In this note, we consider the parabolic Anderson model on R + × R , driven by a Gaussian noise which is fractional in time with index H 0 > 1 / 2 and fractional in space with index 0 < H < 1 / 2 such that H 0 + H > 3 / 4. Under a general condition on the initial data, we prove the existence and uniqueness of the mild solution and obtain its exponential upper bounds in time for all p -th moments with p ≥ 2.
{"title":"Parabolic Anderson model with rough noise in space and rough initial conditions","authors":"R. Balan, Le Chen, Yiping Ma","doi":"10.1214/22-ecp506","DOIUrl":"https://doi.org/10.1214/22-ecp506","url":null,"abstract":"In this note, we consider the parabolic Anderson model on R + × R , driven by a Gaussian noise which is fractional in time with index H 0 > 1 / 2 and fractional in space with index 0 < H < 1 / 2 such that H 0 + H > 3 / 4. Under a general condition on the initial data, we prove the existence and uniqueness of the mild solution and obtain its exponential upper bounds in time for all p -th moments with p ≥ 2.","PeriodicalId":50543,"journal":{"name":"Electronic Communications in Probability","volume":" ","pages":""},"PeriodicalIF":0.5,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49497306","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this note we provide an alternative proof of the fact that subcritical bootstrap percolation models have a positive critical probability in any dimension. The proof relies on a recent extension [18] of the classical framework of Toom [20]. This approach is not only simpler than the original multi-scale renormalisation proof of the result in two and more dimensions [1, 2], but also gives significantly better bounds. As a byproduct, we improve the best known bounds for the stability threshold of Toom’s North-East-Center majority rule cellular automaton.
{"title":"Subcritical bootstrap percolation via Toom contours","authors":"Ivailo Hartarsky, R. Szab'o","doi":"10.1214/22-ecp496","DOIUrl":"https://doi.org/10.1214/22-ecp496","url":null,"abstract":"In this note we provide an alternative proof of the fact that subcritical bootstrap percolation models have a positive critical probability in any dimension. The proof relies on a recent extension [18] of the classical framework of Toom [20]. This approach is not only simpler than the original multi-scale renormalisation proof of the result in two and more dimensions [1, 2], but also gives significantly better bounds. As a byproduct, we improve the best known bounds for the stability threshold of Toom’s North-East-Center majority rule cellular automaton.","PeriodicalId":50543,"journal":{"name":"Electronic Communications in Probability","volume":" ","pages":""},"PeriodicalIF":0.5,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43100680","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We study the large-time and small-time asymptotic behaviors of the spectral heat content for time-changed stable processes, where the time change belongs to a large class of inverse subordinators. For the large-time behavior, the spectral heat content decays polynomially with the decay rate determined by the Laplace exponent of the underlying subordinator, which is in sharp contrast to the exponential decay observed in the case when the time change is a subordinator. On the other hand, the small-time behavior exhibits three different decay regimes, where the decay rate is determined by both the Laplace exponent and the index of the stable process.
{"title":"Large-time and small-time behaviors of the spectral heat content for time-changed stable processes","authors":"Kei Kobayashi, Hyunchul Park","doi":"10.1214/22-ecp478","DOIUrl":"https://doi.org/10.1214/22-ecp478","url":null,"abstract":"We study the large-time and small-time asymptotic behaviors of the spectral heat content for time-changed stable processes, where the time change belongs to a large class of inverse subordinators. For the large-time behavior, the spectral heat content decays polynomially with the decay rate determined by the Laplace exponent of the underlying subordinator, which is in sharp contrast to the exponential decay observed in the case when the time change is a subordinator. On the other hand, the small-time behavior exhibits three different decay regimes, where the decay rate is determined by both the Laplace exponent and the index of the stable process.","PeriodicalId":50543,"journal":{"name":"Electronic Communications in Probability","volume":" ","pages":""},"PeriodicalIF":0.5,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47368804","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The Stein couplings of Chen and Roellin (2010) vastly expanded the range of applications for which coupling constructions in Stein's method for normal approximation could be applied, and subsumed both Stein's classical exchangeable pair, as well as the size bias coupling. A further simple generalization includes zero bias couplings, and also allows for situations where the coupling is not exact. The zero bias versions result in bounds for which often tedious computations of a variance of a conditional expectation is not required. An example to the Lightbulb process shows that even though the method may be simple to apply, it may yield improvements over previous results that had achieved bounds with optimal rates and small, explicit constants.
