This paper concerns the parabolic Anderson equation ∂u ∂t = 1 2 ∆u+ u ∂d+1WH ∂t∂x1 · · · ∂xd generated by a (d + 1)-dimensional fractional noise with the Hurst parameter H = (H0, H1, · · · , Hd) with special interest in the setting that some of H0, · · · , Hd are less than half. In the recent work [9], the case of the spatial roughness has been investigated. To put the last piece of the puzzle in place, this work investigates the case when H0 < 1/2 with the concern on solvability, Feynman-Kac’s moment formula and intermittency of the system. Key-words: parabolic Anderson equation, Dalang’s condition, fractional, rough and critical Gaussian noises, Feynman-Kac’s representation, Brownian motion, moment asymptotics AMS subject classification (2010): 60F10, 60H15, 60H40, 60J65, 81U10. ∗Research partially supported by the Simons Foundation #585506. 1
{"title":"Parabolic Anderson model with a fractional Gaussian noise that is rough in time","authors":"Xia Chen","doi":"10.1214/19-aihp983","DOIUrl":"https://doi.org/10.1214/19-aihp983","url":null,"abstract":"This paper concerns the parabolic Anderson equation ∂u ∂t = 1 2 ∆u+ u ∂d+1WH ∂t∂x1 · · · ∂xd generated by a (d + 1)-dimensional fractional noise with the Hurst parameter H = (H0, H1, · · · , Hd) with special interest in the setting that some of H0, · · · , Hd are less than half. In the recent work [9], the case of the spatial roughness has been investigated. To put the last piece of the puzzle in place, this work investigates the case when H0 < 1/2 with the concern on solvability, Feynman-Kac’s moment formula and intermittency of the system. Key-words: parabolic Anderson equation, Dalang’s condition, fractional, rough and critical Gaussian noises, Feynman-Kac’s representation, Brownian motion, moment asymptotics AMS subject classification (2010): 60F10, 60H15, 60H40, 60J65, 81U10. ∗Research partially supported by the Simons Foundation #585506. 1","PeriodicalId":7902,"journal":{"name":"Annales De L Institut Henri Poincare-probabilites Et Statistiques","volume":"36 1","pages":"792-825"},"PeriodicalIF":1.5,"publicationDate":"2020-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"82202479","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"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 natural nonparametric estimator of a Wasserstein type cost between two distinct continuous distributions Fand G on R. The estimator is based on the order statistics of a sample having marginals F, G and any joint distribution. We prove a central limit theorem under general conditions relating the tails and the cost function. In particular, these conditions are satisfied by Wasserstein distances of order p>1and compatible classical probability distributions.
{"title":"A Central Limit Theorem for Wasserstein type distances between two distinct univariate distributions","authors":"Philippe Berthet, J. Fort, T. Klein","doi":"10.1214/19-aihp990","DOIUrl":"https://doi.org/10.1214/19-aihp990","url":null,"abstract":"In this article we study the natural nonparametric estimator of a Wasserstein type cost between two distinct continuous distributions Fand G on R. The estimator is based on the order statistics of a sample having marginals F, G and any joint distribution. We prove a central limit theorem under general conditions relating the tails and the cost function. In particular, these conditions are satisfied by Wasserstein distances of order p>1and compatible classical probability distributions.","PeriodicalId":7902,"journal":{"name":"Annales De L Institut Henri Poincare-probabilites Et Statistiques","volume":"34 1","pages":"954-982"},"PeriodicalIF":1.5,"publicationDate":"2020-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86479892","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Errata for Perturbation by non-local operators","authors":"Zhen-Qing Chen, Jie-Ming Wang","doi":"10.1214/20-aihp1045","DOIUrl":"https://doi.org/10.1214/20-aihp1045","url":null,"abstract":"","PeriodicalId":7902,"journal":{"name":"Annales De L Institut Henri Poincare-probabilites Et Statistiques","volume":"81 1","pages":"760-763"},"PeriodicalIF":1.5,"publicationDate":"2020-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74880770","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We improve the lower bound on worst case trace reconstruction from Ω(n5/4logn) to Ω(n3/2log7n). As a consequence, we improve the lower bound on average case trace reconstruction from Ω(log9/4n loglogn) to Ω(log5/2n(loglogn)7).
