# Math Help - Brownian motion

1. ## Brownian motion

Could someone help me with this:
Let $\ B_{t}\$ be Brownian motion and $\ t_{0}> 0\$.Prove that $B_{t_{0}+t}-B_{t_{0}}\$ is Brownian motion.

2. 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 $B$.