Some new results provide opportunities to ensure the exponential convergence to a unique quasistationary distribution in the total variation norm, for quite general strong Markov processes. Specifically, non-reversible processes with discontinuous trajectories are concerned, which seems to be a substantial breakthrough. Considering jumps driven by Poisson Point Processes in four different applications, we intend to illustrate the potential of these results and motivate an original yet apparently very technical criterion. Such criterion is expected to be much easier to verify than an implied property essential for our proof, namely a comparison of the asymptotic extinction rate between different initial conditions. Keywords : continuous-time and continuous-space Markov process , jumps , quasi-stationary distribution , survival capacity , Q-process , Harris recurrence 查看全文>>