{"title":"New lower bounds for trace reconstruction","authors":"Zachary Chase","doi":"10.1214/20-AIHP1089","DOIUrl":"https://doi.org/10.1214/20-AIHP1089","url":null,"abstract":"We improve the lower bound on worst case trace reconstruction from Ω(n5/4logn) to Ω(n3/2log7n). As a consequence, we improve the lower bound on average case trace reconstruction from Ω(log9/4n loglogn) to Ω(log5/2n(loglogn)7).","PeriodicalId":7902,"journal":{"name":"Annales De L Institut Henri Poincare-probabilites Et Statistiques","volume":"20 1","pages":"627-643"},"PeriodicalIF":1.5,"publicationDate":"2019-05-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"82008074","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Support theorem for a singular SPDE: The case of gPAM","authors":"K. Chouk, P. Friz","doi":"10.1214/16-AIHP800","DOIUrl":"https://doi.org/10.1214/16-AIHP800","url":null,"abstract":"","PeriodicalId":7902,"journal":{"name":"Annales De L Institut Henri Poincare-probabilites Et Statistiques","volume":"32 1","pages":"202-219"},"PeriodicalIF":1.5,"publicationDate":"2018-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"83536850","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
. Pólya trees form a popular class of prior distributions used in Bayesian nonparametrics. For some choice of parameters, Pólya trees are prior distributions on density functions. In this paper we carry out a frequentist analysis of the induced posterior distributions in the density estimation model. We investigate the contraction rate of Pólya tree posterior densities in terms of the supremum loss and study the limiting shape distribution. A nonparametric Bernstein–von Mises theorem is established, as well as a Bayesian Donsker theorem for the posterior cumulative distribution function. Résumé. Les arbres de Pólya constituent une classe de lois a priori très utilisée en bayésien non-paramétrique. Pour certains choix de paramètres, les arbres de Pólya induisent des lois à densité. Nous menons une analyse fréquentiste des lois a posteriori bayésiennes correspondantes dans le modèle d’estimation de densité. La concentration a posteriori des densités–arbre de Pólya est étudiée en terme de la norme–sup et nous déterminons la loi a posteriori limite après renormalisation. Un théorème de Bernstein– von Mises non-paramétrique est établi, ainsi qu’un théorème de Donsker bayésien pour la fonction de répartition a posteriori.
． Pólya树形成了贝叶斯非参数中常用的一类先验分布。对于某些参数的选择，Pólya树是密度函数的先验分布。本文对密度估计模型中的诱导后验分布进行了频域分析。我们从最大损失的角度研究了Pólya树后密度的收缩率，并研究了极限形状分布。建立了后验累积分布函数的非参数Bernstein-von Mises定理和Bayesian Donsker定理。的简历。Pólya的树木组成了一类先验的、利用的、薪金和非薪金。倒一些精选的parparires, les arbres de Pólya industrial des lois densit。目前的情况是，我们可以分析一下，从统计数据的角度来看，从统计数据的角度来看，从统计数据的角度来看，从统计数据的角度来看，从统计数据的角度来看。浓缩-浓缩-浓缩-浓缩-浓缩-浓缩-浓缩-浓缩-浓缩-浓缩-浓缩-浓缩-浓缩-浓缩-浓缩-浓缩-浓缩-浓缩-浓缩-浓缩-浓缩-浓缩-浓缩-浓缩-浓缩-浓缩-浓缩-浓缩-浓缩-浓缩-浓缩-浓缩-浓缩-浓缩-浓缩-浓缩伯恩斯坦-米塞斯的非帕拉玛姆-拉西姆的非帕拉玛姆-拉西姆的非帕拉玛斯-拉西姆的非帕拉玛斯-拉西姆的非帕拉玛-拉西姆的非帕拉玛-拉西姆的非帕拉玛-拉西姆的非帕拉玛-拉西姆。
{"title":"Pólya tree posterior distributions on densities","authors":"I. Castillo","doi":"10.1214/16-AIHP784","DOIUrl":"https://doi.org/10.1214/16-AIHP784","url":null,"abstract":". Pólya trees form a popular class of prior distributions used in Bayesian nonparametrics. For some choice of parameters, Pólya trees are prior distributions on density functions. In this paper we carry out a frequentist analysis of the induced posterior distributions in the density estimation model. We investigate the contraction rate of Pólya tree posterior densities in terms of the supremum loss and study the limiting shape distribution. A nonparametric Bernstein–von Mises theorem is established, as well as a Bayesian Donsker theorem for the posterior cumulative distribution function. Résumé. Les arbres de Pólya constituent une classe de lois a priori très utilisée en bayésien non-paramétrique. Pour certains choix de paramètres, les arbres de Pólya induisent des lois à densité. Nous menons une analyse fréquentiste des lois a posteriori bayésiennes correspondantes dans le modèle d’estimation de densité. La concentration a posteriori des densités–arbre de Pólya est étudiée en terme