We have $$ \begin{equation*} w(x=\pm \infty,t)=0 \hspace{1cm} \frac{\partial^{n}w(x,t)}{\partial x^n}|_{x=\pm\infty} = 0, \end{equation*} $$ implying that when we study the time-derivative \( \partial\langle x(t)\rangle/\partial t \), we obtain after integration by parts and using Eq. (5) $$ \frac{\partial \langle x\rangle}{\partial t} = \int_{-\infty}^{\infty}x\frac{\partial w(x,t)}{\partial t}dx= D\int_{-\infty}^{\infty}x\frac{\partial^2w(x,t)}{\partial x^2}dx, $$ leading to $$ \frac{\partial \langle x\rangle}{\partial t} = Dx\frac{\partial w(x,t)}{\partial x}|_{x=\pm\infty}- D\int_{-\infty}^{\infty}\frac{\partial w(x,t)}{\partial x}dx. $$