再排列的随机热方程
François Delarue1, William R P Hammersley1
1Laboratoire J.A. Dieudonné, CNRS, Université Côte d'Azur, Nice, France.
概括
这项研究构建了一个强大的费勒半组,用于概率测量,将函数映射到具有可整化的膨胀的连续的利普希茨函数. 该方法使用重新排列的随机热方程和欧勒方案进行稳健的数学分析.
科学领域:
- 随机分析 随机分析
- 可能性理论概率理论.
- 部分微分方程 部分微分方程
背景情况:
- 费勒半组是研究马尔科夫过程的基础.
- 构建具有特定功能分析属性的半组,如利普希茨连续性,具有挑战性.
- 随机热方程为模拟随机进化提供了一个框架.
研究的目的:
- 在概率测量空间上明确构建一个强大的费勒半组.
- 为了确保半组地图将可测量的函数与Lipschitz连续函数相界限.
- 分析Lipschitz常数在小时间内的行为.
主要方法:
- 使用由彩色噪声驱动的重新排列的随机热方程.
- 采用欧勒方案,在平面动力学和重新排列操作之间交替运行.
- 从Bismut-Elworthy-Li公式中适应利普希茨属性分析的技术.
主要成果:
- 介绍了一个强大的费勒半组的新构造.
- 半组表现出所需的映射属性到利普希茨连续函数中.
- 利普希茨常数在很短的时间内证明了可整合的膨胀行为.
结论:
- 欧勒拟议的方案被证明是严格的.
- 建立了包括反射项在内的限制反射方程的一致理论.
- 这种构造成功地产生了一个强大的费勒半组,具有特定的功能分析特性.
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