arXiv 1504.048912015-04-19math.PR

算子标度高斯随机场的不变原理

Invariance principles for operator-scaling Gaussian random fields

Hermine Biermé, Olivier Durieu, Yizao Wang

摘要：

最近，Hammond和Sheffield引入了一个相关随机游动模型，该模型可扩展到具有长程依赖性的分数布朗运动。在本文中，我们考虑一个自然的推广，这个模型的尺寸$d\geq 2$。定义了一个$\mathbb Z^d$ -指标随机场，其依赖关系由一个顶点为$\mathbb Z^d$的随机图所控制，并研究了矩形集上随机场部分和的标度极限.一个有趣的现象出现了：取决于矩形集合沿不同方向增加的速度，在极限中产生不同的随机场。特别地，存在临界状态，其中极限随机场是算子缩放的并且继承离散模型的完全依赖结构，而在其它状态中，极限随机场具有至少一个方向，该方向具有不变增量或独立增量，不再反映离散模型中的依赖结构。极限随机场形成了一类一般的算子标度高斯随机场。研究了它们的增量和路径性质。

英文摘要：

Recently, Hammond and Sheffield introduced a model of correlated random walks that scale to fractional Brownian motions with long-range dependence. In this paper, we consider a natural generalization of this model to dimension $d\geq 2$. We define a $\mathbb Z^d$-indexed random field with dependence relations governed by an underlying random graph with vertices $\mathbb Z^d$, and we study the scaling limits of the partial sums of the random field over rectangular sets. An interesting phenomenon appears: depending on how fast the rectangular sets increase along different directions, different random fields arise in the limit. In particular, there is a critical regime where the limit random field is operator-scaling and inherits the full dependence structure of the discrete model, whereas in other regimes the limit random fields have at least one direction that has either invariant or independent increments, no longer reflecting the dependence structure in the discrete model. The limit random fields form a general class of operator-scaling Gaussian random fields. Their increments and path properties are investigated.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

摘要：

英文摘要：

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