退化Lévy噪声的弱正则化及其应用
Weak regularization by degenerate Lévy noise and its applications
AI总结：
本论文研究退化Lévy噪声的弱正则化，通过前向和后向parametrix方法建立Kolmogorov方程的Schauder估计和SDE的弱适定性，并探讨其在带摩擦动力学中的应用。
摘要：
在退化设定下对噪声正则化现象进行一般性介绍之后，本博士论文的第一部分聚焦于在弱Hörmander型条件下，为两类不同的Kolmogorov方程建立Schauder估计，这是一种用于证明随机微分方程（SDEs）适定性的有效分析工具，其系数位于具有多重正则性指数的适当各向异性Hölder空间中。第一类考虑由仅作用于部分分量的对称稳定算子控制的非线性系统。我们的证明方法依赖于通过Duhamel型公式基于前向parametrix展开的微扰方法。由于退化设定带来的低正则化性质，我们还利用了对Besov范数的某些控制，以处理非线性微扰。作为第一部分的扩展，我们还提出了与由更广泛的稳定类算子（如相对论稳定算子或Lamperti稳定算子）驱动的退化Ornstein-Uhlenbeck算子相关的Schauder估计。利用后向parametrix方法，本工作的第二部分旨在在系数满足最小Hölder正则性假设下，建立由同类稳定类过程驱动的退化SDE链的弱适定性。作为我们方法的副产品，我们还提出了对相关规范过程具有独立研究意义的Krylov型估计。最后，我们通过适当的反例展示了确保SDE弱适定性的正则性指数的（几乎）尖锐阈值的存在性。结合与带摩擦的动力学相关的一些可能应用，我们最后研究了退化Kolmogorov算子在Lp和Hölder范数下二阶微扰的稳定性。
英文摘要：
After a general introduction about the regularization by noise phenomenon in the degenerate setting, the first part of this PhD thesis focuses at establishing the Schauder estimates, a useful analytical tool to prove also the well-posedness of stochastic differential equations (SDEs), for two different classes of Kolmogorov equations under a weak Hörmander-like condition, whose coefficients lie in suitable anisotropic Hölder spaces with multi-indices of regularity. The first class considers a nonlinear system controlled by a symmetric stable operator acting only on some components. Our method of proof relies on a perturbative approach based on forward parametrix expansions through Duhamel-type formulas. Due to the low regularizing properties given by the degenerate setting, we also exploit some controls on Besov norms, in order to deal with the non-linear perturbation. As an extension of the first one, we also present Schauder estimates associated with a degenerate Ornstein-Uhlenbeck operator driven by a larger class of stable-like operators, like the relativistic or the Lamperti stable one. Exploiting a backward parametrix approach, the second part of this work aims at establishing the weak well-posedness for a degenerate chain of SDEs driven by the same class of stable-like processes, under the assumptions of the minimal Hölder regularity on the coefficients. As a by-product of our method, we also present Krylov-type estimates of independent interest for the associated canonical process. Finally, we show through suitable counter-examples the existence of an (almost) sharp threshold on the regularity exponents ensuring the weak well-posedness for the SDE. In connection with some possible applications to kinetic dynamics with friction, we conclude by investigating the stability of second-order perturbations for degenerate Kolmogorov operators in Lp and Hölder norms.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
