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.