Existence of flows for linear Fokker-Planck-Kolmogorov equations and its connection to well-posedness

Abstract

Let the coefficients $a_{ij}$ and $b_i$, $i,j\le d$, of the linear Fokker-Planck-Kolmogorov equation (FPK-eq.) $${\partial }_t{\mu }_t={\partial }_i{\partial }_j(a_{ij}{\mu }_t)-{\partial }_i(b_i{\mu }_t)$$ be Borel measurable, bounded and continuous in space. Assume that for every $s\in [0,T]$ and every Borel probability measure $\nu$ on ${\mathbb{R}}^d$ there is at least one solution $\mu =({\mu }_t{)}_{t\in [s,T]}$ to the FPK-eq. such that ${\mu }_s=\nu$ and $t\mapsto {\mu }_t$ is continuous w.r.t. the topology of weak convergence of measures. We prove that in this situation, one can always select one solution ${\mu }^{s,\nu }$ for each pair $(s,\nu )$ such that this family of solutions fulfills $${\mu }_t^{s,\nu }={\mu }_t^{r,{\mu }_r^{s,\nu }}\text{ for all }0\le s\le r\le t\le T,$$which one interprets as a flow property of this solution family. Moreover, we prove that such a flow of solutions is unqiue if and only if the FPK-eq. is well-posed.