In this work, we establish the existence of large deviation principles of random walk in strongly mixing environments. The quenched and annealed rate functions have the same zero set whose shape is either a singleton point or a line segment, with an illustrative example communicated and given by F. Rassoul-Agha. Whenever the level of disorder is controlled, the two rate functions agree on compact set at the boundary under mixing conditions and in the interior under finite-dependence condition. Along the line we also indicate that under slightly more refined conditions, there is a phase transition of the difference of two rate functions.
{"title":"Large deviations of quenched and annealed random walk in mixing random environment","authors":"Jiaming Chen","doi":"arxiv-2409.06581","DOIUrl":"https://doi.org/arxiv-2409.06581","url":null,"abstract":"In this work, we establish the existence of large deviation principles of\u0000random walk in strongly mixing environments. The quenched and annealed rate\u0000functions have the same zero set whose shape is either a singleton point or a\u0000line segment, with an illustrative example communicated and given by F.\u0000Rassoul-Agha. Whenever the level of disorder is controlled, the two rate\u0000functions agree on compact set at the boundary under mixing conditions and in\u0000the interior under finite-dependence condition. Along the line we also indicate\u0000that under slightly more refined conditions, there is a phase transition of the\u0000difference of two rate functions.","PeriodicalId":501245,"journal":{"name":"arXiv - MATH - Probability","volume":"28 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142212037","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 probability distribution of the number of common zeros of a system of $m$ random $n$-variate polynomials over a finite commutative ring $R$. We compute the expected number of common zeros of a system of polynomials over $R$. Then, in the case that $R$ is a field, under a necessary-and-sufficient condition on the sample space, we show that the number of common zeros is binomially distributed.
{"title":"The number of solutions of a random system of polynomials over a finite field","authors":"Ritik Jain","doi":"arxiv-2409.06866","DOIUrl":"https://doi.org/arxiv-2409.06866","url":null,"abstract":"We study the probability distribution of the number of common zeros of a\u0000system of $m$ random $n$-variate polynomials over a finite commutative ring\u0000$R$. We compute the expected number of common zeros of a system of polynomials\u0000over $R$. Then, in the case that $R$ is a field, under a\u0000necessary-and-sufficient condition on the sample space, we show that the number\u0000of common zeros is binomially distributed.","PeriodicalId":501245,"journal":{"name":"arXiv - MATH - Probability","volume":"65 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142212041","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 construction of invariant measures associated with higher order conservation laws of the intermediate long wave equation (ILW) and their convergence properties in the deep-water and shallow-water limits. By exploiting its complete integrability, we first carry out detailed analysis on the construction of appropriate conservation laws of ILW at the $H^frac k2$-level for each $k in mathbb{N}$, and establish their convergence to those of the Benjamin-Ono equation (BO) in the deep-water limit and to those of the Korteweg-de Vries equation (KdV) in the shallow-water limit. In particular, in the shallow-water limit, we prove rather striking 2-to-1 collapse of the conservation laws of ILW to those of KdV. Such 2-to-1 collapse is novel in the literature and, to our knowledge, this is the first construction of a complete family of shallow-water conservation laws with non-trivial shallow-water limits. We then construct an infinite sequence of generalized Gibbs measures for ILW associated with these conservation laws and prove their convergence to the corresponding (invariant) generalized Gibbs measures for BO and KdV in the respective limits. Finally, for $k ge 3$, we establish invariance of these measures under ILW dynamics, and also convergence in the respective limits of the ILW dynamics at each equilibrium state to the corresponding invariant dynamics for BO and KdV constructed by Deng, Tzvetkov, and Visciglia (2010-2015) and Zhidkov (1996), respectively. In particular, in the shallow-water limit, we establish 2-to-1 collapse at the level of the generalized Gibbs measures as well as the invariant ILW dynamics. As a byproduct of our analysis, we also prove invariance of the generalized Gibbs measure associated with the $H^2$-conservation law of KdV, which seems to be missing in the literature.
