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Linearization and a superposition principle for deterministic and stochastic nonlinear Fokker-Planck-Kolmogorov equations

We prove a superposition principle for nonlinear Fokker-Planck-Kolmogorov equations on Euclidean spaces and their corresponding linearized first-order continuity equation over the space of Borel (sub-)probability measures. As a consequence, we obtain equivalence of existence and uniqueness results for these equations. Moreover, we prove an analogous result for stochastically perturbed Fokker-Planck-Kolmogorov equations. To do so, we particularly show that such stochastic equations for measures are, similarly to the deterministic case, intrinsically related to linearized second-order equations on the space of Borel (sub-)probability measures.


Introduction
In this work we are concerned with nonlinear Fokker-Planck-Kolmogorov equations (FPK-equations) on R d , both deterministic ∂ t µ t = L * t,µt µ t , t ∈ [0, T ], and perturbed by a first-order stochastic term driven by a finite-dimensional Wiener process W ∂ t µ t = L * t,µt µ t − div(σ(t, µ t ))dW t , t ∈ [0, T ], with solutions being continuous curves of subprobability measures µ t ∈ SP. Here, L * denotes the formal dual of a second-order differential operator acting on sufficiently smooth functions ϕ : with coefficients a and b depending on (t, x) ∈ [0, T ] × R d and (in general non-locally) on the solution µ t . These equations are to be understood in distributional sense, see Definition 3.1 and 4.1. The nonlinearity arises from the dependence of L and σ on the solution itself, which renders the theory of existence and uniqueness of such equations significantly more difficult compared to the linear case. For a thorough introduction to the field, we refer to [5] and the references therein. As shown in [17], the deterministic nonlinear equation (NLFPK) is naturally associated to a first-order linear continuity equation on P(SP), the space of Borel probability measures on SP, of type in the sense of distributions, with the linear operator L acting on sufficiently smooth real functions on SP via the gradient operator ∇ SP on SP as Precise information on this operator and equation (SP-CE) are given in Section 3, in particular in Definition 3.5 and the paragraph preceding it. Our first main result, Theorem 3.7 states that each weakly continuous solution (Γ t ) t≤T to (SP-CE) is a superposition of solutions to (NLFPK), i.e. (denoting by e t the canonical projection e t : (µ t ) t≤T → µ t ) for some probability measure η concentrated on solution curves to (NLFPK) in a suitable sense. We also treat the stochastic case in a similar fashion. More precisely, in Section 4 we establish a new correspondence between the stochastic equation for measures (SNLFPK) and a corresponding second-order equation for curves (Γ t ) t≤T in P(SP) of type where, roughly, L t = L t + second-order perturbation. The second-order term stems from the stochastic perturbation of (SNLFPK) and will be geometrically interpreted in terms of a (formal) notion of the Levi-Civita connection on SP. The second main result of this work, Theorem 4.8, is then the stochastic generalization of the deterministic case: For any solution (Γ t ) t≤T to (SP-FPK), there exists a solution process (µ t ) t≤T to (SNLFPK) on some probability space such that µ t has distribution Γ t . We stress that in both cases, we do not require any regularity of the coefficients.
Let us embed these results into the general research in this direction. Let b t (·) : R d → R d be an inhomogeneous vector field and consider the (nonlinear) ODE and the linear continuity equation for curves of Borel (probability) measures on R d understood in distributional sense. In the seminal paper [1], L. Ambrosio showed the following: Any (probability) solution (µ t ) t≤T to (CE) with an appropriate global integrability condition is a superposition of solution curves to (ODE), i.e. there exists a (probability) measure η on the space of continuous paths with values in the state space of (ODE), C([0, T ], R d ), which is concentrated on solutions to (ODE) such that η • e −1 t = µ t , t ≤ T. This allows to transfer existence and uniqueness results between the linear equation (CE) and the nonlinear (ODE). However, the linear equation must be studied on an infinite-dimensional space of (probability) measures. The analogy to our deterministic result from Section 3 is as follows: (ODE) is replaced by (NLFPK), which, in spirit of this analogy, we interpret as a differential equation on the manifold-like state space SP. Likewise, (CE) is replaced by (SP-CE) and our first main result Theorem 3.7 may be understood as the analogue of Ambrosio's result to the present setting. By passing from (NLFPK) to (SP-CE), we linearize the equation. Concerning the stochastic case, consider a stochastic differential equation on R d dX t = b(t, X t )dt +ã(t, X t )dB t , t ∈ [0, T ].
