L∞-convergence to a quasi-stationary distribution

For general absorbed Markov processes Xt with absorption time τ∂ and quasi-stationary distribution (QSD) π, we introduce a Dobrushin-type criterion providing for exponential convergence in L∞(π) of the density dLμ(Xt|τ∂>t) dπ. We establish this for all initial conditions μ, possibly singular with respect to π, under an additional “anti-Dobrushin” condition. These criteria are general - our results provide for convergence to a QSD where it was previously an open problem for any notion of convergence. Moreover these probabilistic criteria allow us to compare the distribution conditioned on survival at a fixed time with the QSD, uniformly over all initial conditions and over the whole space. This provides for a parabolic boundary Harnack inequality, with an explicit expression for the constant. We thereby obtain a parabolic boundary Harnack inequality under parabolic Hörmander conditions. We apply the above to the following examples (with absorption): degenerate diffusions satisfying parabolic Hörmander conditions, 1+1-dimensional Langevin dynamics, random diffeomorphisms, 2-dimensional neutron transport dynamics, and certain classes of piecewise-deterministic Markov process. Our proofs rely on consideration of the time-reversal of an absorbed Markov process.