Past exam of the stochastic differential equations and Brownian motion in complex analysis course of the University of Oxford / 2010 / A / 2 / a

Standard Brownian motion is centered, so $m_t=0$. If $s\le t$, write $B_t=B_s+(B_t-B_s)$; the final increment is independent of $B_s$ and has mean zero. ThusThis is the Brownian covariance kernel. Together with joint Gaussianity and continuous paths, it gives the Gaussian-process characterization of Brownian motion.