(SDE) By Itô's formula, the one-dimensional marginals µ t of any (weak) martingale solution X solve the corresponding linear FPK-equation where L lin is a linear second-order diffusion operator with coefficients b and 1 2ãã T . Conversely, a superposition principle has successively been developed in increasingly general frameworks (cf. [9,14,21,6]): Under mild global integrability assumptions, for every weakly continuous solution curve of probability measures (µ t ) t≤T to (FPK), there exists a (weak) martingale solution X to (SDE) with one-dimensional marginals (µ t ) t≤T , thereby providing an equivalence between solutions to (SDE) and (FPK), which offers a bridge between probabilistic and analytic approaches to diffusion processes. As in the deterministic case, the transition from (SDE) to (FPK) provides a linearization, while at the same time it transfers the equation to a much higher dimensional state space. Concerning our stochastic result Theorem 4.8, we replace the stochastic equation on R d by the stochastic equation for measures (SNLFPK) and the corresponding second-order equation for measures (FPK) by (SP-FPK) and prove an analogous superposition result for solutions to the latter equation. The proofs of both the deterministic and stochastic result rely on superposition principles for differential equations on R ∞ and the corresponding continuity equation (for the deterministic case) and for martingale solutions and FPK-equations on R ∞ (for the stochastic case) by Ambrosio and Trevisan ([2], [20]). The key technique is to transfer (SP-CE) and (NLFPK) (and, similarly, (SP-FPK) and (SNLFPK) for the stochastic case) to suitable equations on R ∞ via a homeomorphism between SP and R ∞ (replaced by ℓ 2 for the stochastic case, in order to handle the stochastic integral). Moreover, our results also blend into the theory of distribution dependent stochastic differential equations, also called McKean-Vlasov equations, i.e. stochastic equations on Euclidean space of type see the classical papers [15,10,18] as well as the more recent works [12,11,8]. Here, L Xt denotes the distribution of X t and is not to be confused with the operators L t,µ and L t from above. As in the nondistribution dependent case, where the curve of marginals of any solution to (SDE) solves an equation of type (FPK), a similar observation holds here: Each solution X to (DDSDE) provides a solution to a nonlinear FPK-equation of type (NLFPK) via µ t = L Xt and a corresponding superposition principle holds analogously to the linear case as well ( [4,3]). However, while for (SDE) the passage to (FPK) provides a complete linearization, the situation is different for equations of type (NLFPK). This stems from the observation that (DDSDE) is an equation with two sources of nonlinearity. Hence, it seems natural to linearize (NLFPK) once more in order to obtain a linear equation, which is related to (DDSDE) and (NLFPK) in a natural way. By the results of [17], this linear equation is of type (SP-CE). Similar considerations prevail in the stochastic case, where one considers equations of type (DDSDE) with an additional source of randomness (we shall not pursue this direction in this work).
On the one hand, the superposition principles of Theorem 3.7 and Theorem 4.8 provide new structural results for nonlinear FPK-equations and its corresponding linearized equations on the space of probability measures over SP, involving a geometric interpretation of the latter. On the other hand, it is our future plan to further study the geometry of SP as initiated in [17] and this work to develop an analysis on such infinite-dimensional manifold-like spaces, which allows to solve linear equations of type (SP-CE) and (SP-FPK) on such spaces. By means of the results of this work, one can then lift such solutions to solutions to the nonlinear equations for measures (NLFPK) and (SNLFPK), thereby obtaining new existence results for these nonlinear equations for measures.
We point out that although our main aim is to lift weakly continuous solutions to (SP-CE) and (SP-FPK) concentrated on probability measures to a measure on the space of continuous probability measure-valued paths (µ t ) t≤T , for technical reasons we more generally develop our results for vaguely continuous subprobability solutions (i.e. µ t ∈ SP). We comment on the advantages of this approach in Remark 3.9 for the deterministic case and note that similar arguments prevail in the stochastic case as well. However, due to the global integrability assumptions we consider, we are able to obtain results for probability solutions as desired.
