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

TL;DR

This paper explores pairs of complementary transmission conditions for snapping-out Brownian motions and demonstrates their convergence to skew Brownian motion, providing new semigroup-theoretic insights into this approximation process.

Contribution

It introduces and analyzes pairs of complementary transmission conditions for snapping-out Brownian motions and links them to skew Brownian motion through a semigroup-theoretic framework.

Findings

Transmission conditions are complementary and characterize snapping-out Brownian motions.

As parameters tend to infinity, snapping-out Brownian motions converge to skew Brownian motion.

The transmission condition for skew Brownian motion is complementary to a specific boundary condition.

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 \[ f(0-) = - f(0+), \beta f'(0+)=- \alpha f'(0-). \]