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性质，这些过程是低记忆力的随机过程（知道现在、未来和过去是独立的）。马尔可夫过程的类别是。相当大，我主要对分段确定性马尔可夫过程（PDMP）和谱负Lévy过程（SNLP）的研究感兴趣，后者是具有独立且平稳增量的马尔可夫过程，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，我研究的是单个线性轨迹的速度-位置随机过程的长期行为。过程以及相互作用的过程系统的情况下，由于位置和速度之间的强烈依赖性，即使使用分析工具进行研究，这些过程也很难研究，这使得锯齿形和台球的研究非常具有挑战性。流程是非常好的“玩具”模型，可以考虑更复杂的模型。我也对马尔可夫过程的波动感兴趣，由于它们的特性，Lévy 过程非常适合，我的目标是继续开发 Lévy 过程的结果。将结果扩展到其他过程，例如 Lévy 过程的马尔可夫变换 事实上，有大量关于 SNLP 波动的文献，特别是在精算科学的破产理论中，但类似的问题也出现在生物学中，特别是在毒理学中。 Lévy 过程不再适用于对这些现象进行建模。