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

Let the coefficients a ij and b i , i , j ≤ d , of the linear Fokker–Planck–Kolmogorov equation (FPK-eq.) ∂ t μ t = ∂ i ∂ j ( a ij μ t ) - ∂ i ( b i μ t ) be Borel measurable, bounded and continuous in space. Assume that for every s ∈ [ 0 , T ] and every Borel probability measure ν on R d there is at least one solution μ = ( μ t ) t ∈ [ s , T ] to the FPK-eq. such that μ s = ν and t ↦ μ 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 μ s , ν for each pair ( s , ν ) such that this family of solutions fulfills μ t s , ν = μ t r , μ r s , ν for all 0 ≤ s ≤ r ≤ t ≤ T , which one interprets as a flow property of this solution family. Moreover, we prove that such a flow of solutions is unique if and only if the FPK-eq. is well-posed.