A second order analysis of McKean–Vlasov semigroups
A second order analysis of McKean–Vlasov semigroups
复制标题
McKean-Vlasov 半群的二阶分析
DOI：
--
复制
发表时间：
2019
期刊：
影响因子：
--
通讯作者：
P. Moral
中科院分区：
文献类型：
--
作者：
M. Arnaudon;P. Moral
我们提出了一个二阶微分方程来分析McKean-Vlasov扩散分布半群的正则性和稳定性。这种方法提供了二阶泰勒型扩展与剩余的发展半群以及与这类非线性扩散的随机流。给出了与这些展开式相关的积分微分算子的梯度和Hessian的Bismut-Elworthy-Li公式。文章还提供了明确的戴森-菲利普斯展开和这些积分微分算子的范数的精细分析。在一些自然的和容易验证的正则性条件下，我们得到了一系列关于时间范围的指数衰减不等式。我们说明了这些结果的影响与Alekseev-Gr{o}bner引理的二阶扩展的非线性测度值半群和相互作用扩散流。这种二阶微扰分析提供了几种混沌特性w.r.t.时间参数，包括偏差、波动误差估计以及指数浓度不等式。
We propose a second order differential calculus to analyze the regularity and the stability properties of the distribution semigroup associated with McKean-Vlasov diffusions. This methodology provides second order Taylor type expansions with remainder for both the evolution semigroup as well as the stochastic flow associated with this class of nonlinear diffusions. Bismut-Elworthy-Li formulae for the gradient and the Hessian of the integro-differential operators associated with these expansions are also presented. The article also provides explicit Dyson-Phillips expansions and a refined analysis of the norm of these integro-differential operators. Under some natural and easily verifiable regularity conditions we derive a series of exponential decays inequalities with respect to the time horizon. We illustrate the impact of these results with a second order extension of the Alekseev-Gr{o}bner lemma to nonlinear measure valued semigroups and interacting diffusion flows. This second order perturbation analysis provides direct proofs of several uniform propagation of chaos properties w.r.t. the time parameter, including bias, fluctuation error estimate as well as exponential concentration inequalities.
关于随机微分方程的 Ità-Alekseev-Gröbner 公式
DOI：
10.17185/duepublico/70569
发表时间：
2019
期刊：
影响因子：
--
作者：
A. Hudde
通讯作者：
A. Hudde