Captive Jump Processes

TL;DR

This paper introduces captive jump processes, a new class of stochastic processes confined within time-dependent boundaries, with potential applications in physics such as quantum tunneling and electron transitions.

Contribution

It explicitly constructs captive jump processes, extending captive diffusion models to include jumps and complex boundary conditions, with detailed examples and simulations.

Findings

Captive jump processes are confined within time-dependent boundaries.

They can model phenomena like quantum tunneling and electron transitions.

Simulations demonstrate dynamics within circular boundaries.

Abstract

We explicitly construct so-called captive jump processes. These are stochastic processes in continuous time, whose dynamics are confined by a time-inhomogeneous bounded domain. The drift and volatility of the captive processes depend on the domain's boundaries which in turn restricts the state space of the process. We show how an insurmountable shell may contain the captive jump process while the process may jump within the restricted domain. In a further development, we also show how - within a confined domain - inner time-dependent corridors can be introduced, which a captive jump process may leave only if the jumps reach far enough, while nonetheless being unable to ever penetrate the outer confining shell. Captive jump processes generalize the recently developed captive diffusion processes. In the case where a captive jump-diffusion is a continuous martingale or a pure-jump process, the uppermost confining boundary is non-decreasing, and the lowermost confining boundary is non-increasing. Captive jump processes could be considered to model phenomena such as electrons transitioning from one orbit (valence shell) to another, and quantum tunneling where stochastic wave-functions can "penetrate" boundaries (i.e., walls) of potential energy. We provide concrete, worked-out examples and simulations of the dynamics of captive jump processes in a geometry with circular boundaries, for demonstration.