Construct Siegmund duality for the random acceleration process

Develop an explicit Siegmund duality for the random acceleration process (integrated Brownian motion) where the acceleration is Gaussian white noise, by constructing a dual process with hard-wall boundaries and demonstrating a rigorous relation between exit probabilities of the original process and cumulative spatial distributions of the dual, both at finite time and in the stationary regime when it exists.

Background

The duality framework in this thesis assumes that the driving noise has a stationary distribution, enabling Fokker-Planck-based derivations of equivalences between first-passage and spatial observables. While this covers many important models, the random acceleration process falls outside these assumptions because its acceleration is white noise and the effective driving noise is non-stationary.

The authors explicitly note that extending their duality construction to this case remains unresolved, marking it as a natural and challenging frontier for generalizing the methodology beyond its current assumptions.