{"title":"抛物型SPDEs的耗散II:解的振荡和衰减","authors":"D. Khoshnevisan, Kunwoo Kim, C. Mueller","doi":"10.1214/22-aihp1289","DOIUrl":null,"url":null,"abstract":"We consider a stochastic heat equation of the type, $\\partial_t u = \\partial^2_x u + \\sigma(u)\\dot{W}$ on $(0\\,,\\infty)\\times[-1\\,,1]$ with periodic boundary conditions and on-degenerate positive initial data, where $\\sigma:\\mathbb{R} \\to\\mathbb{R}$ is a non-random Lipschitz continuous function and $\\dot{W}$ denotes space-time white noise. If additionally $\\sigma(0)=0$ then the solution is known to be strictly positive; see Mueller '91. In that case, we prove that the oscillation of the logarithm of the solution decays sublinearly as time tends to infinity. Among other things, it follows that, with probability one, all limit points of $t^{-1}\\, \\sup_{x\\in[-1,1]}\\, \\log u(t\\,,x)$ and $t^{-1}\\, \\inf_{x\\in[-1,1]}\\, \\log u(t\\,,x)$ must coincide. As a consequence of this fact, we prove that, when $\\sigma$ is linear, there is a.s. only one such limit point and hence the entire path decays almost surely at an exponential rate.","PeriodicalId":42884,"journal":{"name":"Annales de l Institut Henri Poincare D","volume":"108 1","pages":""},"PeriodicalIF":1.5000,"publicationDate":"2021-10-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"3","resultStr":"{\"title\":\"Dissipation in parabolic SPDEs II: Oscillation and decay of the solution\",\"authors\":\"D. Khoshnevisan, Kunwoo Kim, C. Mueller\",\"doi\":\"10.1214/22-aihp1289\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"We consider a stochastic heat equation of the type, $\\\\partial_t u = \\\\partial^2_x u + \\\\sigma(u)\\\\dot{W}$ on $(0\\\\,,\\\\infty)\\\\times[-1\\\\,,1]$ with periodic boundary conditions and on-degenerate positive initial data, where $\\\\sigma:\\\\mathbb{R} \\\\to\\\\mathbb{R}$ is a non-random Lipschitz continuous function and $\\\\dot{W}$ denotes space-time white noise. If additionally $\\\\sigma(0)=0$ then the solution is known to be strictly positive; see Mueller '91. In that case, we prove that the oscillation of the logarithm of the solution decays sublinearly as time tends to infinity. Among other things, it follows that, with probability one, all limit points of $t^{-1}\\\\, \\\\sup_{x\\\\in[-1,1]}\\\\, \\\\log u(t\\\\,,x)$ and $t^{-1}\\\\, \\\\inf_{x\\\\in[-1,1]}\\\\, \\\\log u(t\\\\,,x)$ must coincide. As a consequence of this fact, we prove that, when $\\\\sigma$ is linear, there is a.s. only one such limit point and hence the entire path decays almost surely at an exponential rate.\",\"PeriodicalId\":42884,\"journal\":{\"name\":\"Annales de l Institut Henri Poincare D\",\"volume\":\"108 1\",\"pages\":\"\"},\"PeriodicalIF\":1.5000,\"publicationDate\":\"2021-10-13\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"3\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Annales de l Institut Henri Poincare D\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/10.1214/22-aihp1289\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q2\",\"JCRName\":\"PHYSICS, MATHEMATICAL\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Annales de l Institut Henri Poincare D","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.1214/22-aihp1289","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"PHYSICS, MATHEMATICAL","Score":null,"Total":0}
引用次数: 3
摘要
我们考虑一个随机热方程,$\partial_t u = \partial^2_x u + \sigma(u)\dot{W}$在$(0\,,\infty)\times[-1\,,1]$上具有周期边界条件和不退化的正初始数据,其中$\sigma:\mathbb{R} \to\mathbb{R}$是一个非随机Lipschitz连续函数,$\dot{W}$表示时空白噪声。如果另外$\sigma(0)=0$,则已知解是严格正的;参见穆勒'91。在这种情况下,我们证明了当时间趋于无穷时,解的对数振荡呈次线性衰减。除其他事项外,可以得出,在概率为1的情况下,$t^{-1}\, \sup_{x\in[-1,1]}\, \log u(t\,,x)$和$t^{-1}\, \inf_{x\in[-1,1]}\, \log u(t\,,x)$的所有极限点必须重合。作为这一事实的结果,我们证明,当$\sigma$是线性的,只有一个这样的极限点,因此整个路径几乎肯定以指数速率衰减。
Dissipation in parabolic SPDEs II: Oscillation and decay of the solution
We consider a stochastic heat equation of the type, $\partial_t u = \partial^2_x u + \sigma(u)\dot{W}$ on $(0\,,\infty)\times[-1\,,1]$ with periodic boundary conditions and on-degenerate positive initial data, where $\sigma:\mathbb{R} \to\mathbb{R}$ is a non-random Lipschitz continuous function and $\dot{W}$ denotes space-time white noise. If additionally $\sigma(0)=0$ then the solution is known to be strictly positive; see Mueller '91. In that case, we prove that the oscillation of the logarithm of the solution decays sublinearly as time tends to infinity. Among other things, it follows that, with probability one, all limit points of $t^{-1}\, \sup_{x\in[-1,1]}\, \log u(t\,,x)$ and $t^{-1}\, \inf_{x\in[-1,1]}\, \log u(t\,,x)$ must coincide. As a consequence of this fact, we prove that, when $\sigma$ is linear, there is a.s. only one such limit point and hence the entire path decays almost surely at an exponential rate.