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分数科尔摩戈罗夫方程与单一的对照控制终端条件
Helena Kremp1, Nicolas Perkowski2
1Institut für Analysis und Scientific Computing, Technische Universität Wien, Wiedner Hauptstraße 8-10, 1040 Wien, Austria.
Journal of theoretical probability
|March 10, 2025
概括
这项研究引入了一种新型的对控制式解决方案空间,用于具有奇点漂移和终端条件的逆向分数科尔摩戈罗夫方程. 这种方法统一了单一和非单一数据,提供了更好的规律性和一般化先前的研究.
科学领域:
- 随机局部微分方程 随机局部微分方程
- 分数微积分的计算.
- 非线性分析 非线性分析
背景情况:
- 目前对逆向分数科尔摩戈罗夫方程的研究往往需要对漂移和终端条件的强有力的规律性假设.
- 之前的作品,如Cannizzaro和Chouk (2018) 和Kremp和Perkowski (2022),解决了特定的案例,但缺乏对单一数据的统一理论.
- "年轻制度"和"修改副产品"是这些方程当前分析工具的局限性.
研究的目的:
- 开发一个统一的解决方案理论的后向分数科尔摩戈罗夫方程与单一的贝索夫漂移和单一的终端条件.
- 将现有方法的适用性扩展到"年轻制度"之外的漂流.
- 引入一个新的抛物线控制的解决方案空间,它固有的提供抛物线规律性.
主要方法:
- 引入对漂移进行增强假设以处理低规律性.
- 利用对照控制的终端条件.
- 开发一个新的抛物线控制的解决方案空间,避免了对"修改后备产品"的需求.
主要成果:
- 建立了一个通用的解决方案理论,该理论包括单一和非单一数据在一个单一框架内.
- 拟议的抛物线控制的解决方案空间为解决方案产生抛物线时间和空间规律.
- 该方法成功地处理了超越"年轻人制度"的偏移,克服了以前研究的局限性.
结论:
- 这项研究在分析具有挑战性数据奇点的逆向分数科尔摩戈罗夫方程方面取得了重大进展.
- 开发的抛物线控制的替代和解决空间提供了一个强大的和多功能工具,适用于更广泛的线性偏微分方程类.
- 这项工作为未来研究更复杂的分数动态和相关的PDEs铺平了道路.
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