Long-time behaviour and fluctuations of Markov processes

马尔可夫过程的长期行为和波动

• 批准号: RGPIN-2020-07239
• 负责人: GUÉRIN, HÉLENE
• 金额: $ 1.31万
• 依托单位: Université du Québec à Montréal
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2021
• 资助国家: 加拿大
• 起止时间: 2021-01-01 至 2022-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=741040
• 关键词: Long; time; behaviour; fluctuations; Markov

My research focus on properties of Markov processes, which are stochastic processes with low memory (knowing the present, the future and the past are independent). These processes are frequently used to model biological, physical or actuarial phenomenons. The class of Markov processes is quite large and I mainly interested in the study of Piecewise Deterministic Markov Processes (PDMP) and Spectrally Negative Lévy Processes (SNLP), the latter being Markov processes with independent and stationary increments and no positive jumps. The questions of the long-time behaviour and of the fluctuations are central in Markov theory. More precisely, I work on the long-time behaviour of speed-position stochastic processes with piecewise linear trajectories, in the case of a single process and in the case of an interacting system of processes. Those processes are difficult to study due to the strong dependence between the position and the velocity, even when they studied by analytical tools, which makes the study really challenging. The zigzag and the billiard processes are quite good "toy" models to then consider more complex models. I am also interested in the fluctuations of Markov processes. Due to their properties, the Lévy processes are well adapted and my aim is to continue to develop results for Lévy processes, but also to extend the results to other processes such as Markovian transformations of Lévy processes. Indeed, there is a wide literature on the fluctuations of SNLPs, in particular in ruin theory in actuarial science, but similar questions emerge in biology, particularly in toxicology, where the Lévy processes are no more adapted to model those phenomenons.

我的研究重点是马尔可夫过程的性质，这是一个具有低记忆的随机过程（知道现在，未来和过去是独立的）。这些过程经常被用来模拟生物、物理或精算现象。马尔可夫过程的类别是相当大的，我主要对分段确定性马尔可夫过程（PDMP）和频谱负马尔可夫过程（SNLP）的研究感兴趣，后者是具有独立和平稳增量且没有正跳跃的马尔可夫过程。长期行为和波动的问题是马尔可夫理论的核心。更准确地说，我研究的是速度-位置随机过程的长期行为，在单个过程和过程相互作用系统的情况下，具有分段线性轨迹。这些过程很难研究，因为位置和速度之间有很强的依赖性，即使用分析工具研究也是如此，这使得研究非常具有挑战性。之字形和台球过程是非常好的“玩具”模型，然后再考虑更复杂的模型。我对马尔可夫过程的波动也很感兴趣。由于它们的特性，lsamvy过程具有很好的适应性，我的目标是继续开发lsamvy过程的结果，同时也将结果扩展到其他过程，例如lsamvy过程的马尔可夫变换。事实上，关于单核苷酸多态性波动的文献很多，特别是精算科学的破产理论，但在生物学，特别是毒理学中出现了类似的问题，在那里，lsamvy过程不再适用于模拟这些现象。