On uniquely solvable Fokker–Planck–Kolmogorov equations

In this paper we obtain broad sufficient conditions for the existence
of probability solutions to the Cauchy problem for
Fokker–Planck–Kolmogorov equations on the real line without using Lyapunov
functions. In the multidimensional case, we prove that if
the Fokker–Planck–Kolmogorov equation for an elliptic operator $L$has a probability solution $\sigma$, and the Cauchy problem for this
equation has a unique probability solution for every initial probability
distribution, then there exists a strongly continuous Markov operator semigroup
on the space $L^1(\sigma)$ with respect to which the measure $\sigma$is invariant and the generator of which extends the operator $L$.
We give an answer to the long-standing question about existence of
a sub-Markov semigroup different from the canonical semigroup with
the generator extending $L$.

Fokker–Planck–Kolmogorov equation, Cauchy problem, stationary equation