The organization of this paper is as follows. After introducing general notation and recalling basic properties of the spaces P and SP in Section 2, Section 3 contains the deterministic case, i.e. the superposition principle between solutions to (SP-CE) and (NLFPK). Here, the main result is Theorem 3.7. We use this result to prove an open conjecture of [17] (cf. Proposition 3.12) and present several consequences. In Section 4, we treat the stochastic case for equations of type (SNLFPK), the main result being Theorem 4.8.
Acknowledgements Financial support by the German Science Foundation DFG (IRTG 2235) is gratefully acknowledged.

Notation and Preliminaries
We introduce notation and repeat basic facts on spaces and topologies of measures.

Notation
For a measure space (X , A, µ) and a measurable function ϕ : X → R, we set µ(ϕ) := X ϕ(x)dµ(x) whenever the integral is well-defined. For x ∈ X , we denote by δ x the Dirac measure in x, i.e. δ x (A) = 1 if and only if x ∈ A and δ x (A) = 0 else. For a topological space X with Borel σ-algebra B(X) we denote the set of continuous bounded functions by C b (X), the set of Borel probability measures on X by P(X) and write for each g ∈ G implies µ = ν for any two finite Borel measures µ, ν on R d . For x, y ∈ R d , the usual inner product is denoted by x · y and, with slight abuse of notation, we also denote by x · y = k≥1 x k y k the inner product in ℓ 2 (the Hilbert space of square-summable real-valued sequences x = (x k ) k≥1 ). For ϕ ∈ C b (R d ), we set ||ϕ|| ∞ := sup x∈R d |ϕ(x)|. If ϕ has first-and second-order partial derivatives, we denote them by ∂ i ϕ and ∂ 2 ij ϕ for i, j ≤ d.
We use notation for function spaces as follows. For k ∈ N 0 , C k b (R d ) denotes the subset of functions ϕ in C b (R d ) with continuous, bounded partial derivatives up to order k, with the usual norm denotes the subset of all such ϕ with compact support; for k = 0, we write C c (R d ) instead. For n ≥ 1, p ≥ 1 and a measure µ on B(R d ), we denote by L p (R d , R n ; µ) the space of Borel functions ϕ : where || · || denotes the standard Euclidean norm on R n . For p = 2, ·, · L 2 (R d ,R n ;µ) denotes the usual inner product on the Hilbert space L 2 (R d , R n ; µ). For T > 0 and a topological space Y , we write C T Y for the set of continuous functions ϕ : [0, T ] → Y . By S + d we denote the space of symmetric, positive-semidefinite d × d-matrices with real entries.

Probability measures
For a topological space X, we endow P(X) with the topology of weak convergence of measures, i.e. the initial topology of the maps µ → µ(ϕ), ϕ ∈ C b (X). If X is Polish, then so is P(X).

Subprobability measures
By SP we denote the set of all Borel subprobability measures on R d , i.e. µ ∈ SP if and only if µ is a non-negative measure on B(R d ) with µ(R d ) ≤ 1. Throughout, we endow SP with the vague topology, i.e. the initial topology of the maps µ → µ(g), g ∈ C c (R d ). Hence, a sequence (µ n ) n≥1 converges to µ in SP if and only if µ n (g) −→ n→∞ µ(g) for each g ∈ C c (R d ). Its Borel σ-algebra is denoted by B(SP). In particular, P(SP), the set of Borel probability measures on SP, is a topological space with the weak topology of probability measures on (SP, B(SP)). The Riesz-Markov representation theorem yields that SP with the vague topology coincides with the positive half of the closed unit ball of the dual space of C c (R d ) with the weak*-topology. Hence SP with the vague topology is compact. It is also Polish and µ → µ(R d ) is vaguely lower semicontinuous, see [13,Ch.4.1]. In particular, P ∈ B(SP). Recall that B(P) = B(SP) ↾P . Hence, in the sequel we may consider measures Γ ∈ P(P) as elements in P(SP) with mass on P.
