We study a stochastic version of the classical Becker-Doring model, a well-known kinetic model for cluster formation that predicts the existence of a long-lived metastable state before a thermodynamically unfavorable nucleation occurs, leading to a phase transition phenomena. This continuous-time Markov chain model has received little attention, compared to its deterministic differential equations counterpart. We show that the stochastic formulation leads to a precise and quantitative description of stochastic nucleation events thanks to an exponentially ergodic quasi-stationary distribution for the process conditionally on nucleation has not yet occurred.
{"title":"Quasi-stationary distribution and metastability for the stochastic Becker-Döring model","authors":"Erwan Hingant, R. Yvinec","doi":"10.1214/21-ecp411","DOIUrl":"https://doi.org/10.1214/21-ecp411","url":null,"abstract":"We study a stochastic version of the classical Becker-Doring model, a well-known kinetic model for cluster formation that predicts the existence of a long-lived metastable state before a thermodynamically unfavorable nucleation occurs, leading to a phase transition phenomena. This continuous-time Markov chain model has received little attention, compared to its deterministic differential equations counterpart. We show that the stochastic formulation leads to a precise and quantitative description of stochastic nucleation events thanks to an exponentially ergodic quasi-stationary distribution for the process conditionally on nucleation has not yet occurred.","PeriodicalId":8470,"journal":{"name":"arXiv: Probability","volume":"12 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-08-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86994068","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}
We study the fluctuations, as $d,nto infty$, of the Wishart matrix $mathcal{W}_{n,d}= frac{1}{d} mathcal{X}_{n,d} mathcal{X}_{n,d}^{T} $ associated to a $ntimes d$ random matrix $mathcal{X}_{n,d}$ with non-Gaussian entries. We analyze the limiting behavior in distribution of $mathcal{W}_{n,d}$ in two situations: when the entries of $mathcal{X}_{n,d}$ are independent elements of a Wiener chaos of arbitrary order and when the entries are partially correlated and belong to the second Wiener chaos. In the first case, we show that the (suitably normalized) Wishart matrix converges in distribution to a Gaussian matrix while in the correlated case, we obtain its convergence in law to a diagonal non-Gaussian matrix. In both cases, we derive the rate of convergence in the Wasserstein distance via Malliavin calculus and analysis on Wiener space.
{"title":"Limiting behavior of large correlated Wishart matrices with chaotic entries","authors":"S. Bourguin, Charles-Philippe Diez, C. Tudor","doi":"10.3150/20-BEJ1266","DOIUrl":"https://doi.org/10.3150/20-BEJ1266","url":null,"abstract":"We study the fluctuations, as $d,nto infty$, of the Wishart matrix $mathcal{W}_{n,d}= frac{1}{d} mathcal{X}_{n,d} mathcal{X}_{n,d}^{T} $ associated to a $ntimes d$ random matrix $mathcal{X}_{n,d}$ with non-Gaussian entries. We analyze the limiting behavior in distribution of $mathcal{W}_{n,d}$ in two situations: when the entries of $mathcal{X}_{n,d}$ are independent elements of a Wiener chaos of arbitrary order and when the entries are partially correlated and belong to the second Wiener chaos. In the first case, we show that the (suitably normalized) Wishart matrix converges in distribution to a Gaussian matrix while in the correlated case, we obtain its convergence in law to a diagonal non-Gaussian matrix. In both cases, we derive the rate of convergence in the Wasserstein distance via Malliavin calculus and analysis on Wiener space.","PeriodicalId":8470,"journal":{"name":"arXiv: Probability","volume":"22 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-08-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"78474973","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 : 2020-08-03DOI: 10.1016/j.spa.2021.08.010
M. Salins
