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

Abstract

Let the coefficients $a_{ij}$ and $b_i$, $i,j \leq d$, 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 $$\mu^{s,\nu}_t = \mu^{r,\mu^{s,\nu}_r}_t \text{ for all }0 \leq s \leq r \leq t \leq T,$$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.