In contrast to weak convergence in P, vague convergence in SP can be characterized by countably many functions in a sense made precise by Lemma 3.3. The fact that this is not true for weak convergence in P is the main reason why we formulate all equations for subprobability measures, although we are mainly interested in the case of probability solutions. More details in this direction are stated in Remark 3.9.

Superposition Principle for deterministic nonlinear Fokker-Planck-Kolmogorov Equations
Fix T > 0 throughout, let each component of the coefficients holds and for each ϕ ∈ C 2 (ii) A probability solution to (NLFPK) is a curve (µ t ) t≤T ⊆ P fulfilling (3) and (4) such that t → µ t is weakly continuous.
Since vaguely continuous curves of measures are in particular Borel curves, all integrals in the above definition are defined. Below we shortly refer to subprobability and probability solutions and keep in mind the respective continuity conditions. In the literature, more general notions of solutions to (NLFPK) are considered, such as (possibly discontinuous) curves of signed, bounded measures [5]. However, in this work, we restrict attention to continuous (sub-)probability solutions. In presence of the global integrability condition (3), we make the following observation. (i) Any subprobability solution (µ t ) t≤T with µ 0 ∈ P is a probability solution. Indeed, to prove this it suffices to show µ t (R d ) = 1 for each t ≤ T . Since (µ t ) t≤T fulfills (4), it suffices to choose a sequence ϕ l , l ≥ 1, from C 2 c (R d ) with the following properties: 0 ≤ ϕ l ր 1 pointwise such that ∂ i ϕ l −→ l→∞ 0, ∂ 2 ij ϕ l −→ l→∞ 0 pointwise with all first and second order derivatives bounded by some M < +∞ uniformly in l ≥ 1 and x ∈ R d . Considering (4) for the limit l → ∞, we obtain, by (3) and dominated convergence, and hence the claim.
(ii) By the above argument, one shows that for any subprobability solution, (4) holds for each Geometric approach to SP For our goals, it is preferable to consider SP as a manifold-like space. We refer the reader to the appendix in [17], where for the space of probability measures P the tangent spaces T µ P = L 2 (R d , R d ; µ) and a suitable test function class F C 2 b (P), have been introduced. Further, based on these choices, a natural pointwise definition of the gradient ∇ P F as a section in the tangent bundle for F as above is given by which is shown to be independent of the representation of F in terms of f and ϕ i . The setting in the present paper is nearly identical, but we consider the manifold-like space SP with the vague topology instead of P with the weak topology as in [17], because SP is embedded in R ∞ in the following sense.
be dense in C 2 c (R d ) with respect to || · || C 2 b such that no g i is constantly 0. Clearly, any such set of functions is dense in C c (R d ) with respect to uniform convergence and measure-determining. Such sets of functions are sufficiently extensive to characterize the topology of SP as well as solutions to (NLFPK): Let G be any set of functions with the properties mentioned above and let (µ n ) n≥1 ⊆ SP. Then, (ii) A vaguely continuous curve (µ t ) t≤T ⊆ SP, which fulfills (3), is a subprobability solution to (NLFPK) if and only if (4) holds for each g i ∈ G in place of ϕ.
for all sufficiently large n ≥ 1.
(ii) Let ϕ ∈ C 2 c (R d ) be approximated uniformly up to second-order derivatives by a sequence {g i k } k≥1 from G. Considering (4) for such g i k and letting k → ∞, the result follows by dominated convergence, which applies due to (3).
Considering SP as a (infinite-dimensional) manifold-like topological space, any set of functions G as above provides a global chart (i.e., an atlas consisting of a single chart) for SP, as it yields an embedding SP ⊆ R ∞ (cf. Lemma 3.4). Consider R ∞ as a Polish space with the topology of pointwise convergence and the range G(SP) ⊆ R ∞ of G as introduced below with its subspace topology. We write C T G(SP) for the set of all elements in C T R ∞ with values in G(SP). For u ∈ [0, T ], we denote by e u the canonical projection on C T SP e u : (µ t ) t≤T → µ u and, likewise, by e ∞ u the projection on C T R ∞ . Subsequently, without further mentioning, we consider the spaces C T SP and C T R ∞ with σ-algebras respectively. These algebras coincide with the Borel σ-algebras with respect to the topology of uniform convergence (because both SP and R ∞ are Polish). Also, consider C T G(SP) with the natural subspace σ-algebra of B(C T R ∞ ). We refer to these σ-algebras as the canonical σ-algebras on the respective spaces and denote the set of probability measures on the respective σ-algebras by P(C T SP) and P(C T R ∞ ).
be a set of functions as in (6).
