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Quantitative propagation of chaos for mean field Markov decision process with common noise 具有公共噪声的平均场Markov决策过程混沌的定量传播
IF 1.4 3区 数学
Electronic Journal of Probability Pub Date : 2022-07-26 DOI: 10.1214/23-ejp978
M'ed'eric Motte, H. Pham
{"title":"Quantitative propagation of chaos for mean field Markov decision process with common noise","authors":"M'ed'eric Motte, H. Pham","doi":"10.1214/23-ejp978","DOIUrl":"https://doi.org/10.1214/23-ejp978","url":null,"abstract":"We investigate propagation of chaos for mean field Markov Decision Process with common noise (CMKV-MDP), and when the optimization is performed over randomized open-loop controls on infinite horizon. We first state a rate of convergence of order $M_N^gamma$, where $M_N$ is the mean rate of convergence in Wasserstein distance of the empirical measure, and $gamma in (0,1]$ is an explicit constant, in the limit of the value functions of $N$-agent control problem with asymmetric open-loop controls, towards the value function of CMKV-MDP. Furthermore, we show how to explicitly construct $(epsilon+mathcal{O}(M_N^gamma))$-optimal policies for the $N$-agent model from $epsilon$-optimal policies for the CMKV-MDP. Our approach relies on sharp comparison between the Bellman operators in the $N$-agent problem and the CMKV-MDP, and fine coupling of empirical measures.","PeriodicalId":50538,"journal":{"name":"Electronic Journal of Probability","volume":" ","pages":""},"PeriodicalIF":1.4,"publicationDate":"2022-07-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46459344","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
A Markov process for a continuum infinite particle system with attraction 具有吸引力的连续无限粒子系统的Markov过程
IF 1.4 3区 数学
Electronic Journal of Probability Pub Date : 2022-07-15 DOI: 10.1214/23-ejp952
Y. Kozitsky, M. Rockner
{"title":"A Markov process for a continuum infinite particle system with attraction","authors":"Y. Kozitsky, M. Rockner","doi":"10.1214/23-ejp952","DOIUrl":"https://doi.org/10.1214/23-ejp952","url":null,"abstract":"An infinite system of point particles placed in $mathds{R}^d$ is studied. The particles are of two types; they perform random walks in the course of which those of distinct types repel each other. The interaction of this kind induces an effective multi-body attraction of the same type particles, which leads to the multiplicity of states of thermal equilibrium in such systems. The pure states of the system are locally finite counting measures on $mathds{R}^d$. The set of such states $Gamma^2$ is equipped with the vague topology and the corresponding Borel $sigma$-field. For a special class $mathcal{P}_{rm exp}$ of probability measures defined on $Gamma^2$, we prove the existence of a family ${P_{t,mu}: tgeq 0, mu in mathcal{P}_{rm exp}}$ of probability measures defined on the space of c{`a}dl{`a}g paths with values in $Gamma^2$, which is a unique solution of the restricted martingale problem for the mentioned stochastic dynamics. Thereby, the corresponding Markov process is specified.","PeriodicalId":50538,"journal":{"name":"Electronic Journal of Probability","volume":" ","pages":""},"PeriodicalIF":1.4,"publicationDate":"2022-07-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43286029","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
Transience and anchored isoperimetric dimension of supercritical percolation clusters 超临界渗流团簇的瞬态和锚定等周维度
IF 1.4 3区 数学
Electronic Journal of Probability Pub Date : 2022-07-11 DOI: 10.1214/23-ejp905
Tom Hutchcroft
