论文标题
关于McKean-Vlasov方程的不变概率度量的稳定性
On the stability of the invariant probability measures of McKean-Vlasov equations
论文作者
论文摘要
我们研究了一些McKean-Vlasov随机微分方程的长期行为，该方程用于模拟大量相互作用剂的演变。我们提供条件，以确保不变概率度量的局部稳定性。狮子衍生物以一种新颖的方式使用来获得我们的稳定性标准。我们在$ \ mathbb {r}^d $上获得非本地McKean-Vlasov方程的结果，以及在卷积中给出相互作用内核的圆环上的McKean-Vlasov方程的结果。在$ \ mathbb {r}^d $上，我们证明了分析功能的根的位置决定了稳定性。在圆环上，我们的稳定性标准涉及相互作用内核的傅立叶系数。在这两种情况下，我们都证明了Wasserstein Metric $ W_1 $的收敛性，并具有指数的收敛速度。
We study the long-time behavior of some McKean-Vlasov stochastic differential equations used to model the evolution of large populations of interacting agents. We give conditions ensuring the local stability of an invariant probability measure. Lions derivatives are used in a novel way to obtain our stability criteria. We obtain results for non-local McKean-Vlasov equations on $\mathbb{R}^d$ and for McKean-Vlasov equations on the torus where the interaction kernel is given by a convolution. On $\mathbb{R}^d$, we prove that the location of the roots of an analytic function determines the stability. On the torus, our stability criterion involves the Fourier coefficients of the interaction kernel. In both cases, we prove the convergence in the Wasserstein metric $W_1$ with an exponential rate of convergence.
