论文标题
在分数驱动随机微分方程的（非）固定密度上
On the (Non-)Stationary Density of Fractional-Driven Stochastic Differential Equations
论文作者
论文摘要
我们调查了由任何Hurst参数驱动的固定度量$π$ SDE的SDE $π$，并确定$π$承认平滑的Lebesgue密度遵守高斯型下限和上限。这些证明是基于对固定密度的新颖表示，该密度以Wiener-liouville桥的形式进行，这被证明具有独立的兴趣：我们表明，它还允许在非平稳密度上获得高斯界限，这在添加剂设置中扩展了以前已知的范围。此外，我们研究了SDE的参数依赖性版本，并证明了固定密度的平滑性，该密度是在参数和空间坐标中共同的。因此，我们重新审视了李和西伯的平均原理[Ann。应用。概率。 32（5）（2022）]并删除有关限制系数的临时假设。避免在我们的论点中使用malliavin conculus，我们可以根据最少的规律性要求证明结果。
We investigate the stationary measure $π$ of SDEs driven by additive fractional noise with any Hurst parameter and establish that $π$ admits a smooth Lebesgue density obeying both Gaussian-type lower and upper bounds. The proofs are based on a novel representation of the stationary density in terms of a Wiener-Liouville bridge, which proves to be of independent interest: We show that it also allows to obtain Gaussian bounds on the non-stationary density, which extend previously known results in the additive setting. In addition, we study a parameter-dependent version of the SDE and prove smoothness of the stationary density, jointly in the parameter and the spatial coordinate. With this we revisit the fractional averaging principle of Li and Sieber [Ann. Appl. Probab. 32(5) (2022)] and remove an ad-hoc assumption on the limiting coefficients. Avoiding any use of Malliavin calculus in our arguments, we can prove our results under minimal regularity requirements.
