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-)=\alpha(f(0+)-f(0-)), \quad f'(0+)=\beta(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+) =\alpha f'(0-)+\beta f'(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 [ \alpha f'(0+) = \beta f'(0-), ] that characterizes the skew Brownian motion turns out to be complementary to [ f(0-) = - f(0+), \beta f'(0+)=- \alpha f'(0-). ]