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

Following our previous work on `perpendicular’ boundary conditions, we show that transmission conditions $$f^{\mathrm{\prime }}(0-)=\alpha (f(0+)-f(0-)),f^{\mathrm{\prime }}(0+)=\beta (f(0+)-f(0-)),$$ describing so-called snapping out Brownian motions on the real line, are in a sense complementary to the transmission conditions $$f(0-)=-f(0+),f^{\mathrm{\prime }\mathrm{\prime }}(0+)=\alpha f^{\mathrm{\prime }}(0-)+\beta f^{\mathrm{\prime }}(0+)\mathrm{.}$$ 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 $$\alpha f^{\mathrm{\prime }}(0+)=\beta f^{\mathrm{\prime }}(0-),$$ that characterizes the skew Brownian motion turns out to be complementary to $$f(0-)=-f(0+),\beta f^{\mathrm{\prime }}(0+)=-\alpha f^{\mathrm{\prime }}(0-)\mathrm{.}$$