  Motivated by entropic optimal transport, we investigate an extended notion of
solution to the parabolic equation $( \partial_t + b\cdot \nabla + \Delta _{
a}/2 +V)g
  =0$ with a final boundary condition. It is well-known that the viscosity
solution $g$ of this PDE is represented by the Feynman-Kac formula when the
drift $b$, the diffusion matrix $a$ and the scalar potential $V$ are regular
enough and not growing too fast. In this article, $b$ and $V$ are not assumed
to be regular and their growth is controlled by a finite entropy condition,
allowing for instance $V$ to belong to some Kato class. We show that the
Feynman-Kac formula represents a solution, in an extended sense, to the
parabolic equation. This notion of solution is trajectorial and expressed with
the semimartingale extension of the Markov generator $ b\cdot \nabla + \Delta
_{ a}/2.$ Our probabilistic approach relies on stochastic derivatives,
semimartingales, Girsanov's theorem and the Hamilton-Jacobi-Bellman equation
satisfied by $\log g$.
