A Feynman Approach to Dynamic Rate Markov Processes

Physics inspired mathematics helps us understand the random evolution of Markov processes. For example, the Kolmogorov forward and backward differential equations that govern the dynamics of Markov transition probabilities are analogous to the Schrodinger and Heisenberg pictures of quantum mechanics.

Richard Feynman introduced a time-ordered operator calculus that created a third framework for quantum mechanics and incorporates the two older quantum pictures. Applying these techniques to time-inhomogeneous Markov processes creates a transition probability analysis that reveals their fundamental sample path structure. Moreover, this interpretation can motivate a time-inhomogeneous, asymptotic generalization of steady state analysis.