Apologies if this has been asked before... I came across the following relation:
if $$P(x_2, t_2 \mid x_1, t_1) = \frac{1}{\sqrt{2\pi\sigma^2(t_2-t_1)}}e^{-\frac{(x_2-x_1)^2}{2\sigma^2(t_2-t_1)}}$$ ($t_2 > t_1$) then we have the following identity, with $T>0$: $$P(x_1, T \mid x_0, 0)\delta(x_1-x_2)-P(x_1, T \mid x_0, 0)P(x_2, T \mid x_0, 0)$$ $$=\sigma^2 \int_0^T\int_x P(x, t \mid x_0, 0)\frac{\partial P(x_1, T \mid x, t)}{\partial x}\frac{\partial P(x_2, T \mid x, t)}{\partial x}dxdt$$
I tried several things but none came close to a beginning of a solution...
Thanks a lot!