Quantitative averaging of Markov-modulated additive functionals and Wentzell boundary homogenization

Source abstract

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.