局部极小值下颗粒介质方程的退出时间

J. Tugaut
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引用次数: 1

摘要

.我们感兴趣的是一个非线性偏微分方程:颗粒介质方程。由于我们之前的一些结果[10,11],我们知道在容易检查的假设下,存在一个独特的稳态。我们指出,我们考虑了一种情况,其中con(cid:12)ning势不是全局凸的。根据最近的文章[8,9],我们知道这种稳态存在弱收敛性。然而,我们对收敛速度一无所知。在本文中,我们通过证明关于颗粒介质方程的解第一次离开局部阱的(cid:12)确定性Kramers型定律,向这个方向迈出了第一步。换句话说,我们证明了在指数等价于exp{2(cid:27)2H}的时间内,颗粒介质方程的解被困在局部极小值附近,H是所谓的退出成本。
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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.
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来源期刊
Communications on Stochastic Analysis
Communications on Stochastic Analysis Mathematics-Statistics and Probability
CiteScore
2.40
自引率
0.00%
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0
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
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