From boundary random walks to Feller's Brownian Motions

We establish an invariance principle connecting boundary random walks on $\mathbb{N}$ with Feller’s Brownian motions on $[0,\mathrm{\infty })$. A Feller’s Brownian motion is a Feller process on $[0,\mathrm{\infty })$ whose excursions away from the boundary $0$ coincide with those of a killed Brownian motion, while its behavior at the boundary is characterized by a quadruple $(p_1,p_2,p_3,p_4)$. This class encompasses many classical models, including absorbed, reflected, elastic, and sticky Brownian motions, and further allows boundary jumps from $0$ governed by the measure $p_4$. For any Feller’s Brownian motion that is not purely driven by jumps at the boundary, we construct a sequence of boundary random walks whose appropriately rescaled processes converge weakly to the given Feller’s Brownian motion.