My question comes from page 3 of this manuscript, in which the authors mentioned that if $B_t$ is a $d$-dimensional Brownian motion and we define $$dB’_t = (mathrm{Id} – 2e_te^intercal_t),dB_t,$$ where $e_t$ is defined to be $X_t/|X_t|$ for some $X_t neq mathbf{0}$ and $mathbf{0}$ otherwise. They mentioned that this is the reflection coupling of two Brownian motions and that Levy’s characterization ensures that $B’_t$ is also a Brownian motion. May I know is there any intuitive explanation for the name “reflection coupling”? Also, how does the Levy’s characterization theorem applies to $B’_t$? Thank you very much!