GENERALIZED MASTER EQUATIONS FOR RANDOM WALKS WITH TIME-DEPENDENT JUMP SIZES

In this work, we develop a unified generalized master equation (GME) framework that extends the theory of continuous time random walks to include the cases when the jump sizes may have a delayed dependence on time and are not restricted to any particular class of distributions. We compare and contrast analytical and numerical behavior of the corresponding master equations, including the instantaneous vs. delayed jump dependence on time and exponential vs. Mittag-Leffler interarrival times, with the latter leading to fractional evolution equation. We provide existence and uniqueness proofs for the resulting GMEs.