  This paper tackles the issue of establishing a lower-bound on the asymptotic
ratio of survival probabilities between two different initial conditions,
asymptotically in time for a given Markov process with extinction. Such a
comparison is a crucial step in recent techniques for proving exponential
convergence to a quasi-stationary distribution. We introduce a weak form of the
Harnack inequality as the essential ingredient for such a comparison. This
property is actually a consequence of the convergence property we intend to
prove. Its complexity appears as the price to pay for the level of flexibility
required by our applications. We show in our illustrations how simply and
efficiently it can be used nonetheless. As illustrations, we consider two
continuous-time processes on R^d that do not satisfy the classical Harnack
inequalities, even in a local version. The first one is a piecewise
deterministic process while the second is a pure jump process with restrictions
on the directions of its jumps.
