On pairs of complementary transmission conditions and on approximation of skew Brownian motion by snapping-out Brownian motions

Abstract

Following our previous work on `perpendicular' boundary conditions, we show that transmission conditions \[ f'(0-)=α(f(0+)-f(0-)), \quad f'(0+)=β(f(0+)-f(0-)),\] describing so-called snapping out Brownian motions on the real line, are in a sense complementary to the transmission conditions \[f(0-)=-f(0+), \quad f''(0+) =αf'(0-)+βf'(0+). \] As an application of the analysis leading to this result, we also provide a deeper semigroup-theoretic insight into the theorem saying that as the coefficients $\alpha$ and $\beta$ tend to infinity but their ratio remains constant, the snapping-out Brownian motions converge to a skew Brownian motion. In particular, the transmission condition \[ αf'(0+) = βf'(0-), \] that characterizes the skew Brownian motion turns out to be complementary to \[ f(0-) = - f(0+), βf'(0+)=- αf'(0-). \]