论文标题
在低纤维化随机微分方程产生的马尔可夫半群中，定量光谱间隙和均匀的下限
Quantitative spectral gaps and uniform lower bounds in the small noise limit for Markov semigroups generated by hypoelliptic stochastic differential equations
论文作者
论文摘要
我们研究马尔可夫半群家族$ \ {\ nathcal {p} _t^^ε\} _ {ε> 0} $由$ \ \ \ nathbb {r}^d $ in compress incompress incompress incompress incompress incompress incompress， Lorenz-96和Shell Model Sabra。在消失，平衡的噪声和耗散状态下，我们根据小参数$ε$获得了对指数收敛的尖锐（就缩放）定量估计。通过缩放，该机制意味着固定耗散和较大的噪声极限或固定噪声以及消失的耗散限制的相应最佳结果。作为证明的一部分和独立利益的一部分，我们在固定度量的密度上获得了统一的上限和下限。上边界是通过低纤维化的摩擦迭代获得的，该迭代是通过de giorgi型迭代的下限（均在$ε$中均匀）。半群的光谱差异估计是通过弱庞加莱不平等参数以及时间依赖性问题的定量性低纤维化正则化获得的。
We study the convergence rate to equilibrium for a family of Markov semigroups $\{\mathcal{P}_t^ε\}_{ε> 0}$ generated by a class of hypoelliptic stochastic differential equations on $\mathbb{R}^d$, including Galerkin truncations of the incompressible Navier-Stokes equations, Lorenz-96, and the shell model SABRA. In the regime of vanishing, balanced noise and dissipation, we obtain a sharp (in terms of scaling) quantitative estimate on the exponential convergence in terms of the small parameter $ε$. By scaling, this regime implies corresponding optimal results both for fixed dissipation and large noise limits or fixed noise and vanishing dissipation limits. As part of the proof, and of independent interest, we obtain uniform-in-$ε$ upper and lower bounds on the density of the stationary measure. Upper bounds are obtained by a hypoelliptic Moser iteration, the lower bounds by a de Giorgi-type iteration (both uniform in $ε$). The spectral gap estimate on the semigroup is obtained by a weak Poincaré inequality argument combined with quantitative hypoelliptic regularization of the time-dependent problem.
