AI 中文总结
研究大象随机游走的拉普拉斯变换的长时间渐近行为，通过建立递推关系找到由输运型偏微分方程控制的生成函数，借助施瓦茨 - 克里斯托费尔映射实现解析控制，用奇点分析方法估计拉普拉斯变换。
AI 中文摘要
我们推导了大象随机游走（ERW）的拉普拉斯变换的长时间渐近行为。ERW是简单随机游走的推广，过去根据记忆参数影响过程的未来演化。其转移核既非空间齐次也非时间齐次，第n步的转移概率可写为n和时间n - 1时游走位置的函数。其拉普拉斯变换的递推关系使我们能找到由输运型偏微分方程控制的生成函数，其解可用施瓦茨 - 克里斯托费尔映射表示，从而实现精确的解析控制，进而应用奇点分析方法估计拉普拉斯变换。
英文摘要
We derive the long time asymptotic behavior of the Laplace transform of the Elephant Random Walk (ERW). The ERW is a generalization of the Simple Random Walk in which the past influences the future evolution of the process according to some memory parameter. Its transition kernel is neither space nor time homogeneous and its transition probability at the $n$th step can be written as a function of $n$ and the position of the walk at time $n - 1$. A recurrence relation for its Laplace transform allows one to find a generating function governed by a transport-type PDE whose solution can be written in terms of a Schwarz-Christoffel mapping, allowing a precise analytic control. This enables to apply singularity analysis methods to estimate the Laplace transforms.