{"title":"Deep-water and shallow-water limits of statistical equilibria for the intermediate long wave equation","authors":"Andreia Chapouto, Guopeng Li, Tadahiro Oh","doi":"arxiv-2409.06905","DOIUrl":"https://doi.org/arxiv-2409.06905","url":null,"abstract":"We study the construction of invariant measures associated with higher order\u0000conservation laws of the intermediate long wave equation (ILW) and their\u0000convergence properties in the deep-water and shallow-water limits. By\u0000exploiting its complete integrability, we first carry out detailed analysis on\u0000the construction of appropriate conservation laws of ILW at the $H^frac\u0000k2$-level for each $k in mathbb{N}$, and establish their convergence to those\u0000of the Benjamin-Ono equation (BO) in the deep-water limit and to those of the\u0000Korteweg-de Vries equation (KdV) in the shallow-water limit. In particular, in\u0000the shallow-water limit, we prove rather striking 2-to-1 collapse of the\u0000conservation laws of ILW to those of KdV. Such 2-to-1 collapse is novel in the\u0000literature and, to our knowledge, this is the first construction of a complete\u0000family of shallow-water conservation laws with non-trivial shallow-water\u0000limits. We then construct an infinite sequence of generalized Gibbs measures\u0000for ILW associated with these conservation laws and prove their convergence to\u0000the corresponding (invariant) generalized Gibbs measures for BO and KdV in the\u0000respective limits. Finally, for $k ge 3$, we establish invariance of these\u0000measures under ILW dynamics, and also convergence in the respective limits of\u0000the ILW dynamics at each equilibrium state to the corresponding invariant\u0000dynamics for BO and KdV constructed by Deng, Tzvetkov, and Visciglia\u0000(2010-2015) and Zhidkov (1996), respectively. In particular, in the\u0000shallow-water limit, we establish 2-to-1 collapse at the level of the\u0000generalized Gibbs measures as well as the invariant ILW dynamics. As a\u0000byproduct of our analysis, we also prove invariance of the generalized Gibbs\u0000measure associated with the $H^2$-conservation law of KdV, which seems to be\u0000missing in the literature.","PeriodicalId":501245,"journal":{"name":"arXiv - MATH - Probability","volume":"14 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142211981","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 article, we give explicit bounds on the Wasserstein and the Kolmogorov distances between random variables lying in the first chaos of the Poisson space and the standard Normal distribution, using the results proved by Last, Peccati and Schulte. Relying on the theory developed in the work of Saulis and Statulevicius and on a fine control of the cumulants of the first chaoses, we also derive moderate deviation principles, Bernstein-type concentration inequalities and Normal approximation bounds with Cram'er correction terms for the same variables. The aforementioned results are then applied to Poisson shot-noise processes and, in particular, to the generalized compound Hawkes point processes (a class of stochastic models, introduced in this paper, which generalizes classical Hawkes processes). This extends the recent results availale in the literature regarding the Normal approximation and moderate deviations.
{"title":"Gaussian Approximation and Moderate Deviations of Poisson Shot Noises with Application to Compound Generalized Hawkes Processes","authors":"Mahmoud Khabou, Giovanni Luca Torrisi","doi":"arxiv-2409.06394","DOIUrl":"https://doi.org/arxiv-2409.06394","url":null,"abstract":"In this article, we give explicit bounds on the Wasserstein and the\u0000Kolmogorov distances between random variables lying in the first chaos of the\u0000Poisson space and the standard Normal distribution, using the results proved by\u0000Last, Peccati and Schulte. Relying on the theory developed in the work of\u0000Saulis and Statulevicius and on a fine control of the cumulants of the first\u0000chaoses, we also derive moderate deviation principles, Bernstein-type\u0000concentration inequalities and Normal approximation bounds with Cram'er\u0000correction terms for the same variables. The aforementioned results are then\u0000applied to Poisson shot-noise processes and, in particular, to the generalized\u0000compound Hawkes point processes (a class of stochastic models, introduced in\u0000this paper, which generalizes classical Hawkes processes). This extends the\u0000recent results availale in the literature regarding the Normal approximation\u0000and moderate deviations.","PeriodicalId":501245,"journal":{"name":"arXiv - MATH - Probability","volume":"28 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142212038","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}
Heshan Aravinda, Yevgeniy Kovchegov, Peter T. Otto, Amites Sarkar
Random coalescent processes and branching processes are two fundamental constructs in the field of stochastic processes, each with a rich history and a wide range of applications. Though developed within distinct contexts, in this note we present a novel connection between a multi-type (vector) multiplicative coalescent process and a multi-type branching process with Poisson offspring distributions. More specifically, we show that the equations that govern the phenomenon of gelation in the vector multiplicative coalescent process are equivalent to the set of equations that yield the extinction probabilities of the corresponding multi-type Poisson branching process. We then leverage this connection with two applications, one in each direction. The first is a new quick proof of gelation in the vector multiplicative coalescent process using a well known result of branching processes, and the second is a new series expression for the extinction probabilities of the multi-type Poisson branching process using results derived from the theory of vector multiplicative coalescence. While the correspondence is fairly straightforward, it illuminates a deep connection between these two paradigms which we hope will continue to reveal new insights and potential for cross-disciplinary research.
