Long-time behaviour and fluctuations of Markov processes

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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.