The result is $$ \frac{\partial \langle x\rangle}{\partial t} = 0. $$ This means in turn that \( \langle x\rangle \) is independent of time. If we choose the initial position \( x(t=0)=0 \), the average displacement \( \langle x\rangle= 0 \). If we link this discussion to a random walk in one dimension with equal probability of jumping to the left or right and with an initial position \( x=0 \), then our probability distribution remains centered around \( \langle x\rangle= 0 \) as function of time.