We consider a nonlinear random walk which, in each time step, is free to choose its own transition probability within a neighborhood (w.r.t. Wasserstein distance) of the transition probability of a fixed L'evy process. In analogy to the classical framework we show that, when passing from discrete to continuous time via a scaling limit, this nonlinear random walk gives rise to a nonlinear semigroup. We explicitly compute the generator of this semigroup and corresponding PDE as a perturbation of the generator of the initial L'evy process.
{"title":"Limits of random walks with distributionally robust transition probabilities","authors":"Daniel Bartl, S. Eckstein, M. Kupper","doi":"10.1214/21-ECP393","DOIUrl":"https://doi.org/10.1214/21-ECP393","url":null,"abstract":"We consider a nonlinear random walk which, in each time step, is free to choose its own transition probability within a neighborhood (w.r.t. Wasserstein distance) of the transition probability of a fixed L'evy process. In analogy to the classical framework we show that, when passing from discrete to continuous time via a scaling limit, this nonlinear random walk gives rise to a nonlinear semigroup. We explicitly compute the generator of this semigroup and corresponding PDE as a perturbation of the generator of the initial L'evy process.","PeriodicalId":8470,"journal":{"name":"arXiv: Probability","volume":"2 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-07-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"89769204","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}
This note is motivated by connections between the online and offline problems of selecting a possibly long subsequence from a Poisson-paced sequence of uniform marks under either a monotonicity or a sum constraint. The offline problem with the sum constraint amounts to counting the Poisson arrivals before their total exceeds a certain level. A precise asymptotics for the mean count is obtained by coupling with a nonlinear pure birth process.
{"title":"On sequential selection and a first passage problem for the Poisson process","authors":"A. Gnedin","doi":"10.1214/21-ECP377","DOIUrl":"https://doi.org/10.1214/21-ECP377","url":null,"abstract":"This note is motivated by connections between the online and offline problems of selecting a possibly long subsequence from a Poisson-paced sequence of uniform marks under either a monotonicity or a sum constraint. The offline problem with the sum constraint amounts to counting the Poisson arrivals before their total exceeds a certain level. A precise asymptotics for the mean count is obtained by coupling with a nonlinear pure birth process.","PeriodicalId":8470,"journal":{"name":"arXiv: Probability","volume":"33 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-07-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"75003846","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}
Here, we consider an SIS epidemic model where the individuals are distributed on several distinct patches. We construct a stochastic model and then prove that it converges to a deterministic model as the total population size tends to infinity. Furthermore, we show the existence and the global stability of a unique endemic equilibrium provided that the migration rates of susceptible and infectious individuals are equal. Finally, we compare the equilibria with those of the homogeneous model, and with those of isolated patches.
{"title":"Stochastic and deterministic SIS patch model","authors":"Ténan Yeo","doi":"10.3934/dcdsb.2021012","DOIUrl":"https://doi.org/10.3934/dcdsb.2021012","url":null,"abstract":"Here, we consider an SIS epidemic model where the individuals are distributed on several distinct patches. We construct a stochastic model and then prove that it converges to a deterministic model as the total population size tends to infinity. Furthermore, we show the existence and the global stability of a unique endemic equilibrium provided that the migration rates of susceptible and infectious individuals are equal. Finally, we compare the equilibria with those of the homogeneous model, and with those of isolated patches.","PeriodicalId":8470,"journal":{"name":"arXiv: Probability","volume":"86 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-07-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"83762267","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}
Pub Date : 2020-07-11DOI: 10.33048/semi.2020.17.082
E. Bulinskaya
We study the distribution of the maximal displacement of particles positions for the whole time of the population existence in the model of critical and subcritical catalytic branching random walk on Z. In particular, we prove that in the case of simple symmetric random walk on Z, the distribution of the maximal displacement has "a heavy tail" decreasing as a function of the power 1/2 or 1, when the branching process is critical or subcritical, respectively. These statements describe new effects which do not arise in the corresponding investigations of the maximal displacement of critical and subcritical branching random walks on Z.
