$L^{\infty}$-convergence to a quasi-stationary distribution

Abstract

For general absorbed Markov processes $(X_t)_{0\leq t<\tau_{\partial}}$ having a quasi-stationary distribution (QSD) $\pi$ and absorption time $\tau_{\partial}$, we introduce a Dobrushin-type criterion providing for exponential convergence in $L^{\infty}(\pi)$ as $t\rightarrow\infty$ of the density $\frac{d\mathcal{L}_{\mu}(X_t\lvert \tau_{\partial}>t)}{d\pi}$. We establish this for all initial conditions $\mu$, possibly mutually singular with respect to $\pi$, under an additional ``anti-Dobrushin'' condition. This relies on inequalities we obtain comparing $\mathcal{L}_{\mu}(X_t\lvert \tau_{\partial}>t)$ with the QSD $\pi$, uniformly over all initial conditions and over the whole space, under the aforementioned conditions. On a PDE level, these probabilistic criteria provide a parabolic boundary Harnack inequality (with an additional caveat) for the corresponding Kolmogorov forward equation. In addition to hypoelliptic settings, these comparison inequalities are thereby obtained in a setting where the corresponding Fokker-Planck equation is first order, with the possibility of discontinuous solutions. As a corollary, we obtain a sufficient condition for a submarkovian transition kernel to have a bounded, positive right eigenfunction, without requiring that any operator is compact. We apply the above to the following examples (with absorption): Markov processes on finite state spaces, degenerate diffusions satisfying parabolic H\"{o}rmander conditions, $1+1$-dimensional Langevin dynamics, random diffeomorphisms, $2$-dimensional neutron transport dynamics, and certain piecewise-deterministic Markov processes. In the last case, convergence to a QSD was previously unknown for any notion of convergence. Our proof is entirely different to earlier work, relying on consideration of the time-reversal of an absorbed Markov process.