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Using moment approximations to study the density of jump driven SDEs 用矩近似法研究跳跃驱动SDE的密度
IF 1.4 3区 数学
Electronic Journal of Probability Pub Date : 2022-01-01 DOI: 10.1214/22-ejp785
V. Bally, L. Caramellino, A. Kohatsu-Higa
{"title":"Using moment approximations to study the density of jump driven SDEs","authors":"V. Bally, L. Caramellino, A. Kohatsu-Higa","doi":"10.1214/22-ejp785","DOIUrl":"https://doi.org/10.1214/22-ejp785","url":null,"abstract":"In order to study the regularity of the density of a solution of a infinite activity jump driven stochastic differential equation we consider the following two-step approximation method. First, we use the solution of the moment problem in order to approximate the small jumps by another whose Lévy measure has finite support. In a second step we replace the approximation of the first two moments by a small noise Brownian motion based on the Assmussen-Rosiński approach. This approximation needs to satisfy certain properties in order to apply the “balance” method which allows the study of densities for the solution process based on Malliavin Calculus for the Brownian motion. Our results apply to situations where the Lévy measure is absolutely continuous with respect to the Lebesgue measure or purely atomic measures or combinations of them.","PeriodicalId":50538,"journal":{"name":"Electronic Journal of Probability","volume":" ","pages":""},"PeriodicalIF":1.4,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44858931","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Convex hulls of stable random walks 稳定随机游走的凸包
IF 1.4 3区 数学
Electronic Journal of Probability Pub Date : 2022-01-01 DOI: 10.1214/22-ejp826
W. Cygan, Nikola Sandri'c, Stjepan vSebek
{"title":"Convex hulls of stable random walks","authors":"W. Cygan, Nikola Sandri'c, Stjepan vSebek","doi":"10.1214/22-ejp826","DOIUrl":"https://doi.org/10.1214/22-ejp826","url":null,"abstract":"We consider convex hulls of random walks whose steps belong to the domain of attraction of a stable law in Rd . We prove convergence of the convex hull in the space of all convex and compact subsets ofRd , equipped with the Hausdorff distance, towards the convex hull spanned by a path of the limit stable Lévy process. As an application, we establish convergence of (expected) intrinsic volumes under some mild moment/structure assumptions posed on the random walk.","PeriodicalId":50538,"journal":{"name":"Electronic Journal of Probability","volume":" ","pages":""},"PeriodicalIF":1.4,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43232019","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 1
Conservative random walk 保守随机漫步
IF 1.4 3区 数学
Electronic Journal of Probability Pub Date : 2022-01-01 DOI: 10.1214/22-ejp863
J. Englander, S. Volkov
{"title":"Conservative random walk","authors":"J. Englander, S. Volkov","doi":"10.1214/22-ejp863","DOIUrl":"https://doi.org/10.1214/22-ejp863","url":null,"abstract":"Recently, in [\"The coin-turning walk and its scaling limit\", Electronic Journal of Probability, 25 (2020)], the ``coin-turning walk'' was introduced on ${mathbb Z}$. It is a non-Markovian process where the steps form a (possibly) time-inhomogeneous Markov chain. In this article, we follow up the investigation by introducing analogous processes in ${mathbb Z}^d$, $dge 2$: at time $n$ the direction of the process is ``updated'' with probability $p_n$; otherwise the next step repeats the previous one. We study some of the fundamental properties of these walks, such as transience/recurrence and scaling limits. Our results complement previous ones in the literature about ``correlated'' (or ``Newtonian'') and ``persistent'' random walks.","PeriodicalId":50538,"journal":{"name":"Electronic Journal of Probability","volume":" ","pages":""},"PeriodicalIF":1.4,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46422214","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 2
Large deviations for Gibbs ensembles of the classical Toda chain 经典户田链的吉布斯系综的大偏差
IF 1.4 3区 数学
Electronic Journal of Probability Pub Date : 2022-01-01 DOI: 10.1214/22-ejp771
A. Guionnet, Ronan Memin
