On a domain $(0;T) \times (0;L)$, I have the equation
$$0 = \partial_t f - \frac{\sigma^2}{2}\cdot \partial_{xx}^2 f + \partial_x(p \cdot f),$$
where $p$ is fixed real function on the domain. I also have the boundary conditions $f(t,0) = 0$ and $f(0,x) = F(x)$, which is a known non-negative function.
Can I, under any conditions (which do not force $p$ to be zero everywhere), guarantee that any solution $f$ to this equation is non-negative or at least bounded below by a certain negative constant?