{"title":"Zero bias enhanced Stein couplings","authors":"L. Goldstein","doi":"10.1214/22-ECP504","DOIUrl":"https://doi.org/10.1214/22-ECP504","url":null,"abstract":"The Stein couplings of Chen and Roellin (2010) vastly expanded the range of applications for which coupling constructions in Stein's method for normal approximation could be applied, and subsumed both Stein's classical exchangeable pair, as well as the size bias coupling. A further simple generalization includes zero bias couplings, and also allows for situations where the coupling is not exact. The zero bias versions result in bounds for which often tedious computations of a variance of a conditional expectation is not required. An example to the Lightbulb process shows that even though the method may be simple to apply, it may yield improvements over previous results that had achieved bounds with optimal rates and small, explicit constants.","PeriodicalId":50543,"journal":{"name":"Electronic Communications in Probability","volume":" ","pages":""},"PeriodicalIF":0.5,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44555929","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We study a regularization by noise phenomenon for the continuous parabolic Anderson model with a potential shifted along paths of fractional Brownian motion. We demonstrate that provided the Hurst parameter is chosen sufficiently small, this shift allows to establish well-posedness and stability to the corresponding problem - without the need of renormalization - in any dimension. We moreover provide a robustified Feynman-Kac type formula for the unique solution to the regularized problem building upon regularity estimates for the local time of fractional Brownian motion as well as non-linear Young integration.
{"title":"Regularization by random translation of potentials for the continuous PAM and related models in arbitrary dimension","authors":"F. Bechtold","doi":"10.1214/22-ecp490","DOIUrl":"https://doi.org/10.1214/22-ecp490","url":null,"abstract":"We study a regularization by noise phenomenon for the continuous parabolic Anderson model with a potential shifted along paths of fractional Brownian motion. We demonstrate that provided the Hurst parameter is chosen sufficiently small, this shift allows to establish well-posedness and stability to the corresponding problem - without the need of renormalization - in any dimension. We moreover provide a robustified Feynman-Kac type formula for the unique solution to the regularized problem building upon regularity estimates for the local time of fractional Brownian motion as well as non-linear Young integration.","PeriodicalId":50543,"journal":{"name":"Electronic Communications in Probability","volume":" ","pages":""},"PeriodicalIF":0.5,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44368701","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We analyse the eigenvectors of the adjacency matrix of the Erdős-Rényi graph on N vertices with edge probability d N . We determine the full region of delocalization by determining the critical values of d log N down to which delocalization persists: for d log N > 1 log 4−1 all eigenvectors are completely delocalized, and for d log N > 1 all eigenvectors with eigenvalues away from the spectral edges are completely delocalized. Below these critical values, it is known [1, 3] that localized eigenvectors exist in the corresponding spectral regions.
我们分析了Erdős-Rényi图在N个顶点上的邻接矩阵的特征向量，边缘概率为d N。我们通过确定d log N的临界值来确定脱域的整个区域:对于d log N > 1 log 4−1，所有特征向量都是完全脱域的，对于d log N > 1，所有特征值远离谱边的特征向量都是完全脱域的。在这些临界值以下，我们知道[1,3]在相应的光谱区域存在局域特征向量。
{"title":"The completely delocalized region of the Erdős-Rényi graph","authors":"Johannes Alt, Raphael Ducatez, A. Knowles","doi":"10.1214/22-ecp450","DOIUrl":"https://doi.org/10.1214/22-ecp450","url":null,"abstract":"We analyse the eigenvectors of the adjacency matrix of the Erdős-Rényi graph on N vertices with edge probability d N . We determine the full region of delocalization by determining the critical values of d log N down to which delocalization persists: for d log N > 1 log 4−1 all eigenvectors are completely delocalized, and for d log N > 1 all eigenvectors with eigenvalues away from the spectral edges are completely delocalized. Below these critical values, it is known [1, 3] that localized eigenvectors exist in the corresponding spectral regions.","PeriodicalId":50543,"journal":{"name":"Electronic Communications in Probability","volume":" ","pages":""},"PeriodicalIF":0.5,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47542430","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
. Consider the set of Borel probability measures on R k and endow it with the topology of weak convergence. We show that the subset of all probability measures which belong to the domain of attraction of some multivariate extreme value distributions is dense and of the first Baire category. In addition, the analogue result holds in the context of free probability theory.
