Feynman-Kac formula under a finite entropy condition

TL;DR

This paper extends the Feynman-Kac formula to solutions of certain parabolic PDEs with irregular coefficients under a finite entropy condition, using probabilistic methods involving semimartingales and stochastic calculus.

Contribution

It introduces an extended notion of solution for parabolic equations with irregular coefficients, broadening the applicability of the Feynman-Kac representation under finite entropy constraints.

Findings

Feynman-Kac formula represents solutions under finite entropy conditions.

Extended solutions are trajectorial and involve semimartingale extensions.

Probabilistic approach uses stochastic derivatives, Girsanov's theorem, and Hamilton-Jacobi-Bellman equations.

Abstract

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 $\log g$.