{"title":"Gelation in Vector Multiplicative Coalescence and Extinction in Multi-Type Poisson Branching Processes","authors":"Heshan Aravinda, Yevgeniy Kovchegov, Peter T. Otto, Amites Sarkar","doi":"arxiv-2409.06910","DOIUrl":"https://doi.org/arxiv-2409.06910","url":null,"abstract":"Random coalescent processes and branching processes are two fundamental\u0000constructs in the field of stochastic processes, each with a rich history and a\u0000wide range of applications. Though developed within distinct contexts, in this\u0000note we present a novel connection between a multi-type (vector) multiplicative\u0000coalescent process and a multi-type branching process with Poisson offspring\u0000distributions. More specifically, we show that the equations that govern the\u0000phenomenon of gelation in the vector multiplicative coalescent process are\u0000equivalent to the set of equations that yield the extinction probabilities of\u0000the corresponding multi-type Poisson branching process. We then leverage this\u0000connection with two applications, one in each direction. The first is a new\u0000quick proof of gelation in the vector multiplicative coalescent process using a\u0000well known result of branching processes, and the second is a new series\u0000expression for the extinction probabilities of the multi-type Poisson branching\u0000process using results derived from the theory of vector multiplicative\u0000coalescence. While the correspondence is fairly straightforward, it illuminates\u0000a deep connection between these two paradigms which we hope will continue to\u0000reveal new insights and potential for cross-disciplinary research.","PeriodicalId":501245,"journal":{"name":"arXiv - MATH - Probability","volume":"18 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142212043","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 convergence of a generic tamed Euler-Maruyama (EM) scheme for the kinetic type stochastic differential equations (SDEs) (also known as second order SDEs) with singular coefficients in both weak and strong probabilistic senses. We show that when the drift exhibits a relatively low regularity compared to the state of the art, the singular system is well-defined both in the weak and strong probabilistic senses. Meanwhile, the corresponding tamed EM scheme is shown to converge at the rate of 1/2 in both the weak and the strong senses.
{"title":"Quantitative approximation of stochastic kinetic equations: from discrete to continuum","authors":"Zimo Hao, Khoa Lê, Chengcheng Ling","doi":"arxiv-2409.05706","DOIUrl":"https://doi.org/arxiv-2409.05706","url":null,"abstract":"We study the convergence of a generic tamed Euler-Maruyama (EM) scheme for\u0000the kinetic type stochastic differential equations (SDEs) (also known as second\u0000order SDEs) with singular coefficients in both weak and strong probabilistic\u0000senses. We show that when the drift exhibits a relatively low regularity\u0000compared to the state of the art, the singular system is well-defined both in\u0000the weak and strong probabilistic senses. Meanwhile, the corresponding tamed EM\u0000scheme is shown to converge at the rate of 1/2 in both the weak and the strong\u0000senses.","PeriodicalId":501245,"journal":{"name":"arXiv - MATH - Probability","volume":"11 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142212052","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}
For the iterations of $xmapsto |x-theta|$ random functions with Lipschitz number one, we represent the dynamics as a Markov chain and prove its convergence under mild conditions. We also demonstrate that the Wasserstein metric of any two measures will not increase after the corresponding induced iterations for measures and identify conditions under which a polynomial convergence rate can be achieved in this metric. We also consider an associated nonlinear operator on the space of probability measures and identify its fixed points through an detailed analysis of their characteristic functions.
{"title":"On the iterations of some random functions with Lipschitz number one","authors":"Yingdong Lu, Tomasz Nowicki","doi":"arxiv-2409.06003","DOIUrl":"https://doi.org/arxiv-2409.06003","url":null,"abstract":"For the iterations of $xmapsto |x-theta|$ random functions with Lipschitz\u0000number one, we represent the dynamics as a Markov chain and prove its\u0000convergence under mild conditions. We also demonstrate that the Wasserstein\u0000metric of any two measures will not increase after the corresponding induced\u0000iterations for measures and identify conditions under which a polynomial\u0000convergence rate can be achieved in this metric. We also consider an associated\u0000nonlinear operator on the space of probability measures and identify its fixed\u0000points through an detailed analysis of their characteristic functions.","PeriodicalId":501245,"journal":{"name":"arXiv - MATH - Probability","volume":"4 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142212045","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}
The asymmetric switch process is a binary stochastic process that alternates between the values one and minus one, where the distribution of the time in these states may differ. In this sense, the process is asymmetric, and this paper extends previous work on symmetric switch processes. Two versions of the process are considered: a non-stationary one that starts with either the one or minus one at time zero and a stationary version constructed from the non-stationary one. Characteristics of these two processes, such as the expected values and covariance, are investigated. The main results show an equivalence between the monotonicity of the expected value functions and the distribution of the intervals having a stochastic representation in the form of a sum of random variables, where the number of terms follows a geometric distribution. This representation has a natural interpretation as a model in which switching attempts may fail at random. From these results, conditions are derived when these characteristics lead to valid interval distributions, which is vital in applications.
