Mode-Weighted Transport Certificates for State-Dependent Reflected Switching Diffusions
2026-08-03 • Computational Engineering, Finance, and Science
Computational Engineering, Finance, and Science
AI summaryⓘ
The authors study systems that switch between different behaviors depending on the current state, where some behaviors might normally spread out but the entire system still becomes more concentrated over time. They create a way to measure how these systems contract by combining spatial factors and switching penalties, applicable in flat spaces and bounded regions with reflective boundaries. Using coupling techniques, they establish mathematical conditions that guarantee the system becomes closer together at an exponential rate. They also show how to check these conditions practically and demonstrate their approach with examples.
state-dependent switching diffusioncontraction in distributioncouplingMarkov semigroupFokker-Planck equationnormal reflectiontransport costgenerator inequalitiesexponential contraction
Authors
Yutong Zhu, Ye Zhang
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.