de la norme–sup et nous déterminons la loi a posteriori limite après renormalisation. Un théorème de Bernstein– von Mises non-paramétrique est établi, ainsi qu’un théorème de Donsker bayésien pour la fonction de répartition a posteriori.","PeriodicalId":7902,"journal":{"name":"Annales De L Institut Henri Poincare-probabilites Et Statistiques","volume":"119 1","pages":"2074-2102"},"PeriodicalIF":1.5,"publicationDate":"2017-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"73639574","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Limit theorems for the left random walk on $operatorname{GL}_{d}(mathbb{R})$","authors":"C. Cuny, J. Dedecker, Christophe Jan","doi":"10.1214/16-AIHP773","DOIUrl":"https://doi.org/10.1214/16-AIHP773","url":null,"abstract":"","PeriodicalId":7902,"journal":{"name":"Annales De L Institut Henri Poincare-probabilites Et Statistiques","volume":"45 1","pages":"1839-1865"},"PeriodicalIF":1.5,"publicationDate":"2017-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"76715329","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this paper, for $mu$ and $nu$ two probability measures on $mathbb{R}^d$ with finite moments of order $rhoge 1$, we define the respective projections for the $W_rho$-Wasserstein distance of $mu$ and $nu$ on the sets of probability measures dominated by $nu$ and of probability measures larger than $mu$ in the convex order. The $W_2$-projection of $mu$ can be easily computed when $mu$ and $nu$ have finite support by solving a quadratic optimization problem with linear constraints. In dimension $d=1$, Gozlan et al.~(2018) have shown that the projections do not depend on $rho$. We explicit their quantile functions in terms of those of $mu$ and $nu$. The motivation is the design of sampling techniques preserving the convex order in order to approximate Martingale Optimal Transport problems by using linear programming solvers. We prove convergence of the Wasserstein projection based sampling methods as the sample sizes tend to infinity and illustrate them by numerical experiments.
{"title":"Sampling of probability measures in the convex order by Wasserstein projection","authors":"A. Alfonsi, Jacopo Corbetta, B. Jourdain","doi":"10.1214/19-AIHP1014","DOIUrl":"https://doi.org/10.1214/19-AIHP1014","url":null,"abstract":"In this paper, for $mu$ and $nu$ two probability measures on $mathbb{R}^d$ with finite moments of order $rhoge 1$, we define the respective projections for the $W_rho$-Wasserstein distance of $mu$ and $nu$ on the sets of probability measures dominated by $nu$ and of probability measures larger than $mu$ in the convex order. The $W_2$-projection of $mu$ can be easily computed when $mu$ and $nu$ have finite support by solving a quadratic optimization problem with linear constraints. In dimension $d=1$, Gozlan et al.~(2018) have shown that the projections do not depend on $rho$. We explicit their quantile functions in terms of those of $mu$ and $nu$. The motivation is the design of sampling techniques preserving the convex order in order to approximate Martingale Optimal Transport problems by using linear programming solvers. We prove convergence of the Wasserstein projection based sampling methods as the sample sizes tend to infinity and illustrate them by numerical experiments.","PeriodicalId":7902,"journal":{"name":"Annales De L Institut Henri Poincare-probabilites Et Statistiques","volume":"62 1","pages":"1706-1729"},"PeriodicalIF":1.5,"publicationDate":"2017-09-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"82782972","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We consider a jump type diffusion X = (Xt) with infinitesimal generator given by Lψ(x) = 1/2 ∑ a ij (x) ∂2 ψ(x)/∂x i ∂x j + g(x)∇ψ(x) + ∫ (ψ(x + c(z, x)) − ψ(x))γ(z, x)µ(dz) where µ is of infinite total mass. We prove Harris recurrence of X using a regeneration scheme which is entirely based on the jumps of the process. Moreover we state explicit conditions in terms of the coefficients of the process allowing to control the speed of convergence to equilibrium in terms of deviation inequalities for integrable additive functionals.
