arXiv 1606.060122016-06math.PR

扩散和McKean-Vlasov过程的定量Harris型定理

Quantitative Harris type theorems for diffusions and McKean-Vlasov processes

Andreas Eberle, Arnaud Guillin, Raphael Zimmer

摘要：

考虑\begin{align*} dX_t\ =\ b(X_t)dt\, +\, dB_t. \end{align*}型$\mathbb{R}^d$值扩散过程.在几何漂移条件下，建立了转移核在Kantorovich（$L^1$ Wasserstein）距离上的显式常数压缩.我们的结果是Hairer和Mattingly的Harris定理的推广。特别地，它们不依赖于小集合条件。相反，我们结合李雅普诺夫函数与反射耦合和凹距离函数。我们检索常数是明确的参数，可以计算出很少的努力，从单侧利普希茨条件的漂移系数和增长的一个选定的李雅普诺夫函数。结果包括指数收敛加权全变差规范，梯度界，遍历平均的界限，和Kantorovich收缩的非线性McKean-Vlasov扩散的情况下，足够弱，但不一定有界的非线性。我们还建立了定量的界限，假设一个亚几何漂移条件的亚几何遍历。

英文摘要：

We consider $\mathbb{R}^d$-valued diffusion processes of type \begin{align*} dX_t\ =\ b(X_t)dt\, +\, dB_t. \end{align*} Assuming a geometric drift condition, we establish contractions of the transitions kernels in Kantorovich ($L^1$ Wasserstein) distances with explicit constants. Our results are in the spirit of Hairer and Mattingly's extension of Harris' Theorem. In particular, they do not rely on a small set condition. Instead we combine Lyapunov functions with reflection coupling and concave distance functions. We retrieve constants that are explicit in parameters which can be computed with little effort from one-sided Lipschitz conditions for the drift coefficient and the growth of a chosen Lyapunov function. Consequences include exponential convergence in weighted total variation norms, gradient bounds, bounds for ergodic averages, and Kantorovich contractions for nonlinear McKean-Vlasov diffusions in the case of sufficiently weak but not necessarily bounded nonlinearities. We also establish quantitative bounds for sub-geometric ergodicity assuming a sub-geometric drift condition.

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