(i) The map G, depending on G, is a homeomorphism between SP and its range G(SP) (hence, formally, a global chart for SP).
is measurable and one-to-one with measurable inverse J −1 : Proof. (i) The continuity of G is obvious by definition of the vague topology on SP and since G ⊆ C c (R d ). Since SP is compact with respect to the vague topology, compactness of G(SP) ⊆ R ∞ follows. G is measure-determining on R d , which implies that G is one-to-one. Since by definition continuity of G −1 follows from Lemma 3.3 (i). The final assertion follows, since for G ′ as in the assertion, V : (ii) Since G is one-to-one and measurable, so is J. Clearly, C T G(SP) is the range of J and hence J : C T SP → C T G(SP) is a bijection between standard Borel spaces (the latter, because SP and G(SP) with the respective topologies are Polish). This yields the measurability of J −1 . Finally, carries the subspace topology inherited from R ∞ .
By part (i) of the previous lemma it is justified to fix a set G = {g i , i ≥ 1} for the remainder of the section. In order to switch between test functions on SP and R ∞ in an equivalent way, we slightly deviate from the test function class presented in [17] (see (5)) and, instead, consider is made for consistency with the stochastic case later on only. We summarize our geometric interpretation of SP, which is of course still a close adaption of the ideas presented in [17]: For the manifold-like space SP, we consider smooth test functions F ∈ F C 2 b (G), with G being fixed as in (6). For each µ ∈ SP, we have the tangent space T µ SP = L 2 (R d , R d ; µ) and the gradient . . . , µ(g n ) as a section in the tangent bundle T SP, which is independent of the representation of F . Adding to the approach of SP as a manifold-like space, the global chart G as in (8) embeds SP into R ∞ . However, we do not rigorously treat SP as a (Fréchet-)manifold and consider the embedding SP ⊆ R ∞ merely as a tool to transfer (NLFPK) and its corresponding continuity equation to equivalent equations over R ∞ , as outlined below.

The continuity equation (SP-CE)
As mentioned in the introduction, we study the linear continuity equation associated to (NLFPK) as derived in [17], which is a first-order equation for curves of measures on SP. More precisely, in analogy to the derivation in [17], it is readily seen that any subprobability solution (µ t ) t≤T to (NLFPK) induces a curve of elements in P(SP), We rewrite (9) this is just the linear continuity equation (SP-CE). The term a∇ has rigorous meaning only, if a has sufficiently regular components in order to put the derivative ∇ on a via integration by parts, which we do not assume at any point. Considering SP as a manifold-like space, one may formally regard to a∇ + b as a time-dependent section in the tangent bundle T SP.
The choice of G as in (6) implies that any solution in the above sense fulfills (12) even for each for the larger class of test functions considered in [17] (upon extending their domain from P to SP). In particular, this notion of solution is independent of G. The main result of this chapter, Theorem 3.7, states that any solution to (SP-CE) as in Definition 3.5 arises as a superposition of solutions to (NLFPK). Note that for ν ∈ SP, uniqueness of solutions (Γ t ) t≤T to (SP-CE) with Γ 0 = δ ν implies uniqueness of subprobability solutions (µ t ) t≤T to (NLFPK) with µ 0 = ν.

Transferring (NLFPK) and (SP-CE) to R ∞
We use the global chart G : SP → R ∞ and the map J of Lemma 3.4 to reformulate both (NLFPK) and which is Borel measurable by Lemma 3.4. Next, consider the differential equation which turns out to be the suitable analogue to (NLFPK) on R ∞ . Analogously, the corresponding continuity equation for curves of Borel probability measuresΓ t on R ∞ , i.e.
with∇ as introduced below, is the natural analogue of the linear continuity equation (SP-CE).
