State-dependent jump processes: Itô-Stratonovich interpretations, potential, and transient solutions

The abrupt changes that are ubiquitous in physical systems often are well characterized by shot noise with a state-dependent recurrence frequency and jump amplitude. For such state-dependent behavior, we derive the transition probability for both the Itô and Stratonovich jump interpretations and subsequently use the transition probability to pose a master equation for the jump process. For exponentially distributed inputs, we present a class of transient solutions, as well as a generic steady-state solution in terms of a potential function and the Pope-Ching formula. These results allow us to describe state-dependent jumps in a double-well potential for steady-state particle dynamics, as well as transient salinity dynamics forced by state-dependent jumps. Both examples showcase a stochastic description that is more general than the limiting case of Brownian motion to which the jump process defaults in the limit of infinitely frequent and small jumps. Accordingly, our analysis may be used to explore a continuum of stochastic behavior from infrequent, large jumps to frequent, small jumps approaching a diffusion process.