Regularity of laws and ergodicity of hypoelliptic SDEs driven by rough paths
Regularity of laws and ergodicity of hypoelliptic SDEs driven by rough paths
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DOI：
10.1214/12-aop777
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发表时间：
2011-04
影响因子：
2.3
通讯作者：
Martin Hairer;N. Pillai
中科院分区：
文献类型：
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作者：
Martin Hairer;N. Pillai
我们考虑由粗糙路径驱动的微分方程，并研究其规律性及其长时间行为。特别地，我们专注于当驱动噪声是一个粗糙的路径值分数布朗运动的Hurst参数H2（1，1）的情况下。我们在这项工作中的贡献是双重的。首先，当驱动向量场满足Hormander的“李括号条件”时，我们给出了Malliavin矩阵的逆的明确的定量界.途中，我们提供了一个新的“确定性”版本的诺里斯引理驱动的粗糙路径的微分方程。这个结果，加上线性化方程有矩的假设，将产生关于勒贝格测度的跃迁定律有光滑密度。我们的第二个主要结果表明，在Hormander条件下，由分数布朗运动驱动的粗糙微分方程的解具有适当的强Feller性质.在一个标准的可控性条件下，这意味着他们承认一个唯一的固定的解决方案，是物理的意义上，它不“展望未来”。
We consider differential equations driven by rough paths and study the regularity of the laws and their long time behavior. In particular, we focus on the case when the driving noise is a rough path valued fractional Brownian motion with Hurst parameter H 2 ( 1 , 1 ). Our contribution in this work is twofold. First, when the driving vector fields satisfy Hormander's c elebrated "Lie bracket condition", we derive explicit quantitative bounds on the i nverse of the Malliavin ma- trix. En route to this, we provide a novel "deterministic" version of Norris's lemma for differential equations driven by rough paths. This result, with the added assumption that the linearized equation has moments, will then yield that the transition laws have a smooth density with respect to Lebesgue measure. Our second main result states that under Hormander's condi tion, the solutions to rough differential equations driven by fractional Brownian motion with H 2 ( 1 , 1 ) enjoy a suitable version of the strong Feller property. Under a standard controllability condition, this implies that they admit a unique stationary solution that is physical in the sense that it does not "look into the future".