The superposition principle for Fokker–Planck–Kolmogorov equations with unbounded coefficients

The superposition principle delivers a probabilistic representation of a solution\break $\{\mu_t\}_{t\in[0, T]}$ of the Fokker–Planck–Kolmogorov equation $\partial_t\mu_t=L^{*}\mu_t$ in terms of a solution $P$ of the martingale problem with operator $L$. We generalize the superposition principle to the case of equations on a domain, examine the transformation of the measure $P$ and the operator $L$ under a change of variables, and obtain new conditions for the validity of the superposition principle under the assumption of the existence of a Lyapunov function for the unbounded part of the drift coefficient.