Long-time behaviour and fluctuations of Markov processes

我的研究重点是马尔可夫过程的性质，马尔可夫过程是具有低记忆的随机过程（知道现在，未来和过去是独立的）。这些过程经常用于模拟生物、物理或精算现象。马尔可夫过程的类是相当大的，我主要感兴趣的研究分段确定性马尔可夫过程（Pestrian）和谱负Lévy过程（SNLP），后者是马尔可夫过程的独立和平稳的增量和没有积极的跳跃。长期行为和涨落问题是马尔可夫理论的核心。 更确切地说，我的工作的速度-位置随机过程的分段线性轨迹的长期行为，在一个单一的过程的情况下，在一个相互作用的系统的过程。由于位置和速度之间的强烈依赖性，这些过程很难研究，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很好的“玩具”模型，然后考虑更复杂的模型。 我对马尔可夫过程的波动也很感兴趣。由于它们的性质，Lévy过程很好地适应，我的目标是继续发展Lévy过程的结果，但也将结果扩展到其他过程，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，特别是精算学中的破产理论，但类似的问题出现在生物学中，特别是在毒理学中，Lévy过程不再适用于模拟这些现象。

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.