局部紧空间上Feller过程经验分布轨线的大偏差
Large deviations of the trajectory of empirical distributions of Feller processes on locally compact spaces
摘要：
本文研究局部紧度量空间上时齐Feller过程独立副本经验分布轨迹的大偏差行为。在我们能找到一个合适的核作为Feller过程的生成元的条件下，我们可以在这个核上定义测度的绝对连续轨线的概念.此外，我们定义的线性生成元和拉格朗日作为其勒让德变换的哈密尔顿。我们证明了大偏差原则，并表明，率函数可以分解为一个率函数的初始时间和一个积分的拉格朗日，有限的绝对连续的轨迹的措施。我们将这个结果应用于R^d上的扩散和Lévy过程，任意局部紧空间上具有有界跳核的纯跳过程以及离散相互作用粒子系统。对于扩散过程，该定理部分地扩展了非相互作用副本的Dawson和Gärtner定理，即它只对时间齐次过程成立，但另一方面它对具有退化扩散矩阵的过程成立。
英文摘要：
We study the large deviation behaviour of the trajectories of empirical distributions of independent copies of time-homogeneous Feller processes on locally compact metric spaces. Under the condition that we can find a suitable core for the generator of the Feller process, we are able to define a notion of absolutely continuous trajectories of measures in terms of some topology on this core. Also, we define a Hamiltonian in terms of the linear generator and a Lagrangian as its Legendre transform. We prove the large deviation principle and show that the rate function can be decomposed as a rate function for the initial time and an integral over the Lagrangian, finite only for absolutely continuous trajectories of measures. We apply this result for diffusion and Lévy processes on R^d, for pure jump processes with bounded jump kernel on arbitrary locally compact spaces and for discrete interacting particle systems. For diffusion processes, the theorem partly extends the Dawson and Gärtner theorem for non-interacting copies in the sense that it only holds for time-homogeneous processes, but on the other hand it holds for processes with degenerate diffusion matrix.
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