{"title":"关于一类亚临界约束的Fokker-Planck方程","authors":"G. Toscani, M. Zanella","doi":"10.4171/rlm/944","DOIUrl":null,"url":null,"abstract":"We study the relaxation to equilibrium for a class linear onedimensional Fokker–Planck equations characterized by a particular subcritical confinement potential. An interesting feature of this class of Fokker–Planck equations is that, for any given probability density e(x), the diffusion coefficient can be built to have e(x) as steady state. This representation of the equilibrium density can be fruitfully used to obtain one-dimensional Wirtinger-type inequalities and to recover, for a sufficiently regular density e(x), a polynomial rate of convergence to equilibrium. Numerical results then confirm the theoretical analysis, and allow to conjecture that convergence to equilibrium with positive rate still holds for steady states characterized by a very slow polynomial decay at infinity.","PeriodicalId":54497,"journal":{"name":"Rendiconti Lincei-Matematica e Applicazioni","volume":" ","pages":""},"PeriodicalIF":0.6000,"publicationDate":"2021-03-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"3","resultStr":"{\"title\":\"On a class of Fokker–Planck equations with subcritical confinement\",\"authors\":\"G. Toscani, M. Zanella\",\"doi\":\"10.4171/rlm/944\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"We study the relaxation to equilibrium for a class linear onedimensional Fokker–Planck equations characterized by a particular subcritical confinement potential. An interesting feature of this class of Fokker–Planck equations is that, for any given probability density e(x), the diffusion coefficient can be built to have e(x) as steady state. This representation of the equilibrium density can be fruitfully used to obtain one-dimensional Wirtinger-type inequalities and to recover, for a sufficiently regular density e(x), a polynomial rate of convergence to equilibrium. Numerical results then confirm the theoretical analysis, and allow to conjecture that convergence to equilibrium with positive rate still holds for steady states characterized by a very slow polynomial decay at infinity.\",\"PeriodicalId\":54497,\"journal\":{\"name\":\"Rendiconti Lincei-Matematica e Applicazioni\",\"volume\":\" \",\"pages\":\"\"},\"PeriodicalIF\":0.6000,\"publicationDate\":\"2021-03-20\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"3\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Rendiconti Lincei-Matematica e Applicazioni\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://doi.org/10.4171/rlm/944\",\"RegionNum\":4,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q3\",\"JCRName\":\"MATHEMATICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Rendiconti Lincei-Matematica e Applicazioni","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.4171/rlm/944","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"MATHEMATICS","Score":null,"Total":0}
On a class of Fokker–Planck equations with subcritical confinement
We study the relaxation to equilibrium for a class linear onedimensional Fokker–Planck equations characterized by a particular subcritical confinement potential. An interesting feature of this class of Fokker–Planck equations is that, for any given probability density e(x), the diffusion coefficient can be built to have e(x) as steady state. This representation of the equilibrium density can be fruitfully used to obtain one-dimensional Wirtinger-type inequalities and to recover, for a sufficiently regular density e(x), a polynomial rate of convergence to equilibrium. Numerical results then confirm the theoretical analysis, and allow to conjecture that convergence to equilibrium with positive rate still holds for steady states characterized by a very slow polynomial decay at infinity.
期刊介绍:
The journal is dedicated to the publication of high-quality peer-reviewed surveys, research papers and preliminary announcements of important results from all fields of mathematics and its applications.