论文标题
fokker-Planck方程的收敛速度与限制漂移
Decay rates of convergence for Fokker-Planck equations with confining drift
论文作者
论文摘要
我们考虑在整个欧几里得空间中的fokker-planck方程，在征收过程的驱动下，在限制漂移的作用下，如经典的Ornstein-ulhenbeck模型。我们引入了一种新的PDE方法，以在征费过程的某些扩散率条件下，随着时间的流逝，随着时间的流逝，将指数或次指数的衰减速率（随着时间的流逝）作为模型示例。我们的方法取决于伴随问题的长期振荡估计，并适用于局部和非局部扩散的（可能的叠加），以及强烈或弱限制的漂移。我们的结果从统一的角度扩展了许多以前基于不同分析或概率方法的作品，并具有多个有趣的联系。一方面，我们在用于长期行为的（非线性）PDE方法之间建立了联系，而汉密尔顿 - 雅各比方程的长期行为与fokker-planck方程的衰减估计值之间的链接；另一方面，我们为某些振荡衰减估计值提供了一种纯粹的分析方法，这些方法仅通过概率耦合方法才能获得。
We consider Fokker-Planck equations in the whole Euclidean space, driven by Levy processes, under the action of confining drifts, as in the classical Ornstein-Ulhenbeck model. We introduce a new PDE method to get exponential or sub-exponential decay rates, as time goes to infinity, of zero average solutions, under some diffusivity condition on the Levy process, which includes the fractional Laplace operator as a model example. Our approach relies on the long time oscillation estimates of the adjoint problem and applies to (the possible superposition of) both local and nonlocal diffusions, as well as to strongly or weakly confining drifts. Our results extend, with a unifying perspective, many previous works based on different analytic or probabilistic methods, with several interesting connections. On one hand, we make a link between the (nonlinear) PDE methods used for the long time behavior of Hamilton-Jacobi equations and the decay estimates of Fokker-Planck equations; on another hand, we give a purely analytical approach towards some oscillation decay estimates which were obtained so far only with probabilistic coupling methods.
