We simplify the above by introducing the moments

$$ M_n=\frac{1}{\tau}\int_{-\infty}^{\infty} \xi^nW(\mathbf{x}+\xi,\tau|\mathbf{x})d\xi= \frac{\langle [\Delta x(\tau)]^n\rangle}{\tau}, $$

resulting in

$$ \frac{\partial W(\mathbf{x},s|\mathbf{x}_0)}{\partial s}= \sum_{n=1}^{\infty}\frac{(-\xi)^n}{n!}\frac{\partial^n}{\partial x^n} \left[W(\mathbf{x},s|\mathbf{x}_0)M_n\right]. $$