at large scale. We investigate which of the limits of the steady states of the KPZ equation obtained recently by G. Barraquand and P. Le Doussal can be represented by Markov processes in spatial variable under an additional restriction on the range of parameters.","PeriodicalId":49244,"journal":{"name":"Alea-Latin American Journal of Probability and Mathematical Statistics","volume":" ","pages":""},"PeriodicalIF":0.7,"publicationDate":"2021-09-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47834799","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We propose and study a certain discrete time counterpart of the classical Feynman--Kac semigroup with a confining potential in countable infinite spaces. For a class of long range Markov chains which satisfy the direct step property we prove sharp estimates for functions which are (sub-, super-)harmonic in infinite sets with respect to the discrete Feynman--Kac operators. These results are compared with respective estimates for the case of a nearest-neighbour random walk which evolves on a graph of finite geometry. We also discuss applications to the decay rates of solutions to equations involving graph Laplacians and to eigenfunctions of the discrete Feynman--Kac operators. We include such examples as non-local discrete Schr"odinger operators based on fractional powers of the nearest-neighbour Laplacians and related quasi-relativistic operators. Finally, we analyse various classes of Markov chains which enjoy the direct step property and illustrate the obtained results by examples.
{"title":"Decay of harmonic functions for discrete time Feynman–Kac operators with confining potentials","authors":"W. Cygan, K. Kaleta, Mateusz 'Sliwi'nski","doi":"10.30757/alea.v19-44","DOIUrl":"https://doi.org/10.30757/alea.v19-44","url":null,"abstract":"We propose and study a certain discrete time counterpart of the classical Feynman--Kac semigroup with a confining potential in countable infinite spaces. For a class of long range Markov chains which satisfy the direct step property we prove sharp estimates for functions which are (sub-, super-)harmonic in infinite sets with respect to the discrete Feynman--Kac operators. These results are compared with respective estimates for the case of a nearest-neighbour random walk which evolves on a graph of finite geometry. We also discuss applications to the decay rates of solutions to equations involving graph Laplacians and to eigenfunctions of the discrete Feynman--Kac operators. We include such examples as non-local discrete Schr\"odinger operators based on fractional powers of the nearest-neighbour Laplacians and related quasi-relativistic operators. Finally, we analyse various classes of Markov chains which enjoy the direct step property and illustrate the obtained results by examples.","PeriodicalId":49244,"journal":{"name":"Alea-Latin American Journal of Probability and Mathematical Statistics","volume":" ","pages":""},"PeriodicalIF":0.7,"publicationDate":"2021-09-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44983588","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The Arboreal gas model on a finite graph $G$ is the Bernoulli bond percolation on $G$ conditioned on the event that the sampled subgraph is a forest. In this short note we study the arboreal gas on a regular tree wired at the leaves and obtain a comprehensive description of the weak limit of this model.
{"title":"Forests on wired regular trees","authors":"G. Ray, Ben Xiao","doi":"10.30757/alea.v19-42","DOIUrl":"https://doi.org/10.30757/alea.v19-42","url":null,"abstract":"The Arboreal gas model on a finite graph $G$ is the Bernoulli bond percolation on $G$ conditioned on the event that the sampled subgraph is a forest. In this short note we study the arboreal gas on a regular tree wired at the leaves and obtain a comprehensive description of the weak limit of this model.","PeriodicalId":49244,"journal":{"name":"Alea-Latin American Journal of Probability and Mathematical Statistics","volume":" ","pages":""},"PeriodicalIF":0.7,"publicationDate":"2021-08-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45502245","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this article, we study the existence of an absorbing-state phase transition of an Abelian process that generalises the Activated Random Walk (ARW). Given a vertex transitive $G=(V,E)$, we associate to each site $x in V$ a capacity $w_x ge 0$, which describes how many inactive particles $x$ can hold, where ${w_x}_{x in V}$ is a collection of i.i.d random variables. When $G$ is an amenable graph, we prove that if $mathbb E[w_x]0$. Moreover, in the former case, we provide bounds for the critical density that match the ones available in the classical Activated Random Walk.
{"title":"Absorbing-state phase transition and activated random walks with unbounded capacities","authors":"L. Chiarini, Alexandre O. Stauffer","doi":"10.30757/alea.v19-46","DOIUrl":"https://doi.org/10.30757/alea.v19-46","url":null,"abstract":"In this article, we study the existence of an absorbing-state phase transition of an Abelian process that generalises the Activated Random Walk (ARW). Given a vertex transitive $G=(V,E)$, we associate to each site $x in V$ a capacity $w_x ge 0$, which describes how many inactive particles $x$ can hold, where ${w_x}_{x in V}$ is a collection of i.i.d random variables. When $G$ is an amenable graph, we prove that if $mathbb E[w_x]<infty$, the model goes through an absorbing state phase transition and if $mathbb E[w_x]=infty$, the model fixates for all $lambda>0$. Moreover, in the former case, we provide bounds for the critical density that match the ones available in the classical Activated Random Walk.","PeriodicalId":49244,"journal":{"name":"Alea-Latin American Journal of Probability and Mathematical Statistics","volume":" ","pages":""},"PeriodicalIF":0.7,"publicationDate":"2021-08-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43456298","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Ikkei Hotta, W. Mlotkowski, Noriyoshi Sakuma, Yuki Ueda
Inspired by the notion of quasi-infinite divisibility (QID), we introduce and study the class of freely quasi-infinitely divisible (FQID) distributions on $mathbb{R}$, i.e. distributions which admit the free L'{e}vy-Khintchine-type representation with signed L'{e}vy measure. We prove several properties of the FQID class, some of them in contrast to those of the QID class. For example, a FQID distribution may have negative Gaussian part, and the total mass of its signed L'{e}vy measure may be negative. Finally, we extend the Bercovici-Pata bijection, providing a characteristic triplet, with the L'{e}vy measure having nonzero negative part, which is at the same time classical and free characteristic triplet.
{"title":"On freely quasi-infinitely divisible distributions","authors":"Ikkei Hotta, W. Mlotkowski, Noriyoshi Sakuma, Yuki Ueda","doi":"10.30757/alea.v20-34","DOIUrl":"https://doi.org/10.30757/alea.v20-34","url":null,"abstract":"Inspired by the notion of quasi-infinite divisibility (QID), we introduce and study the class of freely quasi-infinitely divisible (FQID) distributions on $mathbb{R}$, i.e. distributions which admit the free L'{e}vy-Khintchine-type representation with signed L'{e}vy measure. We prove several properties of the FQID class, some of them in contrast to those of the QID class. For example, a FQID distribution may have negative Gaussian part, and the total mass of its signed L'{e}vy measure may be negative. Finally, we extend the Bercovici-Pata bijection, providing a characteristic triplet, with the L'{e}vy measure having nonzero negative part, which is at the same time classical and free characteristic triplet.","PeriodicalId":49244,"journal":{"name":"Alea-Latin American Journal of Probability and Mathematical Statistics","volume":" ","pages":""},"PeriodicalIF":0.7,"publicationDate":"2021-07-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45168236","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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