Path Integral Methods and Source Propagators for Linear Transport Operators

A linear transport calculation can tell us how much reaches an observation without remem- bering where it entered. Its propagator contains that information. We show how to read it in three ways: carry a label forward with each source, carry the observation backward with the adjoint, or sample the paths joining the two. The operator may contain drift, diffusion, nonlocal exchange, conversion among internal states, and linear gain or loss. A prescribed pos- itive total field gives a fourth computational view: backward paths can follow the ancestry of the field. When the field disagrees with the transport equation, its residual can be assigned a separate signed label or shared proportionally among the existing labels. We derive the two recursions and their exact path estimators. Every sampled path accounts for its whole con- tribution, although its individual labels still fluctuate. The derivations are exact for positive finite-dimensional propagators; the continuous representations require a well-defined evolution, admissible boundaries, and sufficient support of the guiding field. Neither linearity nor a known total determines how an unexplained residual should be assigned.