论文标题
Wasserstein收缩和高温下椭圆扩散的庞加莱不平等现象
Wasserstein contraction and Poincaré inequalities for elliptic diffusions at high temperature
论文作者
论文摘要
我们考虑$ \ Mathbb r^d $上的椭圆扩散过程。假设漂移合同在紧凑型集合之外的距离，我们证明，在足够高的温度下，与该过程相关的马尔可夫半组是$ \ Mathcal W_2 $ WASSERSTEIN距离的收缩，这意味着庞加莱的不平等程度是其不变度量的不平等。结果不需要可逆性，也不需要不变度度量的明确表达，并且估计值对维度具有急剧的依赖性。然后，将参数的某些变化用于研究该过程在其漂移方面不变度量的稳定性，其次是相互作用粒子的系统，得出了无维度的庞加莱不平等标准和非线性McKean-McKean-vlasov类型过程的标准。
We consider elliptic diffusion processes on $\mathbb R^d$. Assuming that the drift contracts distances outside a compact set, we prove that, at a sufficiently high temperature, the Markov semi-group associated to the process is a contraction of the $\mathcal W_2$ Wasserstein distance, which implies a Poincaré inequality for its invariant measure. The result doesn't require neither reversibility nor an explicit expression of the invariant measure, and the estimates have a sharp dependency on the dimension. Some variations of the arguments are then used to study, first, the stability of the invariant measure of the process with respect to its drift and, second, systems of interacting particles, yielding a criterion for dimension-free Poincaré inequalities and quantitative long-time convergence for non-linear McKean-Vlasov type processes.