{"title":"Characteristics of asymmetric switch processes with independent switching times","authors":"Henrik Bengtsson, Krzysztof Podgorski","doi":"arxiv-2409.05641","DOIUrl":"https://doi.org/arxiv-2409.05641","url":null,"abstract":"The asymmetric switch process is a binary stochastic process that alternates\u0000between the values one and minus one, where the distribution of the time in\u0000these states may differ. In this sense, the process is asymmetric, and this\u0000paper extends previous work on symmetric switch processes. Two versions of the\u0000process are considered: a non-stationary one that starts with either the one or\u0000minus one at time zero and a stationary version constructed from the\u0000non-stationary one. Characteristics of these two processes, such as the\u0000expected values and covariance, are investigated. The main results show an\u0000equivalence between the monotonicity of the expected value functions and the\u0000distribution of the intervals having a stochastic representation in the form of\u0000a sum of random variables, where the number of terms follows a geometric\u0000distribution. This representation has a natural interpretation as a model in\u0000which switching attempts may fail at random. From these results, conditions are\u0000derived when these characteristics lead to valid interval distributions, which\u0000is vital in applications.","PeriodicalId":501245,"journal":{"name":"arXiv - MATH - Probability","volume":"5 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142212056","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 prove existence of a value for two-player zero-sum stopper vs. singular-controller games on finite-time horizon, when the underlying dynamics is one-dimensional, diffusive and bound to evolve in $[0,infty)$. We show that the value is the maximal solution of a variational inequality with both obstacle and gradient constraint and satisfying a Dirichlet boundary condition at $[0,T)times{0}$. Moreover, we obtain an optimal strategy for the stopper. Compared to the existing literature on this topic, we introduce new probabilistic methods to obtain gradient bounds and equi-continuity for the solutions of penalised partial differential equations (PDE) that approximate the variational inequality.
{"title":"Finite-time horizon, stopper vs. singular-controller games on the half-line","authors":"Andrea Bovo, Tiziano De Angelis","doi":"arxiv-2409.06049","DOIUrl":"https://doi.org/arxiv-2409.06049","url":null,"abstract":"We prove existence of a value for two-player zero-sum stopper vs.\u0000singular-controller games on finite-time horizon, when the underlying dynamics\u0000is one-dimensional, diffusive and bound to evolve in $[0,infty)$. We show that\u0000the value is the maximal solution of a variational inequality with both\u0000obstacle and gradient constraint and satisfying a Dirichlet boundary condition\u0000at $[0,T)times{0}$. Moreover, we obtain an optimal strategy for the stopper.\u0000Compared to the existing literature on this topic, we introduce new\u0000probabilistic methods to obtain gradient bounds and equi-continuity for the\u0000solutions of penalised partial differential equations (PDE) that approximate\u0000the variational inequality.","PeriodicalId":501245,"journal":{"name":"arXiv - MATH - Probability","volume":"101 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142212047","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 consider a class of birth/death like process corresponding to coupled biochemical reactions and consider the problem of quantifying the variance of the molecular species in terms of the rates of the reactions. In particular, we address this problem in a configuration where a species is formed with a rate that depends nonlinearly on another spontaneously formed species. By making use of an appropriately formulated expansion based on the Newton series, in conjunction with spectral properties of the master equation, we derive an analytical expression that provides a hard bound for the variance. We show that this bound is exact when the propensities are linear, with numerical simulations demonstrating that this bound is also very close to the actual variance. An analytical expression for the covariance of the species is also derived.
{"title":"Variance bounds for a class of biochemical birth/death like processes via a discrete expansion and spectral properties of the Master equation","authors":"Giovanni Pugliese Carratelli, Ioannis Leastas","doi":"arxiv-2409.05667","DOIUrl":"https://doi.org/arxiv-2409.05667","url":null,"abstract":"We consider a class of birth/death like process corresponding to coupled\u0000biochemical reactions and consider the problem of quantifying the variance of\u0000the molecular species in terms of the rates of the reactions. In particular, we\u0000address this problem in a configuration where a species is formed with a rate\u0000that depends nonlinearly on another spontaneously formed species. By making use\u0000of an appropriately formulated expansion based on the Newton series, in\u0000conjunction with spectral properties of the master equation, we derive an\u0000analytical expression that provides a hard bound for the variance. We show that\u0000this bound is exact when the propensities are linear, with numerical\u0000simulations demonstrating that this bound is also very close to the actual\u0000variance. An analytical expression for the covariance of the species is also\u0000derived.","PeriodicalId":501245,"journal":{"name":"arXiv - MATH - Probability","volume":"46 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142212055","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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