The *position process* $x_t$ satisfies $$
x_t = x_0 + t v_0 + \int_0^t W_s ds \;.
$$ Because $\int_0^t W_s ds \sim \mathcal{N}(0,\frac{1}{3} t^3)$, a simple change of variables shows that $$
x_t \sim \mathcal{N}( x_0 + t v_0, \frac{1}{3} t^3) \;.
$$