Doubling variables and uniqueness of probability solutions to degenerate stationary Kolmogorov equations
V. I. Bogachev, S. V. Shaposhnikov, D. V. Shatilovich
TL;DR
This work establishes new uniqueness criteria for probability solutions to the stationary Kolmogorov equation $L^{*}\mu=0$ under degenerate diffusion matrices, without Hörmander-type smoothness assumptions. By directly applying the doubling-variables method to the Kolmogorov operator and constructing a coupling measure $\pi$ on $\mathbb{R}^d\times\mathbb{R}^d$ with doubled generator $\mathbb{L}$, the authors show that uniqueness holds under a sign condition on $q(x,y)$ or under a relaxed bound involving a weight function $\Lambda$. The proofs hinge on smoothing two solutions, building a Lyapunov framework for the doubled system, and exploiting diagonal concentration to deduce equality of marginals; a logarithmic-inequality variant further broadens applicability. These results extend uniqueness theory for degenerate diffusions beyond classical nondegenerate or Hörmander settings and illuminate the role of coupling constructions in establishing probabilistic uniqueness.
Abstract
We obtain sufficient conditions for the uniqueness of a probability solution to the stationary Kolmogorov equation with a degenerate diffusion matrix. We employ the method of doubling variables known in stochastic analysis directly to the Kolmogorov equation.