Piecewise-Tunneled Captive Processes and Corridored Random Particle Systems

Abstract

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.

On a family of coupled diffusions that can never change their initial order

“…= µ (i+2) (t, ., .) = 0 (22) for all t ∈ T, since every coordinate is a (P, {F X t })-martingale. Hence,…”

“…Such Hermitian systems can be used in solving quadratic optimization problems that can generate order-preserving efficient-frontiers where the controller is interested in producing truncated distributions of optimal solutions dictated by covariance structures bounded in the Loewner sense (see [22] for details) -note that elements in H m×m + ∩ R m×m are covariance matrices. Extending [22], we can now solve optimization problems of the form:…”

“…We note that our framework can be applied in random particle systems that interact with each other based on their order (see [18][19][20][21]), risk-controlled quadratic optimization problems (see, [22]), and sticky microbial dynamics (see, [23][24][25]), amongst others. We leave a more detailed account of these applications for future and focus on establishing the mathematical groundwork in this paper.…”