Uniqueness of the limiting martingale problem for the cycle-valued random walk

Establish uniqueness of the solution to the martingale problem for the limiting generator \mathfrak{A} (defined in Equation (def:mathfrakA)), which characterizes the diffusive limit of the rescaled cycle-valued random walk on triangulations of the flat torus viewed in the space of currents. Proving uniqueness would ensure full convergence of the sequence of rescaled random walks to a single diffusion solving the martingale problem associated with \mathfrak{A}.

Background

The paper introduces a continuous-time chain-valued random walk on simplicial complexes and studies its diffusive scaling limits on regular triangulations of the flat torus. The authors construct the limiting generator \mathfrak{A}, involving the Rham–Hodge operator, and show tightness; they further identify that all limiting values solve the same martingale problem.

However, they do not establish uniqueness of the solution for this martingale problem. Uniqueness is a key step to conclude convergence of the entire sequence of rescaled random walks to a single limit. The generator \mathfrak{A} is explicitly defined in Equation (def:mathfrakA), and its associated martingale problem governs the limit process in the space of currents.