Feynman-Kac formula under a finite entropy condition

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 $\log g$.