# Brownian motion

Check that the process $A_t=B_{t_0+t}-B_{t_0}$ satisfies the characteristic properties of Brownian motion: $A_0=0$, $A_t$ is Gaussian with mean 0 and variance $t$, $A$ has stationary independent increments, and is almost surely continuous (maybe you know another characterization of B.M. from your lecture; then adapt the sentence). All of these properties are immediate consequences of the same properties for $B$.