08-SEP-2025
We study the Feynman–Kac formula in its general form with a killing potential, establishing the probabilistic representation of solutions to the heat equation $\partial_t u = \tfrac{1}{2}\sigma^2 \partial_{xx} u – c(x)\,u$ on a bounded domain with absorbing boundaries. The solution is given by the expectation $u(x,t) = \mathbb{E}\bigl[e^{-\int_0^t c(X_s)\,ds} f(X_t)\,\mathbf{1}_{\{\tau > t\}}\bigr]$, where $\tau$ is the first exit time and the exponential weight is the Feynman path integral with potential $c$. We prove the formula via Itô’s lemma, analyse how the killing rate $c(x)$ suppresses the solution, and establish the connection to the imaginary-time Schrödinger equation. Numerical experiments confirm the probabilistic representation against direct PDE solutions for quadratic, step, and barrier killing potentials.