{"title":"Ergodicity for multidimensional jump diffusions with position dependent jump rate","authors":"E. Löcherbach, V. Rabiet","doi":"10.1214/16-AIHP750","DOIUrl":"https://doi.org/10.1214/16-AIHP750","url":null,"abstract":"We consider a jump type diffusion X = (Xt) with infinitesimal generator given by Lψ(x) = 1/2 ∑ a ij (x) ∂2 ψ(x)/∂x i ∂x j + g(x)∇ψ(x) + ∫ (ψ(x + c(z, x)) − ψ(x))γ(z, x)µ(dz) where µ is of infinite total mass. We prove Harris recurrence of X using a regeneration scheme which is entirely based on the jumps of the process. Moreover we state explicit conditions in terms of the coefficients of the process allowing to control the speed of convergence to equilibrium in terms of deviation inequalities for integrable additive functionals.","PeriodicalId":7902,"journal":{"name":"Annales De L Institut Henri Poincare-probabilites Et Statistiques","volume":"58 1","pages":"1136-1163"},"PeriodicalIF":1.5,"publicationDate":"2017-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"73436271","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
. In this paper, we consider the parabolic Anderson equation that is driven by a Gaussian noise fractional in time and white or fractional in space, and is solved in a mild sense defined by Skorokhod integral. Our objective is the precise moment Lyapunov exponent and high moment asymptotics. As far as the long term asymptotics are concerned, some feature given in our theorems is different from what have been observed in the Stratonovich-regime and in the setting of the white time noise. While the difference disappears when it comes to the high moment asymptotics. To achieve our goal, we introduce a variational inequality and use some newly developed tools such as time-space LDP of Feynman–Kac type, linearization by tangent approximation, together with some techniques developed along the line of probability in Banach spaces. Résumé. lorsque l’on considère les asymptotiques des grands moments. Nos résultats sont obtenus en introduisant une nouvelle inégalité variationnelle, et à l’aide d’outils nouveaux tels qu’un principe de grandes déviations de type Feynman–Kac, la linéarisation par des approximations tangentes, et des techniques inspirées des probabilités dans les espaces de Banach. MSC:
． 本文考虑由时间分数高斯噪声和空间分数高斯噪声驱动的抛物型Anderson方程，用Skorokhod积分定义其在温和意义上的解。我们的目标是精确矩Lyapunov指数和高矩渐近性。就长期渐近性而言，我们的定理中给出的一些特征不同于在Stratonovich-regime和白时间噪声的设置中观察到的特征。而当涉及到高矩渐近性时，差异就消失了。为了实现我们的目标，我们引入了一个变分不等式，并使用了一些新发展的工具，如费曼-卡茨型的时空LDP，切线近似线性化，以及沿巴纳赫空间的概率线发展的一些技术。的简历。Lorsque l 'on考虑les asymptotiques des grandmoments。在引入一种新形式的变异体的情况下，在引入一种新形式的变异体的情况下，在引入一种新的变异体的情况下，在引入一种新的变异体的情况下，在引入一种新的变异体的情况下，在引入一种新的变异体的情况下，在引入一种新的变异体的情况下，在引入一种新的变异体的情况下，在引入一种新的变异体的情况下，在引入一种新的变异体的情况下，在引入一种新的变异体的情况下，在引入一种新的变异体的情况下，在引入一种新的变异体的情况下，在引入一种新的变异体的情况下，在引入一种新的变异体的情况下，在引入一种新的变异体的情况下，在引入一种新的变异体的情况下，硕士:
{"title":"Moment asymptotics for parabolic Anderson equation with fractional time-space noise: In Skorokhod regime","authors":"Xia Chen","doi":"10.1214/15-AIHP738","DOIUrl":"https://doi.org/10.1214/15-AIHP738","url":null,"abstract":". In this paper, we consider the parabolic Anderson equation that is driven by a Gaussian noise fractional in time and white or fractional in space, and is solved in a mild sense defined by Skorokhod integral. Our objective is the precise moment Lyapunov exponent and high moment asymptotics. As far as the long term asymptotics are concerned, some feature given in our theorems is different from what have been observed in the Stratonovich-regime and in the setting of the white time noise. While the difference disappears when it comes to the high moment asymptotics. To achieve our goal, we introduce a variational inequality and use some newly developed tools such as time-space LDP of Feynman–Kac type, linearization by tangent approximation, together with some techniques developed along the line of probability in Banach spaces. Résumé. lorsque l’on considère les asymptotiques des grands moments. Nos résultats sont obtenus en introduisant une nouvelle inégalité variationnelle, et à l’aide d’outils nouveaux tels qu’un principe de grandes déviations de type Feynman–Kac, la linéarisation par des approximations tangentes, et des techniques inspirées des probabilités dans les espaces de Banach. MSC:","PeriodicalId":7902,"journal":{"name":"Annales De L Institut Henri Poincare-probabilites Et Statistiques","volume":"8 1","pages":"819-841"},"PeriodicalIF":1.5,"publicationDate":"2017-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"88806318","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"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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