{"title":"Systems of small-noise stochastic reaction–diffusion equations satisfy a large deviations principle that is uniform over all initial data","authors":"M. Salins","doi":"10.1016/j.spa.2021.08.010","DOIUrl":"https://doi.org/10.1016/j.spa.2021.08.010","url":null,"abstract":"","PeriodicalId":8470,"journal":{"name":"arXiv: Probability","volume":"128 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-08-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"81333185","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 : 2020-08-03DOI: 10.1016/J.SPA.2021.08.004
Andrew L. Allan, Chong Liu, David J. Promel
{"title":"C`adl`ag Rough Differential Equations with Reflecting Barriers","authors":"Andrew L. Allan, Chong Liu, David J. Promel","doi":"10.1016/J.SPA.2021.08.004","DOIUrl":"https://doi.org/10.1016/J.SPA.2021.08.004","url":null,"abstract":"","PeriodicalId":8470,"journal":{"name":"arXiv: Probability","volume":"15 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-08-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"83582610","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 : 2020-08-01DOI: 10.1142/s0219493721500167
Almaz Tesfay, Daniel Tesfay, A. Khalaf, J. Brannan
In this paper, we formulate a stochastic logistic fish growth model driven by both white noise and non-Gaussian noise. We focus our study on the mean time to extinction, escape probability to measure the noise-induced extinction probability and the Fokker-Planck equation for fish population X(t). In the Gaussian case, these quantities satisfy local partial differential equations while in the non-Gaussian case, they satisfy nonlocal partial differential equations. Following a discussion of existence, uniqueness, and stability, we calculate numerical approximations of the solutions of those equations. For each noise model we then compare the behaviors of the mean time to extinction and the solution of the Fokker-Planck equation as growth rate r, carrying capacity K, the intensity of Gaussian noise ${lambda}$, noise intensity ${sigma}$ and stability index ${alpha}$ vary. The MET from the interval (0,1) at the right boundary is finite if ${lambda} {sqrt2}$, the MET from (0,1) at this boundary is infinite. A larger stability index ${alpha}$ is less likely to lead to the extinction of the fish population.
{"title":"Mean exit time and escape probability for the stochastic logistic growth model with multiplicative α-stable Lévy noise","authors":"Almaz Tesfay, Daniel Tesfay, A. Khalaf, J. Brannan","doi":"10.1142/s0219493721500167","DOIUrl":"https://doi.org/10.1142/s0219493721500167","url":null,"abstract":"In this paper, we formulate a stochastic logistic fish growth model driven by both white noise and non-Gaussian noise. We focus our study on the mean time to extinction, escape probability to measure the noise-induced extinction probability and the Fokker-Planck equation for fish population X(t). In the Gaussian case, these quantities satisfy local partial differential equations while in the non-Gaussian case, they satisfy nonlocal partial differential equations. Following a discussion of existence, uniqueness, and stability, we calculate numerical approximations of the solutions of those equations. For each noise model we then compare the behaviors of the mean time to extinction and the solution of the Fokker-Planck equation as growth rate r, carrying capacity K, the intensity of Gaussian noise ${lambda}$, noise intensity ${sigma}$ and stability index ${alpha}$ vary. The MET from the interval (0,1) at the right boundary is finite if ${lambda} {sqrt2}$, the MET from (0,1) at this boundary is infinite. A larger stability index ${alpha}$ is less likely to lead to the extinction of the fish population.","PeriodicalId":8470,"journal":{"name":"arXiv: Probability","volume":"1 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79830311","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 paper, starting from the methodology proposed in Magdziarz and Weron (2011), we develop asymptotic theory for the detection of mixing in Gaussian anomalous diffusion. The assumptions cover a broad family of stochastic processes including fractional Gaussian noise and the fractional Ornstein-Uhlenbeck process. We show that the asymptotic distribution and convergence rates of the detection statistic may be, respectively, Gaussian or non-Gaussian and standard or nonstandard depending on the diffusion exponent. The results pave the way for mixing detection based on a single observed sample path.
