In many books on finance, the PDE solved by Feynman-Kac is often formulated by the following:
$$\begin{aligned} \frac{\partial}{\partial t}u(t,x)+\mathcal{L}u(t,x)&=V(t,x)u(t,x),\\ u(0,x)&=\varphi(x) \end{aligned}$$ where the generator is defined as $\mathcal{L}:=\mu(t,x)\frac{\partial}{\partial x}+\frac{1}{2}\sigma^2\frac{\partial^2}{\partial x^2}$, $\varphi$ and $V$ continuous, $V\geqslant0$. (One-dimensional case, cf. this wikipedia site and e.g. Definition 2.46 of Interest-Rate Management, R. Zagst.) Then the solution to this boundary problem (if exists) can be expressed as $$u(t,x)=\mathbb{E}_{\mathbb{Q}}\Big[\varphi(X_t)\exp\Big(-\int_0^tV(t,X_s)ds\Big)|X_t=x\Big],$$ where $\mathbb{Q}$ is the equivalent martingale measure, that is, the process $(X_t)$ is a martingale under $\mathbb{Q}$.
However, in other books (actually the lectures I have taken, or cf. this wikipedia) it is formulated as:
$$\begin{aligned} \frac{\partial}{\partial t}u(t,x)&=\mathcal{L}u(t,x)-V(t,x)u(t,x),\\ u(0,x)&=\varphi(x) \end{aligned}$$ with the same functions and the same solution. The derivative on $t$ of $u$ stands on different sides of the equation...
My Question: Are these two formulations equivalent? How can we prove it?
