论文标题
$ l^{\ infty} $ - 收敛到准平台分布
$L^{\infty}$-convergence to a quasi-stationary distribution
论文作者
论文摘要
对于一般吸收了马尔可夫的过程$（x_t）_ {0 \ leq t <τ_ {\ partial}} $具有准安置分布（qsd）$π$和吸收时间$τ_ {\ partial} $ $ l^{\ infty}（π）$ as $ t \ rightArrow \ infty \ infty $的密度$ \ frac {d \ Mathcal {l}_μm（x_t \ lvertτ_{\ partial}> t）}} {dπ} $。我们在所有初始条件$μ$的情况下建立了这一点，在额外的``抗Dobrushin''条件下，可能相对于$π$相互奇异。这取决于不等式，我们可以将$ \ MATHCAL {l}_μ（x_t \ lvertτ_{\ partial}> t）与QSD $π$，在所有初始条件下，在整个空间中，在上述条件下，在上述条件下，在整个空间中均匀。在PDE级别上，这些概率标准为相应的Kolmogorov方程提供了抛物线边界不等式（带有其他警告）。除了纤维化设置外，这些比较不等式在相应的Fokker-Planck方程是一阶的情况下还获得了这些比较不等式，并且可能存在不连续的解决方案。作为推论，我们获得了submarkovian过渡内核具有有限的，正右函数的足够条件，而无需任何操作员都紧凑。我们将上述内容应用于以下示例（具有吸收）：马尔可夫在有限状态空间上的过程，不满意的寄生寄生虫条件，$ 1+1 $ 1 $ -Dimensional langevin dynamilics，随机的差异性，$ 2 $ $ 2 $ - 数量的中性动态动力学和某些零件构成的标记。在最后一个情况下，以前对QSD的收敛性以前是任何收敛概念所不明的。我们的证明与早期的工作完全不同，依靠对被吸收的马尔可夫过程的时间倒流。
For general absorbed Markov processes $(X_t)_{0\leq t<τ_{\partial}}$ having a quasi-stationary distribution (QSD) $π$ and absorption time $τ_{\partial}$, we introduce a Dobrushin-type criterion providing for exponential convergence in $L^{\infty}(π)$ as $t\rightarrow\infty$ of the density $\frac{d\mathcal{L}_μ(X_t\lvert τ_{\partial}>t)}{dπ}$. We establish this for all initial conditions $μ$, possibly mutually singular with respect to $π$, under an additional ``anti-Dobrushin'' condition. This relies on inequalities we obtain comparing $\mathcal{L}_μ(X_t\lvert τ_{\partial}>t)$ with the QSD $π$, 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 Hörmander 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.
