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: 2022
• 资助国家: 加拿大
• 起止时间: 2022-01-01 至 2023-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=759320
• 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属性，马尔可夫过程是记忆力较低的随机过程（知道现在，未来和过去是独立的）。这些过程经常用于建模生物学，物理或实际现象。马尔可夫过程的类别非常大，我主要对研究确定性马尔可夫流程（PDMP）和频谱负Lévy过程（SNLP）的研究感兴趣，后者是具有独立和固定增量且无积极跳跃的马尔可夫过程。长期行为和波动的问题在马尔可夫理论中是核心。更确切地说，在单个过程和过程相互作用系统的情况下，我将使用分段线性轨迹的速度随机过程的长期行为。这些过程由于位置和速度之间的强烈依赖，即使通过分析工具进行了研究，因此很难研究这些过程，这使研究确实挑战了。曲折和台球过程是相当不错的“玩具”模型，可以考虑更复杂的模型。我也对马尔可夫流程的波动感兴趣。由于其特性，Lévy过程非常适合，我的目的是继续为Lévy过程开发结果，但也将结果扩展到其他过程，例如Markovian的Lévy流程转换。确实，有关于SNLP波动的广泛文献，尤其是在实际科学中的毁灭理论中，但是在生物学中出现了类似的问题，尤其是在毒理学中，在毒理学中，lévy过程不再适合模拟这些现象。