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

Abstract

State-dependent switching diffusions can contract in law even when individual modes are expansive. We establish a computable, constant-one contraction certificate on ${\mathbb{R}}^n$ and compact convex domains with normal reflection. The mode-weighted transport cost combines spatial separation with a discrete graph penalty and retains spatial dependence when the modes differ. A synchronous Brownian coupling and a maximal coupling of jump clocks in equal modes lead to separate equal- and unequal-mode generator inequalities. We give a complete regularization argument for the nonsmooth cost, including the drift and jump errors, the sign of reflection, and the limiting Itô-Dynkin calculation. A resolvent proof transfers contraction to probability-valued weak solutions of the no-flux Fokker-Planck-Kolmogorov system, while an explicit Foster-Lyapunov condition ensures existence and uniqueness of an invariant law on the whole space. For fixed weight ordering and decay rate, certificate synthesis is a semi-infinite linear feasibility problem; on compact domains, a Lipschitz-buffered finite mesh provides a rigorous test. One-dimensional and planar examples illustrate the certificate and its numerical verification.