Given the Ito diffusion
$$ dX(t)= \mu dt + \sigma dW(t) $$
And a function $F(X(t),t)$, then by Ito Lemma
$$ dF = \left(\mu \frac{ \partial F }{ \partial x } + \frac{ \sigma^2 }{2} \frac{\partial^2 F }{\partial x^2} \right) dt + \sigma \frac{\partial F}{\partial x} dW $$
(Leaving out the arguments for simplicity)
Then the density $p(x,t)$ of the function $F(X(t),t)$ is obtained from the Forward Kolmogorov equation
$$ \frac{\partial p(x,t)}{\partial t} = - \frac{\partial (\mu p(x,t))}{\partial x} + \frac{1}{2}\frac{\partial^2 (\sigma^2 p(x,t))}{\partial x^2} $$
Now to derive the FK, I take the derivative of the expectation and perform a series of integration by parts. During this step I end up with a bunch of integrals which I set to zero.
Are the following equalities/identities true? If so why?
(1) $$ p(x,t) \mu F \left. \right|_{\infty }^{-\infty}=0 $$
(2) $$ p(x,t) \intop\nolimits_{-\infty}^{\infty} \frac{\sigma^2}{2}\frac{\partial^2 F}{\partial x^2} dx =0 $$
(3) $$ p(x,t) \frac{1}{2}\frac{\partial^2 (\sigma^2 p(x,t))}{\partial x^2} F \left. \right|_{-\infty }^{\infty}=0 $$
Thank you in advance!
Notes: I understand that when time goes to infinity, the probability density and the derivatives converge to zero - but I still do not understand why integrating over the state space should also equal to zero. Please leave a comment if I should write out the whole derivation.