{"title":"Large deviations for Gibbs ensembles of the classical Toda chain","authors":"A. Guionnet, Ronan Memin","doi":"10.1214/22-ejp771","DOIUrl":"https://doi.org/10.1214/22-ejp771","url":null,"abstract":"We prove large deviation principles for the distribution of the empirical measure of the eigenvalues of Lax matrices following the Generalized Gibbs ensembles of the classical Toda chain introduced in [9]. We deduce the almost sure convergence of this empirical measure towards a limit which we describe in terms of the limiting empirical measure of Beta-ensembles. Our results apply to general smooth potentials.","PeriodicalId":50538,"journal":{"name":"Electronic Journal of Probability","volume":" ","pages":""},"PeriodicalIF":1.4,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47419818","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 5
The sharp K4-percolation threshold on the Erdős–Rényi random graph 在Erdős-Rényi随机图上有明显的k4渗透阈值
IF 1.4 3区 数学
Electronic Journal of Probability Pub Date : 2022-01-01 DOI: 10.1214/21-ejp710
Brett Kolesnik
{"title":"The sharp K4-percolation threshold on the Erdős–Rényi random graph","authors":"Brett Kolesnik","doi":"10.1214/21-ejp710","DOIUrl":"https://doi.org/10.1214/21-ejp710","url":null,"abstract":"We locate the critical threshold pc ∼ 1/ √ 3n logn at which it becomes likely that the complete graph Kn can be obtained from the Erdős–Rényi graph Gn,p by iteratively completing copies of K4 minus an edge. This refines work of Balogh, Bollobás and Morris that bounds the threshold up to multiplicative constants.","PeriodicalId":50538,"journal":{"name":"Electronic Journal of Probability","volume":" ","pages":""},"PeriodicalIF":1.4,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48342916","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 4
Zooming in at the root of the stable tree 放大稳定树的根部
IF 1.4 3区 数学
Electronic Journal of Probability Pub Date : 2022-01-01 DOI: 10.1214/22-ejp764
Michel Nassif
{"title":"Zooming in at the root of the stable tree","authors":"Michel Nassif","doi":"10.1214/22-ejp764","DOIUrl":"https://doi.org/10.1214/22-ejp764","url":null,"abstract":"We study the shape of the normalized stable Lévy tree T near its root. We show that, when zooming in at the root at the proper speed with a scaling depending on the index of stability, we get the unnormalized Kesten tree. In particular the limit is described by a tree-valued Poisson point process which does not depend on the initial normalization. We apply this to study the asymptotic behavior of additive functionals of the form as max( α, β ) → ∞ , where µ is the mass measure on T , H ( x ) is the height of x and σ r,x (resp. h r,x ) is the mass (resp. height) of the subtree of T above level r containing x . Such functionals arise as scaling limits of additive functionals of the size and height on conditioned Bienaymé-Galton-Watson trees.","PeriodicalId":50538,"journal":{"name":"Electronic Journal of Probability","volume":" ","pages":""},"PeriodicalIF":1.4,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42049743","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
On a front evolution problem for the multidimensional East model 多维East模型的前沿演化问题
IF 1.4 3区 数学
Electronic Journal of Probability Pub Date : 2021-12-29 DOI: 10.1214/22-EJP870
Yannick Couzini'e, F. Martinelli
{"title":"On a front evolution problem for the multidimensional East model","authors":"Yannick Couzini'e, F. Martinelli","doi":"10.1214/22-EJP870","DOIUrl":"https://doi.org/10.1214/22-EJP870","url":null,"abstract":"We consider a natural front evolution problem the East process on $mathbb{Z}^d, dge 2,$ a well studied kinetically constrained model for which the facilitation mechanism is oriented along the coordinate directions, as the equilibrium density $q$ of the facilitating vertices vanishes. Starting with a unique unconstrained vertex at the origin, let $S(t)$ consist of those vertices which became unconstrained within time $t$ and, for an arbitrary positive direction $mathbf x,$ let $v_{max}(mathbf x),v_{min}(mathbf x )$ be the maximal/minimal velocities at which $S(t)$ grows in that direction. If $mathbf x$ is independent of $q$, we prove that $v_{max}(mathbf x)= v_{min}(mathbf x)^{(1+o(1))}=gamma(d) ^{(1+o(1))}$ as $qto 0$, where $gamma(d)$ is the spectral gap of the process on $mathbb{Z}^d$. We also analyse the case in which some of the coordinates of $mathbf x$ vanish as $qto 0$. In particular, for $d=2$ we prove that if $mathbf x$ approaches one of the two coordinate directions fast enough, then $v_{max}(mathbf x)= v_{min}(mathbf x)^{(1+o(1))}=gamma(1) ^{(1+o(1))}=gamma(d)^{d(1+o(1))},$ i.e. the growth of $S(t)$ close to the coordinate directions is dictated by the one dimensional process. As a result the