We first verify the global well-posedness of the impulsive reaction-diffusion equations on infinite lattices. Then we establish that the generated process by the solution operators has a pullback attractor and a family of Borel invariant probability measures. Furthermore, we formulate the definition of statistical solution for the addressed impulsive system and prove the existence. Our results show that the statistical solution of the impulsive system satisfies merely the Liouville type theorem piecewise, and the Liouville type equation for impulsive system will not always hold true on the interval containing any impulsive point.

中文翻译：

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无限格上脉冲反应扩散方程的统计解和分段Liouville定理

我们首先验证了无限网格上脉冲反应扩散方程的整体适定性。然后，我们确定解决方案运营商生成的过程具有回撤吸引子和一系列Borel不变概率测度。此外，我们为所处理的脉冲系统制定了统计解的定义，并证明了该解的存在性。我们的结果表明，脉冲系统的统计解仅满足分段Liouville型定理，并且脉冲系统的Liouville型方程在包含任何脉冲点的区间上并不总是成立。