Brownian Occupation Measures, Compactness and Large Deviations
Brownian Occupation Measures, Compactness and Large Deviations
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布朗占用测度、紧性和大偏差
DOI：
10.1214/15-aop1065
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发表时间：
2014
期刊：
影响因子：
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通讯作者：
S. Varadhan
中科院分区：
文献类型：
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作者：
Chiranjib Mukherjee;S. Varadhan
在证明大偏差估计时，开集的下界和紧集的上限本质上是局部估计。另一方面，闭集的上限是全局的，需要空间的紧致性或指数紧度估计来建立它。在处理 $d$ 维布朗运动的占据测度 $L_t(A)=\frac{1}{t}\int_0^t{\1}_A(W_s) \d s$ 时，它不是正循环，不存在指数紧度的可能性。概率分布空间 $\mathcal {M}_1(\R^d)$ 可以通过用模糊拓扑替换通常的弱收敛拓扑来压缩，其中该空间被视为具有紧支持的连续函数的对偶。这本质上是通过在 $\infty$ 处添加一个点来对 $\R^d$ 进行一点压缩，从而通过允许一些质量逃逸到 $\infty$ 处的点来导致 $\mathcal M_1(\R^d)$ 的压缩。如果仅使用连续且在 $\infty$ 处消失的测试函数，则通过忽略 $\infty$ 处的质量，压缩会导致子概率分布空间 $\mathcal {M}_{\le 1}(\R^d)$。
这种压缩的主要缺点是它忽略了潜在的平移不变性。更明确地说，我们可能对平移群 $\R^d$ 对 $\mathcal M_1(\R^d)$ 作用下的轨道 $\widetilde{\mathcal M}_1=\widetilde{\mathcal M}_1(\R^d)$ 的等价类空间感兴趣。对于一些问题，压缩这个轨道空间是很自然的。我们将提供这样的紧化，证明那里的大偏差原理并给出相关问题的应用。
In proving large deviation estimates, the lower bound for open sets and upper bound for compact sets are essentially local estimates. On the other hand, the upper bound for closed sets is global and compactness of space or an exponential tightness estimate is needed to establish it. In dealing with the occupation measure $L_t(A)=\frac{1}{t}\int_0^t{\1}_A(W_s) \d s$ of the $d$ dimensional Brownian motion, which is not positive recurrent, there is no possibility of exponential tightness. The space of probability distributions $\mathcal {M}_1(\R^d)$ can be compactified by replacing the usual topology of weak convergence by the vague toplogy, where the space is treated as the dual of continuous functions with compact support. This is essentially the one point compactification of $\R^d$ by adding a point at $\infty$ that results in the compactification of $\mathcal M_1(\R^d)$ by allowing some mass to escape to the point at $\infty$. If one were to use only test functions that are continuous and vanish at $\infty$ then the compactification results in the space of sub-probability distributions $\mathcal {M}_{\le 1}(\R^d)$ by ignoring the mass at $\infty$.
The main drawback of this compactification is that it ignores the underlying translation invariance. More explicitly, we may be interested in the space of equivalence classes of orbits $\widetilde{\mathcal M}_1=\widetilde{\mathcal M}_1(\R^d)$ under the action of the translation group $\R^d$ on $\mathcal M_1(\R^d)$. There are problems for which it is natural to compactify this space of orbits. We will provide such a compactification, prove a large deviation principle there and give an application to a relevant problem.