自适应进化中奇异和退化扩散模型的大偏差
Large deviations for singular and degenerate diffusion models in adaptive evolution
摘要：
在种群的达尔文进化过程中，守时是一个重要的现象，即长时期的遗传停滞与短时期的快速进化变化交替。本文对自适应动力学理论中的扩散模型的吸引吸引区域变化序列给出了一个数学解释。这样的结果依赖于扩散过程的大偏差估计。主要困难在于这个扩散过程在空间的孤立点处具有退化的非Lipschitz扩散部分，在相同点处具有非连续漂移部分。然而，我们能够证明这些扩散的强存在性和强马尔可夫性，并给出条件下，路径唯一性保持。接下来，我们证明了一个大偏差原则，涉及的速率函数，它没有标准式的小噪声扩散，由于特定的奇异性的模型。最后，这一结果是用来获得渐近估计所需的时间退出吸引域，并确定该退出点更有可能发生。
英文摘要：
In the course of Darwinian evolution of a population, punctualism is an important phenomenon whereby long periods of genetic stasis alternate with short periods of rapid evolutionary change. This paper provides a mathematical interpretation of punctualism as a sequence of change of basin of attraction for a diffusion model of the theory of adaptive dynamics. Such results rely on large deviation estimates for the diffusion process. The main difficulty lies in the fact that this diffusion process has degenerate and non-Lipschitz diffusion part at isolated points of the space and non-continuous drift part at the same points. Nevertheless, we are able to prove strong existence and the strong Markov property for these diffusions, and to give conditions under which pathwise uniqueness holds. Next, we prove a large deviation principle involving a rate function which has not the standard form of diffusions with small noise, due to the specific singularities of the model. Finally, this result is used to obtain asymptotic estimates for the time needed to exit an attracting domain, and to identify the points where this exit is more likely to occur.
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。
