发表机构
Centro de Modelamiento Matemático (CNRS IRL2807), Universidad de Chile; Institute of Mathematical Finance, Ulm University（智利大学数学建模中心; 乌尔姆大学金融数学研究所）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文研究马尔可夫调制加性泛函的定量平均，给出误差估计并应用于快速切换Wentzell边界均匀化，获得边界条件及Feynman-Kac误差的收敛速率。
AI 中文摘要
设$A$为强马尔可夫过程的正连续加性泛函，$\alpha$为独立的有限状态马尔可夫链。对于有界$h$，我们研究\\[ J_\varepsilon(t)=\int_0^t\bigl(h(\alpha_{s/\varepsilon})-\pi(h)\bigr)\\,dA_s. \\] 若$\sup_y \mathrm E_y A_t\le C_T t^\vartheta$，则对于由基础过程决定的每个有界随机时间（不一定是停时），有\\[ \mathrm E|J_\varepsilon(\tau)|^2=O(\varepsilon^\vartheta),\qquad \mathrm E\sup_{t\le T}|J_\varepsilon(t)|^2 =O\\!\left(\varepsilon^\vartheta(1+|\log\varepsilon|)^2\right). \\] 对于反射布朗运动边界局部时，指数$1/2$在固定时刻是精确的。我们将此原理应用于具有快速切换Wentzell边界动力学的反射布朗运动。均匀化后的边界条件仍为Wentzell类型，其粘性和杀死系数由不变平均给出。对于有界Lipschitz可观测函数，淬火Feynman--Kac误差具有低于$1/8$的任意多项式速率，且在确定性时间上一致并具有期望一致范数。对于具有成比例杀死和粘性的光滑Neumann相容可观测函数，速率提升至$O(\varepsilon^{1/4})$，在时间一致估计中至多相差一个对数因子。我们还构造了切换过程，推导了淬火和退火的后向演化，并通过对偶性将估计转移到前向测度演化。对于前向演化，我们获得了有界Lipschitz距离下的定量收敛，以及光滑可观测函数（包括存活质量）的更优$1/4$速率。数值实验说明了有限尺度下的收敛性。
英文摘要
Let $A$ be a positive continuous additive functional of a strong Markov process and $α$ an independent finite-state Markov chain. For bounded $h$, we study \[ J_\varepsilon(t)=\int_0^t\bigl(h(α_{s/\varepsilon})-π(h)\bigr)\,dA_s . \] If $\sup_y \mathrm E_y A_t\le C_T t^\vartheta$, then for every bounded random time determined by the base process, not necessarily a stopping time, \[ \mathrm E|J_\varepsilon(τ)|^2=O(\varepsilon^\vartheta),\qquad \mathrm E\sup_{t\le T}|J_\varepsilon(t)|^2 =O\!\left(\varepsilon^\vartheta(1+|\log\varepsilon|)^2\right). \] For reflected Brownian boundary local time, the exponent $1/2$ is sharp at fixed time. We apply this principle to reflected Brownian motion with rapidly switching Wentzell boundary dynamics. The homogenized boundary condition is again of Wentzell type, with stickiness and killing coefficients given by invariant averages. For bounded Lipschitz observables, the quenched Feynman--Kac error has every polynomial rate below $1/8$, uniformly over deterministic times and in expected uniform norm. For smooth Neumann-compatible observables with proportional killing and stickiness, the rate improves to $O(\varepsilon^{1/4})$, up to a logarithmic factor in the uniform-in-time estimate. We also construct the switching process, derive quenched and annealed backward evolutions, and transfer the estimates by duality to the forward measure evolution. For the forward evolution, we obtain quantitative convergence in bounded-Lipschitz distance and the sharper $1/4$ rate for smooth observables, including surviving mass. Numerical experiments illustrate the finite-scale convergence.
Comments52 pages, 3 figures
