论文标题
粘性非线性SDE和McKean-Vlasov方程的收敛
Sticky nonlinear SDEs and convergence of McKean-Vlasov equations without confinement
论文作者
论文摘要
我们开发了一种新的方法来研究McKean意义上非线性随机微分方程解决方案的长时间行为，以及对于相应的平均场粒子系统近似值的混乱传播。我们的方法基于两个方程解决方案之间的粘合耦合。我们表明，两个副本之间的距离过程由对一维非线性随机微分方程的解决方案主导，其粘性边界为零。然后仔细分析了新的方程式。特别是，我们表明主导方程式具有相变。在零以零次数是唯一不变的概率度量的制度中，我们证明对一维方程和原始非线性SDE的指数收敛均达到了平衡。同样，通过componentise粘性耦合和与一个尺寸非线性SDE的系统进行比较，与具有粘性边界为零的一个尺寸非线性SDE进行比较，显示了混乱的传播。该方法适用于没有限制潜力的方程式和非梯度类型的相互作用项。
We develop a new approach to study the long time behaviour of solutions to nonlinear stochastic differential equations in the sense of McKean, as well as propagation of chaos for the corresponding mean-field particle system approximations. Our approach is based on a sticky coupling between two solutions to the equation. We show that the distance process between the two copies is dominated by a solution to a one-dimensional nonlinear stochastic differential equation with a sticky boundary at zero. This new class of equations is then analyzed carefully. In particular, we show that the dominating equation has a phase transition. In the regime where the Dirac measure at zero is the only invariant probability measure, we prove exponential convergence to equilibrium both for the one-dimensional equation, and for the original nonlinear SDE. Similarly, propagation of chaos is shown by a componentwise sticky coupling and comparison with a system of one dimensional nonlinear SDEs with sticky boundaries at zero. The approach applies to equations without confinement potential and to interaction terms that are not of gradient type.
