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.) partial derivative(t)mu(t) = partial derivative(i)partial derivative(j)(a(ij)mu(t)) - partial derivative(i)(b(i)mu(t)) be Borel measurable, bounded and continuous in space. Assume that for every s. [0, T] and every Borel probability measure. on Rd there is at least one solution mu = (mu(t))t is an element of([s,T]) to the FPK-eq. such that mu(s) = v and t bar right arrow 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,v) for each pair (s,v) such that this family of solutions fulfills mu(s,v)(t) = mu(r,mu rs,v)(t) 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.
Springer Basel Ag