We prove that joint uniqueness in law and the existence of a strong solution imply pathwise uniqueness for variational solutions to stochastic partial differential equations of the formand show that for such equations uniqueness in law is equivalent to joint uniqueness in law. Here W is a cylindrical Wiener process in a separable Hilbert space U and the equation is considered in a Gelfand triple V ⊆ H ⊆ E, where H is some separable (infinite-dimensional) Hilbert space. This generalizes the corresponding results of A. Cherny for the case of finite-dimensional equations.

Let the coefficients aij and bi, i, j ≤ d, of the linear Fokker-Planck-Kolmogorov equation (FPK-eq.) ∂tµt = ∂i∂j(aijµt) − ∂i(biµt) 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 = (µt) t∈[s,T ] to the FPK-eq. such that µs = ν and t → µt 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 fulfillswhich 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.x)∂ i denotes the second order differential generator associated to A := (a ij ) i,j≤d and b := (b i ) i≤d , also called Kolmogorov operator.Fokker-Planck-Kolmogorov equations have been an active research topic in the past decades. There is a vast literature on general results such as existence, uniqueness and regularity of solutions, also for a more general notion of equations as (1). A thorough analytical introduction into the field is provided by the work [1] of Röckner, Krylov, Bogachev and Shaposhnikov from 2015.Fokker-Planck-Kolmogorov equations also have strong and fruitful connections to probability theory, in particular to the theory of diffusion processes. For example, the transition probabilities of a typical diffusion process in R d with drift b = (b i ) i≤d and diffusion coefficients (a ij ) i,j≤d solve the corresponding Fokker-Planck-Kolmogorov equation, i.e. equation (1). Above that, equation (1) is closely related to the martingale problem associated to the coefficients a ij and b i . More precisely, every continuous solution to the martingale problem provides a narrowly continuous probability solution to equation (1) via its one-dimensional marginals. Conversely, by a socalled superposition principle of Trevisan from 2016, which is an extension of an earlier work by Figalli from 2008, c.f. [5] and [2], respectively, given a narrowly continuous probability solution (µ t ) t∈[s,T ] to equation (1), there exists a continuous solution to the corresponding martingale problem, for which the one-dimensional marginals are given by (µ t ) t∈[s,T ] . A fundamental investigation of martingale problems and its connection to Fokker-Planck-Kolmogorov equations can be found in [4] by Stroock and Varadhan from 2006.

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We provide a method to select flows of solutions to the Cauchy problem for linear and nonlinear Fokker-Planck-Kolmogorov equations (FPK equations) for measures on Euclidean space. In the linear case, our method improves similar results of a previous work of the author. Our consideration of flow selections for nonlinear equations, including the particularly interesting case of Nemytskiitype coefficients, seems to be new. We also characterize the (restricted) well-posedness of FPK equations by the uniqueness of such (restricted) flows. Moreover, we show that under suitable assumptions in the linear case such flows are Markovian, i.e. they fulfill the Chapman-Kolmogorov equations.