Piecewise-Tunneled Captive Processes and Corridored Random Particle Systems

We introduce a family of processes that generalises captive diffusions, whereby the stochastic evolution that remains within a pair of time-dependent boundaries can further be piecewise-tunneled internally. The tunneling effect on the dynamics can be random such that the process has non-zero probability to find itself within any possible tunnel at any given time. We study some properties of these processes and apply them in modelling corridored random particles that can be observed in fluid dynamics and channeled systems. We construct and simulate mean-reverting piecewise-tunneled captive models for demonstration. We also propose a doubly-stochastic system in which the tunnels themselves are generated randomly by another stochastic process that jumps at random times.