{"title":"Transience and anchored isoperimetric dimension of supercritical percolation clusters","authors":"Tom Hutchcroft","doi":"10.1214/23-ejp905","DOIUrl":"https://doi.org/10.1214/23-ejp905","url":null,"abstract":"We establish several equivalent characterisations of the anchored isoperimetric dimension of supercritical clusters in Bernoulli bond percolation on transitive graphs. We deduce from these characterisations together with a theorem of Duminil-Copin, Goswami, Raoufi, Severo, and Yadin that if $G$ is a transient transitive graph then the infinite clusters of Bernoulli percolation on $G$ are transient for $p$ sufficiently close to $1$. It remains open to extend this result down to the critical probability. Along the way we establish two new cluster repulsion inequalities that are of independent interest.","PeriodicalId":50538,"journal":{"name":"Electronic Journal of Probability","volume":" ","pages":""},"PeriodicalIF":1.4,"publicationDate":"2022-07-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48163278","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
On the largest and the smallest singular value of sparse rectangular random matrices 关于稀疏矩形随机矩阵的最大和最小奇异值
IF 1.4 3区 数学
Electronic Journal of Probability Pub Date : 2022-07-07 DOI: 10.1214/23-ejp919
F. Gotze, A. Tikhomirov
{"title":"On the largest and the smallest singular value of sparse rectangular random matrices","authors":"F. Gotze, A. Tikhomirov","doi":"10.1214/23-ejp919","DOIUrl":"https://doi.org/10.1214/23-ejp919","url":null,"abstract":"We derive estimates for the largest and smallest singular values of sparse rectangular $Ntimes n$ random matrices, assuming $lim_{N,ntoinfty}frac nN=yin(0,1)$. We consider a model with sparsity parameter $p_N$ such that $Np_Nsim log^{alpha }N$ for some $alpha>1$, and assume that the moments of the matrix elements satisfy the condition $mathbf E|X_{jk}|^{4+delta}le C<infty$. We assume also that the entries of matrices we consider are truncated at the level $(Np_N)^{frac12-varkappa}$ with $varkappa:=frac{delta}{2(4+delta)}$.","PeriodicalId":50538,"journal":{"name":"Electronic Journal of Probability","volume":" ","pages":""},"PeriodicalIF":1.4,"publicationDate":"2022-07-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42292830","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
On the parabolic Cauchy problem for quantum graphs with vertex noise 关于具有顶点噪声的量子图的抛物型Cauchy问题
IF 1.4 3区 数学
Electronic Journal of Probability Pub Date : 2022-07-05 DOI: 10.1214/23-EJP962
Mih'aly Kov'acs, E. Sikolya
{"title":"On the parabolic Cauchy problem for quantum graphs with vertex noise","authors":"Mih'aly Kov'acs, E. Sikolya","doi":"10.1214/23-EJP962","DOIUrl":"https://doi.org/10.1214/23-EJP962","url":null,"abstract":"We investigate the parabolic Cauchy problem associated with quantum graphs including Lipschitz or polynomial type nonlinearities and additive Gaussian noise perturbed vertex conditions. The vertex conditions are the standard continuity and Kirchhoff assumptions in each vertex. In the case when only Kirchhoff conditions are perturbed, we can prove existence and uniqueness of a mild solution with continuous paths in the standard state space $mathcal{H}$ of square integrable functions on the edges. We also show that the solution is Markov and Feller. Furthermore, assuming that the vertex values of the normalized eigenfunctions of the self-adjoint operator governing the problem are uniformly bounded, we show that the mild solution has continuous paths in the fractional domain space associated with the Hamiltonian operator, $mathcal{H}_{alpha}$ for $alpha<frac{1}{4}$. This is the case when the Hamiltonian operator is the standard Laplacian perturbed by a potential. We also show that if noise is present in both type of vertex conditions, then the problem admits a mild solution with continuous paths in the fractional domain space $mathcal{H}_{alpha}$ with $alpha<-frac{1}{4}$ only. These regularity results are the quantum graph analogues obtained by da Prato and Zabczyk [9] in case of a single interval and classical boundary Dirichlet or Neumann noise.","PeriodicalId":50538,"journal":{"name":"Electronic Journal of Probability","volume":" ","pages":""},"PeriodicalIF":1.4,"publicationDate":"2022-07-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43756337","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
