Killing versus branching: Unexplored facets of diffusive relaxation

We analyze the relaxation dynamics of Feynman-Kac path integral kernel functions, in terms of branching diffusion processes with killing. This amounts to the killing versus branching approach to path integration, which seems to be a novelty in the pathwise description of conditioned Brownian motions and diffusion processes with absorbing boundaries. There, Feynman-Kac kernels appear as building blocks of inferred (Fokker-Planck) transition probability density functions. A consistent probabilistic meaning is hereby provided (killing versus branching time rate) to bounded from below Feynman-Kac potential functions, which instead of being positive-definite (a standard killing paradigm), may take negative values on bounded spatial subdomains (that inflicts trajectory branching).