# stochastic calculus – reflection coupling of Brownian and Levy’s characterization

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!