見本路に依存するジャンプ拡散過程の漸近挙動

取决于样本路径的跳跃扩散过程的渐近行为

• 批准号: 22KF0158
• 负责人: 謝 賓
• 金额: $ 1.41万
• 依托单位: Shinshu University
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for JSPS Fellows
• 财政年份: 2023
• 资助国家: 日本
• 起止时间: 2023-03-08 至 2024-03-31
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-22KF0158/
• 关键词: p-Laplacian; delay; BDG inequality; anisotropic; distribution dependent; transformation; approximation; SPDEs; alpha-stable; Euler-Maruyama

We mainly studied stochastic anisotropic stochastic partial differential equations and approximation of path-distribution dependent SDEs driven by alpha-stable noise.The details are following:(1) We study well-posedness for stochastic anisotropic p-Laplacian reaction-diffusion equation with delay under non-Lipschitz conditions and stochastic anisotropic Navier-Stokes equations with delay. It is worth pointing out that, in contrast to the exist results, the essential difficulty in this work is how to discuss the problem of the uniqueness of solutions for anisotropic SPFDEs under non-Lipschitz condition, other than dealing with the anisotropic one. Under weaker conditions than those imposed in existing literature we show the well-posedness of stochastic partial functional differential equations with anisotropic exponents. Based on the result obtained, we explore the well-posedness of two kinds of typical examples.(2) We show via interacting particle systems an approximation issue on a class of path-distribution dependent SDEs driven by alpha-stable noise, where the drift is singular. The Zvonkin-type approach based on the existing literature does not work for the case of SDEs with multiplicative noises, although the jump diffusion processes can be estimated. Our work involves primarily two parts. The first part shows the well-posedness of path-distribution dependent stochastic differential equations and the corresponding stochastic interacting particle systems. The second part shows approximations of these path-distribution dependent stochastic differential equations.

本文主要研究了随机各向异性随机偏微分方程以及α稳定噪声驱动下依赖于路径分布的随机偏微分方程的逼近问题，具体内容如下：（1）研究了非Lipschitz条件下随机各向异性时滞p-Laplacian反应扩散方程和随机各向异性时滞Navier-Stokes方程的适定性。值得指出的是，与已有的结果相比，本文的主要困难是如何讨论非Lipschitz条件下各向异性随机偏微分方程解的唯一性问题，而不是讨论各向异性随机偏微分方程解的唯一性问题.在较弱的条件下，比现有文献中所施加的，我们证明了具有各向异性指数的随机偏泛函微分方程的适定性。基于所得结果，我们探讨了两类典型例子的适定性。(2)通过相互作用粒子系统，我们展示了一类由α稳定噪声驱动的依赖于路径分布的随机微分方程的近似问题，其中漂移是奇异的。基于现有文献的Zvonkin型方法不适用于带有乘性噪声的偏微分方程，尽管可以估计跳跃扩散过程。我们的工作主要包括两部分。第一部分给出了依赖于路径分布的随机微分方程和相应的随机相互作用粒子系统的适定性。第二部分给出了这些依赖于路径分布的随机微分方程的近似解。