{"title":"The maximum domain of attraction of multivariate extreme value distributions is small","authors":"P. Leonetti, A. K. Chokami","doi":"10.1214/22-ecp501","DOIUrl":"https://doi.org/10.1214/22-ecp501","url":null,"abstract":". Consider the set of Borel probability measures on R k and endow it with the topology of weak convergence. We show that the subset of all probability measures which belong to the domain of attraction of some multivariate extreme value distributions is dense and of the first Baire category. In addition, the analogue result holds in the context of free probability theory.","PeriodicalId":50543,"journal":{"name":"Electronic Communications in Probability","volume":" ","pages":""},"PeriodicalIF":0.5,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44224546","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The paper contains the study of sharp extensions of weak-type estimates for a martingale maximal function. Given 1 < p < ∞ and a pair ( x, y ) of nonnegative numbers satisfying x p ≤ y , we identify the optimal upper bounds for (cid:107)| sup n f n |(cid:107) p, ∞ , for nonnegative martingales f = ( f n ) n ≥ 0 satisfying (cid:107) f (cid:107) 1 = x and (cid:107) f (cid:107) pp = y .
本文研究了一个鞅极大函数的弱型估计的尖锐扩张。给定1＜p＜∞和一对满足x p≤y的非负数（x，y），我们确定了（cid:107）|supn f n|（cid:107）p，∞的最优上界，对于非负鞅f＝（f n）n≥0满足（cid:10 7）f（cid:7）1＝x和（cid:07）f（acid:107）pp＝y。
{"title":"Weak-type estimates for martingale maximal functions","authors":"A. Osȩkowski, Mateusz Wojtas","doi":"10.1214/22-ecp494","DOIUrl":"https://doi.org/10.1214/22-ecp494","url":null,"abstract":"The paper contains the study of sharp extensions of weak-type estimates for a martingale maximal function. Given 1 < p < ∞ and a pair ( x, y ) of nonnegative numbers satisfying x p ≤ y , we identify the optimal upper bounds for (cid:107)| sup n f n |(cid:107) p, ∞ , for nonnegative martingales f = ( f n ) n ≥ 0 satisfying (cid:107) f (cid:107) 1 = x and (cid:107) f (cid:107) pp = y .","PeriodicalId":50543,"journal":{"name":"Electronic Communications in Probability","volume":" ","pages":""},"PeriodicalIF":0.5,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46138797","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"On the central limit theorem for stationary random fields under L1-projective condition","authors":"H. Lin, F. Merlevède, D. Volný","doi":"10.1214/22-ecp486","DOIUrl":"https://doi.org/10.1214/22-ecp486","url":null,"abstract":"","PeriodicalId":50543,"journal":{"name":"Electronic Communications in Probability","volume":" ","pages":""},"PeriodicalIF":0.5,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41419643","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
. In this short note, we prove a central limit theorem for a type of replica overlap of the Brownian directed polymer in a Gaussian random environment, in the low temperature regime and in all dimensions. The proof relies on a superconcentration result for the KPZ equation driven by a spatially mollified noise, which is inspired by the recent work of Chatterjee [14].
{"title":"Gaussian fluctuations of replica overlap in directed polymers","authors":"Yu Gu, T. Komorowski","doi":"10.1214/22-ecp476","DOIUrl":"https://doi.org/10.1214/22-ecp476","url":null,"abstract":". In this short note, we prove a central limit theorem for a type of replica overlap of the Brownian directed polymer in a Gaussian random environment, in the low temperature regime and in all dimensions. The proof relies on a superconcentration result for the KPZ equation driven by a spatially mollified noise, which is inspired by the recent work of Chatterjee [14].","PeriodicalId":50543,"journal":{"name":"Electronic Communications in Probability","volume":" ","pages":""},"PeriodicalIF":0.5,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48840622","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"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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