Extend Siegmund duality to higher dimensions

Establish a multidimensional Siegmund duality framework that connects exit (splitting) probabilities of stochastic processes with absorbing boundaries to cumulative spatial distributions of explicitly constructed dual processes with hard-wall boundaries in dimensions d ≥ 2. Specify suitable boundary geometries (e.g., hypercubes) and the form of the dual dynamics, and prove finite-time and stationary-state relations analogous to those shown in one dimension for run-and-tumble particles, Brownian motion, and resetting processes.

Background

Part V develops Siegmund duality to link first-passage properties (such as exit probabilities) of a one-dimensional process with absorbing boundaries to spatial observables of a dual process with hard walls. The authors construct duals for a broad class of models, including active particles (RTP, AOUP, ABP), random diffusivity models, and processes under stochastic resetting, and they verify the duality both analytically and numerically in one dimension.

However, practical applications often involve higher-dimensional systems where geometry and boundary conditions are more complex. The authors note that while abstract multidimensional formulations exist, there is no explicit, practically useful construction comparable to their 1D results. They identify extending this duality to higher dimensions as a key open direction.