Captive jump processes for bounded random systems with discontinuous dynamics

Abstract

Stochastic captive jump processes are explicitly constructed in continuous time, whose non-linear dynamics are strictly confined by bounded domains that can be time-dependent. By introducing non-anticipative path-dependency, the framework offers the possibility of generating multiple inner tunnels within a master domain, such that a captive jump process is allowed to proceed either within a single inner tunnel or jump in between tunnels without ever penetrating the outermost shell. If a captive jump process is a continuous martingale or a pure-jump process, the uppermost confining boundary is non-decreasing, and the lowermost confining boundary is non-increasing. Under certain conditions, it can be shown that captive jump processes are invariant under monotonic transformations, enabling one to construct and study systems of increasing complexity using simpler building blocks. Amongst many applications, captive jump processes may be considered to model phenomena such as electrons transitioning from one orbit (valence shell) to another, quantum tunnelling where stochastic wave-functions can “penetrate” inner boundaries (i.e., walls) of potential energy, non-linear dynamical systems involving multiple attractors, and sticky concentration behaviour of pathogens in epidemics. We provide concrete, worked-out examples, and numerical simulations for the dynamics of captive jump processes within different geometries as demonstrations.