{"title":"On the maximal displacement of catalytic branching random walk","authors":"E. Bulinskaya","doi":"10.33048/semi.2020.17.082","DOIUrl":"https://doi.org/10.33048/semi.2020.17.082","url":null,"abstract":"We study the distribution of the maximal displacement of particles positions for the whole time of the population existence in the model of critical and subcritical catalytic branching random walk on Z. In particular, we prove that in the case of simple symmetric random walk on Z, the distribution of the maximal displacement has \"a heavy tail\" decreasing as a function of the power 1/2 or 1, when the branching process is critical or subcritical, respectively. These statements describe new effects which do not arise in the corresponding investigations of the maximal displacement of critical and subcritical branching random walks on Z.","PeriodicalId":8470,"journal":{"name":"arXiv: Probability","volume":"122 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-07-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86762939","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}
Pub Date : 2020-07-10DOI: 10.1007/978-3-030-60754-8_4
Marco Aymone, M. Hil'ario, B. D. Lima, V. Sidoravicius
{"title":"Bernoulli Hyperplane Percolation","authors":"Marco Aymone, M. Hil'ario, B. D. Lima, V. Sidoravicius","doi":"10.1007/978-3-030-60754-8_4","DOIUrl":"https://doi.org/10.1007/978-3-030-60754-8_4","url":null,"abstract":"","PeriodicalId":8470,"journal":{"name":"arXiv: Probability","volume":"16 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-07-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"87514642","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}
Pub Date : 2020-07-10DOI: 10.1007/978-3-030-60754-8_19
S. Foss, A. Sakhanenko
{"title":"Structural Properties of Conditioned Random Walks on Integer Lattices with Random Local Constraints","authors":"S. Foss, A. Sakhanenko","doi":"10.1007/978-3-030-60754-8_19","DOIUrl":"https://doi.org/10.1007/978-3-030-60754-8_19","url":null,"abstract":"","PeriodicalId":8470,"journal":{"name":"arXiv: Probability","volume":"11 1","pages":"407-438"},"PeriodicalIF":0.0,"publicationDate":"2020-07-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"87102965","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 note, we show how to provide sharp control on the least singular value of a certain translated linearization matrix arising in the study of the local universality of products of independent random matrices. This problem was first considered in a recent work of Koppel, O'Rourke, and Vu, and compared to their work, our proof is substantially simpler and established in much greater generality . In particular, we only assume that the entries of the ensemble are centered, and have second and fourth moments uniformly bounded away from $0$ and infinity, whereas previous work assumed a uniform subgaussian decay condition and that the entries within each factor of the product are identically distributed. A consequence of our least singular value bound is that the four moment matching universality results for the products of independent random matrices, recently obtained by Koppel, O'Rourke, and Vu, hold under much weaker hypotheses. Our proof technique is also of independent interest in the study of structured sparse matrices.
{"title":"Universality and least singular values of random matrix products: a simplified approach","authors":"Rohit Chaudhuri, Vishesh Jain, N. Pillai","doi":"10.3150/20-BEJ1320","DOIUrl":"https://doi.org/10.3150/20-BEJ1320","url":null,"abstract":"In this note, we show how to provide sharp control on the least singular value of a certain translated linearization matrix arising in the study of the local universality of products of independent random matrices. This problem was first considered in a recent work of Koppel, O'Rourke, and Vu, and compared to their work, our proof is substantially simpler and established in much greater generality . In particular, we only assume that the entries of the ensemble are centered, and have second and fourth moments uniformly bounded away from $0$ and infinity, whereas previous work assumed a uniform subgaussian decay condition and that the entries within each factor of the product are identically distributed. \u0000A consequence of our least singular value bound is that the four moment matching universality results for the products of independent random matrices, recently obtained by Koppel, O'Rourke, and Vu, hold under much weaker hypotheses. Our proof technique is also of independent interest in the study of structured sparse matrices.","PeriodicalId":8470,"journal":{"name":"arXiv: Probability","volume":"38 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-07-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"83527389","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}
Expected ballisticity of a continuous self avoiding walk on hyperbolic spaces $mathbb{H}^d$ is established.
建立了双曲空间$mathbb{H}^d$上连续自避行走的期望弹道性。
{"title":"Hyperbolic self avoiding walk","authors":"I. Benjamini, C. Panagiotis","doi":"10.1214/21-ECP388","DOIUrl":"https://doi.org/10.1214/21-ECP388","url":null,"abstract":"Expected ballisticity of a continuous self avoiding walk on hyperbolic spaces $mathbb{H}^d$ is established.","PeriodicalId":8470,"journal":{"name":"arXiv: Probability","volume":"505 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-07-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"80469752","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}
Pub Date : 2020-07-02DOI: 10.1016/J.JFA.2021.109061
Frederic Weber, Rico Zacher
{"title":"The entropy method under curvature-dimension conditions in the spirit of Bakry-'Emery in the discrete setting of Markov chains","authors":"Frederic Weber, Rico Zacher","doi":"10.1016/J.JFA.2021.109061","DOIUrl":"https://doi.org/10.1016/J.JFA.2021.109061","url":null,"abstract":"","PeriodicalId":8470,"journal":{"name":"arXiv: Probability","volume":"134 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-07-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"77858385","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}
This paper develops a theory for completely random measures in the framework of free probability. A general existence result for free completely random measures is established, and in analogy to the classical work of Kingman it is proved that such random measures can be decomposed into the sum of a purely atomic part and a (freely) infinitely divisible part. The latter part (termed a free Levy basis) is studied in detail in terms of the free Levy-Khintchine representation and a theory parallel to the classical work of Rajput and Rosinski is developed. Finally a Levy-Ito type decomposition for general free Levy bases is established.
{"title":"Completely random measures and Lévy bases in free probability","authors":"Francesca Collet, F. Leisen, S. Thorbjørnsen","doi":"10.1214/21-EJP620","DOIUrl":"https://doi.org/10.1214/21-EJP620","url":null,"abstract":"This paper develops a theory for completely random measures in the framework of free probability. A general existence result for free completely random measures is established, and in analogy to the classical work of Kingman it is proved that such random measures can be decomposed into the sum of a purely atomic part and a (freely) infinitely divisible part. The latter part (termed a free Levy basis) is studied in detail in terms of the free Levy-Khintchine representation and a theory parallel to the classical work of Rajput and Rosinski is developed. Finally a Levy-Ito type decomposition for general free Levy bases is established.","PeriodicalId":8470,"journal":{"name":"arXiv: Probability","volume":"50 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"90000465","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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