region $S(t)$ becomes extremely elongated inside $mathbb{Z}^d_+$. We also establish mixing time cutoff for the chain in finite boxes with minimal boundary conditions. A key ingredient of our analysis is the renormalisation technique of arXiv:1404.7257 to estimate the spectral gap of the East process. Here we extend this technique to get the main asymptotics of a suitable principal Dirichlet eigenvalue of the process.","PeriodicalId":50538,"journal":{"name":"Electronic Journal of Probability","volume":" ","pages":""},"PeriodicalIF":1.4,"publicationDate":"2021-12-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45708576","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 2
Optimal multi-resolvent local laws for Wigner matrices Wigner矩阵的最优多解局部律
IF 1.4 3区 数学
Electronic Journal of Probability Pub Date : 2021-12-27 DOI: 10.1214/22-ejp838
Giorgio Cipolloni, L'aszl'o ErdHos, Dominik Schroder
{"title":"Optimal multi-resolvent local laws for Wigner matrices","authors":"Giorgio Cipolloni, L'aszl'o ErdHos, Dominik Schroder","doi":"10.1214/22-ejp838","DOIUrl":"https://doi.org/10.1214/22-ejp838","url":null,"abstract":"We prove local laws, i.e. optimal concentration estimates for arbitrary products of resolvents of a Wigner random matrix with deterministic matrices in between. We find that the size of such products heavily depends on whether some of the deterministic matrices are traceless. Our estimates correctly account for this dependence and they hold optimally down to the smallest possible spectral scale.","PeriodicalId":50538,"journal":{"name":"Electronic Journal of Probability","volume":" ","pages":""},"PeriodicalIF":1.4,"publicationDate":"2021-12-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41639057","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 13
Scaling limit of stationary coupled Sasamoto-Spohn models 静止耦合Sasamoto-Spohn模型的尺度极限
IF 1.4 3区 数学
Electronic Journal of Probability Pub Date : 2021-12-27 DOI: 10.1214/22-ejp819
Ian Butelmann, Gregorio R. Moreno Flores
{"title":"Scaling limit of stationary coupled Sasamoto-Spohn models","authors":"Ian Butelmann, Gregorio R. Moreno Flores","doi":"10.1214/22-ejp819","DOIUrl":"https://doi.org/10.1214/22-ejp819","url":null,"abstract":"We introduce a family of stationary coupled Sasamoto-Spohn models and show that, in the weakly asymmetric regime, they converge to the energy solution of coupled Burgers equations. Moreover, we show that any system of coupled Burgers equations satisfying the so-called trilinear condition ensuring stationarity can be obtained as the scaling limit of a suitable system of coupled Sasamoto-Spohn models. The core of our proof, which avoids the use of spectral gap estimates, consists in a second order Boltzmann-Gibbs principle for the discrete model.","PeriodicalId":50538,"journal":{"name":"Electronic Journal of Probability","volume":" ","pages":""},"PeriodicalIF":1.4,"publicationDate":"2021-12-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41378418","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 3
Scaling limit of linearly edge-reinforced random walks on critical Galton-Watson trees 临界高尔顿-沃森树上线性边增强随机漫步的缩放极限
IF 1.4 3区 数学
Electronic Journal of Probability Pub Date : 2021-12-22 DOI: 10.1214/23-ejp901
G. Andriopoulos, Eleanor Archer
{"title":"Scaling limit of linearly edge-reinforced random walks on critical Galton-Watson trees","authors":"G. Andriopoulos, Eleanor Archer","doi":"10.1214/23-ejp901","DOIUrl":"https://doi.org/10.1214/23-ejp901","url":null,"abstract":"We prove an invariance principle for linearly edge reinforced random walks on $gamma$-stable critical Galton-Watson trees, where $gamma in (1,2]$ and where the edge joining $x$ to its parent has rescaled initial weight $d(rho, x)^{alpha}$ for some $alpha leq 1$. This corresponds to the recurrent regime of initial weights. We then establish fine asymptotics for the limit process. In the transient regime, we also give an upper bound on the random walk displacement in the discrete setting, showing that the edge reinforced random walk never has positive speed, even when the initial edge weights are strongly biased away from the root.","PeriodicalId":50538,"journal":{"name":"Electronic Journal of Probability","volume":" ","pages":""},"PeriodicalIF":1.4,"publicationDate":"2021-12-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44388687","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
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