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.

主要研究了随机各向异性随机偏微分方程和α稳定噪声驱动的路径分布相关SDE的逼近。具体内容如下：(1)研究了非Lipschitz条件下带时滞的随机各向异性p-Laplacian反应扩散方程和带时滞的随机各向异性Navier-Stokes方程的适定性。值得指出的是，与现有结果相比，本工作的本质难点在于如何讨论非Lipschitz条件下各向异性SPFDE解的唯一性问题，而不是处理各向异性解。在比现有文献中施加的条件更弱的条件下，我们证明了具有各向异性指数的随机偏函数微分方程的适定性。基于所获得的结果，我们探讨了两种典型例子的适定性。(2)我们通过相互作用的粒子系统展示了由α稳定噪声驱动的一类路径分布相关的SDE的近似问题，其中漂移是奇异的。尽管可以估计跳跃扩散过程，但基于现有文献的 Zvonkin 型方法不适用于具有乘性噪声的 SDE 情况。我们的工作主要包括两个部分。第一部分展示了路径分布相关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。第二部分显示了这些路径分布相关的随机微分方程的近似值。