{"title":"局部极小值下颗粒介质方程的退出时间","authors":"J. Tugaut","doi":"10.31390/COSA.12.1.03","DOIUrl":null,"url":null,"abstract":". We are interested in a nonlinear partial differential equation: the granular media one. Thanks to some of our previous results [10, 11], we know that under easily checked assumptions, there is a unique steady state. We point out that we consider a case in which the con(cid:12)ning potential is not globally convex. According to recent articles [8, 9], we know that there is weak convergence towards this steady state. However, we do not know anything about the rate of convergence. In this paper, we make a (cid:12)rst step to this direction by proving a deterministic Kramers’type law concerning the (cid:12)rst time that the solution of the granular media equation leaves a local well. In other words, we show that the solution of the granular media equation is trapped around a local minimum during a time exponentially equivalent to exp { 2 (cid:27) 2 H } , H being the so-called exit-cost.","PeriodicalId":53434,"journal":{"name":"Communications on Stochastic Analysis","volume":" ","pages":""},"PeriodicalIF":0.0000,"publicationDate":"2018-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"1","resultStr":"{\"title\":\"Exit-Time of Granular Media Equation Starting in a Local Minimum\",\"authors\":\"J. Tugaut\",\"doi\":\"10.31390/COSA.12.1.03\",\"DOIUrl\":null,\"url\":null,\"abstract\":\". We are interested in a nonlinear partial differential equation: the granular media one. Thanks to some of our previous results [10, 11], we know that under easily checked assumptions, there is a unique steady state. We point out that we consider a case in which the con(cid:12)ning potential is not globally convex. According to recent articles [8, 9], we know that there is weak convergence towards this steady state. However, we do not know anything about the rate of convergence. In this paper, we make a (cid:12)rst step to this direction by proving a deterministic Kramers’type law concerning the (cid:12)rst time that the solution of the granular media equation leaves a local well. In other words, we show that the solution of the granular media equation is trapped around a local minimum during a time exponentially equivalent to exp { 2 (cid:27) 2 H } , H being the so-called exit-cost.\",\"PeriodicalId\":53434,\"journal\":{\"name\":\"Communications on Stochastic Analysis\",\"volume\":\" \",\"pages\":\"\"},\"PeriodicalIF\":0.0000,\"publicationDate\":\"2018-08-01\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"1\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Communications on Stochastic Analysis\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/10.31390/COSA.12.1.03\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q2\",\"JCRName\":\"Mathematics\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Communications on Stochastic Analysis","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.31390/COSA.12.1.03","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"Mathematics","Score":null,"Total":0}
Exit-Time of Granular Media Equation Starting in a Local Minimum
. We are interested in a nonlinear partial differential equation: the granular media one. Thanks to some of our previous results [10, 11], we know that under easily checked assumptions, there is a unique steady state. We point out that we consider a case in which the con(cid:12)ning potential is not globally convex. According to recent articles [8, 9], we know that there is weak convergence towards this steady state. However, we do not know anything about the rate of convergence. In this paper, we make a (cid:12)rst step to this direction by proving a deterministic Kramers’type law concerning the (cid:12)rst time that the solution of the granular media equation leaves a local well. In other words, we show that the solution of the granular media equation is trapped around a local minimum during a time exponentially equivalent to exp { 2 (cid:27) 2 H } , H being the so-called exit-cost.
期刊介绍:
The journal Communications on Stochastic Analysis (COSA) is published in four issues annually (March, June, September, December). It aims to present original research papers of high quality in stochastic analysis (both theory and applications) and emphasizes the global development of the scientific community. The journal welcomes articles of interdisciplinary nature. Expository articles of current interest will occasionally be published. COSAis indexed in Mathematical Reviews (MathSciNet), Zentralblatt für Mathematik, and SCOPUS