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 \[ f(0-) = - f(0+), \beta f'(0+)=- \alpha f'(0-). \]
- Publication:
-
arXiv e-prints
- Pub Date:
- March 2023
- DOI:
- 10.48550/arXiv.2303.10041
- arXiv:
- arXiv:2303.10041
- Bibcode:
- 2023arXiv230310041B
- Keywords:
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- Mathematics - Probability;
- Mathematics - Functional Analysis;
- 35B06;
- 46E05;
- 47D06;
- 47D07;
- 47D09