Superposition Principle for the Fokker–Planck–Kolmogorov Equations with Unbounded Coefficients

The superposition principle delivers a probabilistic representation of a solution \(\{\mu_t\}_{t\in[0, T]}\) 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.