{"title":"Asymptotic theory for the detection of mixing in anomalous diffusion","authors":"Kui Zhang, G. Didier","doi":"10.1063/5.0023227","DOIUrl":"https://doi.org/10.1063/5.0023227","url":null,"abstract":"In this paper, starting from the methodology proposed in Magdziarz and Weron (2011), we develop asymptotic theory for the detection of mixing in Gaussian anomalous diffusion. The assumptions cover a broad family of stochastic processes including fractional Gaussian noise and the fractional Ornstein-Uhlenbeck process. We show that the asymptotic distribution and convergence rates of the detection statistic may be, respectively, Gaussian or non-Gaussian and standard or nonstandard depending on the diffusion exponent. The results pave the way for mixing detection based on a single observed sample path.","PeriodicalId":8470,"journal":{"name":"arXiv: Probability","volume":"16 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-07-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"82185508","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}
We show that the endpoint large deviation rate function for a continuous-time directed polymer agrees with the rate function of the underlying random walk near the origin in the whole weak disorder phase.
结果表明，连续时间定向聚合物的端点大偏差率函数与整个弱无序相中在原点附近随机游走的速率函数一致。
{"title":"On large deviation rate functions for a continuous-time directed polymer in weak disorder","authors":"Ryoki Fukushima, S. Junk","doi":"10.1214/21-ECP378","DOIUrl":"https://doi.org/10.1214/21-ECP378","url":null,"abstract":"We show that the endpoint large deviation rate function for a continuous-time directed polymer agrees with the rate function of the underlying random walk near the origin in the whole weak disorder phase.","PeriodicalId":8470,"journal":{"name":"arXiv: Probability","volume":"28 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-07-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"83254596","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 : 2020-07-25DOI: 10.1007/978-3-030-83266-7_3
M. Zhitlukhin
{"title":"A Sequential Test for the Drift of a Brownian Motion with a Possibility to Change a Decision","authors":"M. Zhitlukhin","doi":"10.1007/978-3-030-83266-7_3","DOIUrl":"https://doi.org/10.1007/978-3-030-83266-7_3","url":null,"abstract":"","PeriodicalId":8470,"journal":{"name":"arXiv: Probability","volume":"27 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-07-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"73544455","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 : 2020-07-20DOI: 10.1007/springerreference_205692
S. Sagitov
{"title":"Weak Convergence of Probability Measures","authors":"S. Sagitov","doi":"10.1007/springerreference_205692","DOIUrl":"https://doi.org/10.1007/springerreference_205692","url":null,"abstract":"","PeriodicalId":8470,"journal":{"name":"arXiv: Probability","volume":"50 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-07-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"87376959","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 : 2020-07-19DOI: 10.1515/9783110700763-007
Tim Jaschek, M. Murugan
We obtain connectivity of annuli for a volume doubling metric measure Dirichlet space which satisfies a Poincare inequality, a capacity estimate and a fast volume growth condition. This type of connectivity was introduced by Grigor'yan and Saloff-Coste in order to obtain stability results for Harnack inequalities and to study diffusions on manifolds with ends. As an application of our result, we obtain stability of the elliptic Harnack inequality under perturbations of the Dirichlet form with radial type weights.
{"title":"Geometric implications of fast volume growth and capacity estimates","authors":"Tim Jaschek, M. Murugan","doi":"10.1515/9783110700763-007","DOIUrl":"https://doi.org/10.1515/9783110700763-007","url":null,"abstract":"We obtain connectivity of annuli for a volume doubling metric measure Dirichlet space which satisfies a Poincare inequality, a capacity estimate and a fast volume growth condition. This type of connectivity was introduced by Grigor'yan and Saloff-Coste in order to obtain stability results for Harnack inequalities and to study diffusions on manifolds with ends. As an application of our result, we obtain stability of the elliptic Harnack inequality under perturbations of the Dirichlet form with radial type weights.","PeriodicalId":8470,"journal":{"name":"arXiv: Probability","volume":"27 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-07-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74429393","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}
京公网安备 11010802042870号
京ICP备2023020795号-1
扫码关注我们
求助内容：
应助结果提醒方式：