Invariant measures of critical branching random walks in high dimension 高维临界分支随机游动的不变测度
IF 1.4 3区 数学
Electronic Journal of Probability Pub Date : 2022-06-16 DOI: 10.1214/23-ejp906
V. Rapenne
{"title":"Invariant measures of critical branching random walks in high dimension","authors":"V. Rapenne","doi":"10.1214/23-ejp906","DOIUrl":"https://doi.org/10.1214/23-ejp906","url":null,"abstract":"In this work, we characterize cluster-invariant point processes for critical branching spatial processes on R d for all large enough d when the motion law is α -stable or has a finite discrete range. More precisely, when the motion is α -stable with α ≤ 2 and the offspring law µ of the branching process has an heavy tail such that µ ( k ) ∼ k − 2 − β , then we need the dimension d to be strictly larger than the critical dimension α/β . In particular, when the motion is Brownian and the offspring law µ has a second moment, this critical dimension is 2. Contrary to the previous work of Bramson, Cox and Greven in [BCG97] whose proof used PDE techniques, our proof uses probabilistic tools only.","PeriodicalId":50538,"journal":{"name":"Electronic Journal of Probability","volume":" ","pages":""},"PeriodicalIF":1.4,"publicationDate":"2022-06-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49065557","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
Invariance principle for the maximal position process of branching Brownian motion in random environment 随机环境下分支布朗运动最大位置过程的不变性原理
IF 1.4 3区 数学
Electronic Journal of Probability Pub Date : 2022-06-16 DOI: 10.1214/23-ejp956
Haojie Hou, Y-X. Ren, R. Song
{"title":"Invariance principle for the maximal position process of branching Brownian motion in random environment","authors":"Haojie Hou, Y-X. Ren, R. Song","doi":"10.1214/23-ejp956","DOIUrl":"https://doi.org/10.1214/23-ejp956","url":null,"abstract":"In this paper we study the maximal position process of branching Brownian motion in random spatial environment. The random environment is given by a process $xi = left(xi(x)right)_{xinmathbb{R}}$ satisfying certain conditions. We show that the maximum position $M_t$ of particles alive at time $t$ satisfies a quenched strong law of large numbers and an annealed invariance principle.","PeriodicalId":50538,"journal":{"name":"Electronic Journal of Probability","volume":" ","pages":""},"PeriodicalIF":1.4,"publicationDate":"2022-06-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47168518","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
Multifractal analysis of Gaussian multiplicative chaos and applications 高斯乘性混沌的多重分形分析及其应用
IF 1.4 3区 数学
Electronic Journal of Probability Pub Date : 2022-06-14 DOI: 10.1214/22-EJP893
Federico Bertacco
{"title":"Multifractal analysis of Gaussian multiplicative chaos and applications","authors":"Federico Bertacco","doi":"10.1214/22-EJP893","DOIUrl":"https://doi.org/10.1214/22-EJP893","url":null,"abstract":"Let M γ be a subcritical Gaussian multiplicative chaos measure associated with a general log-correlated Gaussian field defined on a bounded domain D ⊂ R d , d ≥ 1. We find an explicit formula for its singularity spectrum by showing that M γ satisfies almost surely the multifractal formalism, i.e","PeriodicalId":50538,"journal":{"name":"Electronic Journal of Probability","volume":" ","pages":""},"PeriodicalIF":1.4,"publicationDate":"2022-06-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42207688","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
Equilibrium perturbations for stochastic interacting systems 随机相互作用系统的平衡摄动
IF 1.4 3区 数学
Electronic Journal of Probability Pub Date : 2022-06-04 DOI: 10.1214/22-ejp900
Lu Xu, Linjie Zhao
