Mode-Weighted Transport Certificates for State-Dependent Reflected Switching Diffusions

Abstract

State-dependent switching diffusions can be contractive in distribution even when some modes are individually expansive. We develop a computable transport-based condition for such contraction on $\mathbb{R}^n$ and on compact convex domains with normal reflection. The transport cost combines mode-dependent spatial weights with a discrete mode penalty while preserving spatial separation for cross-mode pairs. Using synchronous coupling of the Brownian motions and maximal coupling of the state-dependent jump clocks, we derive separate generator inequalities for same-mode and cross-mode configurations. Convex normal reflection contributes a nonpositive finite-variation term, so the same conditions apply to the associated no-flux Fokker-Planck-Kolmogorov system. Their feasibility guarantees global pairwise exponential contraction of the Markov semigroup and weak measure solutions, with unit prefactor and an explicit rate. For a fixed ordering of the mode weights and a prescribed decay rate, the conditions are affine in the spatial weights and graph costs and form a semi-infinite linear feasibility problem. A finite-mesh condition with a Lipschitz margin certifies the inequalities over the full domain. A reflected one-dimensional example validates the distributional computation, and a planar three-mode example demonstrates the synthesis procedure for transition rates depending on both state coordinates.