見本路に依存するジャンプ拡散過程の漸近挙動

主要研究了随机各向异性随机偏微分方程解和α-稳定噪声驱动的路径分布相关随机偏微分方程解的逼近问题。具体内容如下：(1)研究了随机各向异性p-Laplace反应扩散方程在非Lipschitz条件下和随机各向异性N-S方程在非Lipschitz条件下的适定性。值得指出的是，与已有的结果相比，本文的主要困难在于如何讨论各向异性SPFDEs在非Lipschitz条件下解的唯一性问题，而不是处理各向异性的问题。在比现有文献更弱的条件下，证明了具有各向异性指数的随机偏泛函微分方程解的适定性。在所得结果的基础上，我们讨论了两类典型例子的适定性。(2)通过相互作用的粒子系统，我们证明了一类由α-稳定噪声驱动的依赖路径分布的随机微分方程的逼近问题，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随机微分方程，尽管跳跃扩散过程是可以估计的。我们的工作主要包括两个部分。第一部分证明了依赖于路径分布的随机微分方程解的适定性，以及相应的随机相互作用粒子系统。第二部分给出了这些依赖于路径分布的随机微分方程组的近似解。

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.