{"title":"Equilibrium perturbations for stochastic interacting systems","authors":"Lu Xu, Linjie Zhao","doi":"10.1214/22-ejp900","DOIUrl":"https://doi.org/10.1214/22-ejp900","url":null,"abstract":"We consider the equilibrium perturbations for two stochastic systems: the $d$-dimensional generalized exclusion process and the one-dimensional chain of anharmonic oscillators. We add a perturbation of order $N^{-alpha}$ to the equilibrium profile and speed up the process by $N^{1+kappa}$ for parameters $0<kappalealpha$. Under some additional constraints on $kappa$ and $alpha$, we show the perturbed quantities evolve according to the Burgers equation in the exclusion process, and to two decoupled Burgers equations in the anharmonic chain, both in the smooth regime.","PeriodicalId":50538,"journal":{"name":"Electronic Journal of Probability","volume":" ","pages":""},"PeriodicalIF":1.4,"publicationDate":"2022-06-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45627317","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
Prediagnostic Levels of Copper and Zinc and Breast Cancer Risk in the ORDET Cohort. ORDET 队列中诊断前的铜和锌水平与乳腺癌风险。
IF 3.8 3区 数学
Electronic Journal of Probability Pub Date : 2022-06-01 DOI: 10.1158/1055-9965.EPI-21-1252
Valeria Pala, Claudia Agnoli, Adalberto Cavalleri, Sabina Rinaldi, Rosaria Orlandi, Francesco Segrado, Elisabetta Venturelli, Marco Vinceti, Vittorio Krogh, Sabina Sieri
{"title":"Prediagnostic Levels of Copper and Zinc and Breast Cancer Risk in the ORDET Cohort.","authors":"Valeria Pala, Claudia Agnoli, Adalberto Cavalleri, Sabina Rinaldi, Rosaria Orlandi, Francesco Segrado, Elisabetta Venturelli, Marco Vinceti, Vittorio Krogh, Sabina Sieri","doi":"10.1158/1055-9965.EPI-21-1252","DOIUrl":"10.1158/1055-9965.EPI-21-1252","url":null,"abstract":"<p><strong>Background: </strong>Case-control studies show that copper (Cu) is high and zinc (Zn) low in blood and urine of women with breast cancer compared with controls.</p><p><strong>Methods: </strong>To assess whether prediagnostic Cu and Zn are associated with breast cancer risk, OR of breast cancer according to Cu, Zn, and Cu/Zn ratio in plasma and urine was estimated in a nested case-control study within the ORDET cohort, using conditional logistic regression adjusted for multiple variables: First 496 breast cancer cases and matched controls, diagnosed ≥2 years after recruitment (to eliminate reverse causation) were analyzed. Then all eligible cases/controls were analyzed with stratification into years from recruitment to diagnosis.</p><p><strong>Results: </strong>For women diagnosed ≥2 years, compared with lowest tertiles, breast cancer risk was higher in the highest tertile of plasma Cu/Zn ratio (OR, 1.75; 95% CI, 1.21-2.54) and the highest tertile of both plasma and urine Cu/Zn ratio (OR, 2.37; 95% CI, 1.32-4.25). Risk did not vary with ER/PR/HER2 status. For women diagnosed <2 years, high Cu/Zn ratio was strongly associated with breast cancer risk.</p><p><strong>Conclusions: </strong>Our prospective findings suggest that increased Cu/Zn ratio in plasma and urine may be both an early marker of, and a risk factor for, breast cancer development. Further studies are justified to confirm or otherwise our results and to investigate mechanisms.</p><p><strong>Impact: </strong>Our finding that prediagnostic Cu/Zn ratio is a strong risk factor for breast cancer development deserves further investigation and, if confirmed, might open the way to interventions to reduce breast cancer risk in women with disrupted Cu/Zn homeostasis.</p>","PeriodicalId":50538,"journal":{"name":"Electronic Journal of Probability","volume":"6 1","pages":"1209-1215"},"PeriodicalIF":3.8,"publicationDate":"2022-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"88658060","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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