Roughly, these analogies are to be understood in the sense that solutions to (NLFPK) and (SP-CE) can be transferred to solutions to (R ∞ -ODE) and (R ∞ -CE), respectively, via the chart G. We refer to the proof of the main result below for more details. Let denote the canonical projection to the i-th component, set π n = (p 1 , . . . , p n ) and By∇ we denote the gradient-type operator on R ∞ , acting onF Again, the restriction to test functions possessing second-order derivatives is made in order to be consistent with the stochastic (second-order) case later on. and

Main Result: Deterministic case
The following theorem is the main result for the deterministic case.
The proof relies on a superposition principle for measure-valued solution curves of continuity equations on R ∞ and its corresponding differential equation, which we recall in Proposition 3.8 below. More precisely, we proceed in three steps. First, we transfer (Γ t ) t≤T to a solution (Γ t ) t≤T to (R ∞ -CE). Then, by Proposition 3.8 below we obtain a measureη ∈ P(C T R ∞ ) withη • (e ∞ t ) −1 =Γ t , which is concentrated on solution curves to (R ∞ -ODE). Finally, we transferη back to a measures η ∈ P(C T SP) with the desired properties. Below, we denote by F C 1 b (R ∞ ) the set of test functions of same type as in Then, there exists a Borel measuresη ∈ P(C T R ∞ ) concentrated on solutions to (R ∞ -ODE) in the sense of Definition 3.6 (i) such thatη We proceed to the proof of the main result.
Proof of Theorem 3.7: Let Γ = (Γ t ) t≤T be a weakly continuous solution to (SP-CE) as in Definition 3.5.
Step 1: From (SP-CE) to (R ∞ -CE): Set with G as in Lemma 3.4, which corresponds to the fixed set of functions G. Since G is continuous, (Γ t ) t≤T is a weakly continuous curve of Borel subprobability measures on R ∞ . We show that (Γ t ) t≤T solves (R ∞ -CE). Indeed, the integrability condition (14) is fulfilled, since (Γ t ) t≤T fulfills Definition 3.5. Further, since Γ solves (SP-CE), we have for any and hence, abbreviating p k • B(t, ·) by B k t and settingF = f • π n for f as above, we have Comparing with (15), it follows that (Γ t ) t≤T is a solution to (R ∞ -CE) as claimed, because F ∈ F C 2 b (G) was arbitrary and henceF as above is arbitrary in F C 2 b (R ∞ ). By standard approximation, one extends the above equation to test functionsF from F C 1 b (R ∞ ).
Step 2: From (R ∞ -CE) to (R ∞ -ODE): Proposition 3.8 implies the existence of a measureη ∈ P(C T R ∞ ) such that Step 3: From (R ∞ -ODE) to (NLFPK): We show that the measure η :=η • (J −1 ) −1 , with J as in Lemma 3.4 fulfills all desired properties. Indeed, sincē for each t ∈ [0, T ] we deduce thatη • (e ∞ t ) −1 is concentrated on G(SP). By Lemma 3.4, G(SP) ⊆ R ∞ is closed. Since by constructionη is concentrated on continuous curves in R ∞ ,η is concentrated on C T G(SP). Further, C T G(SP) ⊆ C T R ∞ is a measurable set and J −1 : C T G(SP) → C T SP is measurable by Lemma 3.4. Therefore, we may define η ∈ P(C T SP) via It remains to verify η • e −1 t = Γ t for all t ∈ [0, T ] and that η is concentrated on subprobability solutions to (NLFPK). Concerning the first matter, we have t coincide as measurable maps on C T G(SP) and it was shown above that η is concentrated on C T G(SP), we obtain Concerning the second aspect, note that by definition of η andΓ t and by the equality (11) for Γ implies that η is concentrated on vaguely continuous curves t → µ t in SP with the global integrability property (3) such that t → G(µ t ) is a solution to (R ∞ -ODE). Each such curve t → µ t is a subprobability solution to (NLFPK). Indeed, due to µ t ∈ A t dt-a.s., we have and Lemma 3.3 (ii) applies. It remains to prove the additional assertion about probability solutions. To this end, assume Γ 0 is concentrated on P. Then, η(e 0 ∈ P) = 1 and hence the claim follows by Remark 3.2.
The final assertion of the theorem in particular implies: If Γ 0 ∈ P(P) for a weakly continuous solution (Γ t ) t≤T ⊆ P(SP) to (SP-CE), then Γ t ∈ P(P) for each t ≤ T . Of course, this is to be expected due to the global integrability condition in Definition 3.5.
Remark 3.9. Finally, let us explain why we developed the above result for subprobability solutions to (NLFPK) although our principal interest is restricted to probability solutions. If we directly consider solution curves (Γ t ) t≤T to (SP-CE) with Γ t ∈ P(P), we cannot prove that η in Theorem 3.7 is concentrated on C T P (in fact, not even η(C T P) > 0 could be shown). Indeed, inspecting the proof above, one may only prove that η • e −1 t is concentrated on P for each t ≤ T . But since P ⊆ SP is not closed, curves in the support of η may be proper subprobability-valued at single times. The deeper reason for this is that the range G(P) of G as in 3.3 as a map on P with the weak topology is not closed in R ∞ . It seems that one cannot resolve this issue by simply changing the function set G, since there exists no countable set of functions, which allows for a characterization of weak instead of vague convergence as in Lemma 3.3. Since SP with the vague topology is compact and the vague test function class C c (R d ) is separable, it is feasible to carry out the entire development for subprobability measures as above.
We also mention that to our understanding there is no inherent reason why the superposition principle could not be extended to larger spaces of measures (e.g. spaces of signed measures), as long as its topology allows for a suitable identification with R ∞ as in our present case. Our principal motivation from a probabilistic viewpoint was to study curves of probability measures, and we were only forced to extend to SP, the vague closure of P, by the reasons outlined above. In order to replace SP by some larger space of measures M, it seems indispensable that Lemma 3.4 remains true, i.e. that the range of M under a suitable homeomorphism is closed in R ∞ .

Consequences and applications
The following existence-and uniqueness results immediately follow from the superposition principle Theorem 3.7 and provide an equivalence between the nonlinear FPK-equation (NLFPK) and its linearized continuity equation (SP-CE).
Corollary 3.10. Let µ 0 ∈ SP and assume there exists a solution to (SP-CE) with initial condition δ µ0 . Then, there exists a subprobability solution to (NLFPK) with initial condition µ 0 . Moreover, if µ 0 ∈ P, then there exists a probability solution to (NLFPK) with initial condition µ 0 .
Proof. By Theorem 3.7 there exists a probability measure η concentrated on subprobability solutions to (NLFPK) with η • e −1 0 = δ µ0 . Hence, at least one such solution to (NLFPK) with initial condition µ 0 exists. The second assertion is treated similarly.
Corollary 3.11. Let µ 0 ∈ SP and assume there exists at most one vaguely continuous subprobability solution to (NLFPK) with initial condition µ 0 . Then, there exists also at most one weakly continuous solution (Γ t ) t≤T to (SP-CE) with initial condition δ µ0 . If µ 0 ∈ P, then, in the case of existence, Γ t (P) = 1 for each t ∈ [0, T ].

Application to coupled nonlinear-linear Fokker-Planck-Kolmogorov equations
Using the superposition principle, we prove an open conjecture posed in [17]. Let us shortly recapitulate the necessary framework. In [17], the authors consider a coupled nonlinear-linear FPK-equation of type i.e. comparing to our situation the first nonlinear equation is of type (NLFPK) and the second (linear) equation is obtained by "freezing" a solution (µ t ) t≤T to the first equation in the nonlinearity spot of L. For an initial condition (μ,ν) ∈ P × P, (16) is said to have a unique solution, if there exists a unique probability solution (µ t ) t≤T to the first equation in the sense of Definition 3.1 with µ 0 =μ and a unique weakly continuous curve (ν t ) t≤T ⊆ P, which solves the second equation with fixed coefficient µ t with ν 0 =ν (we refer to [17] for more details). The authors associate a linear continuity equation on R d × P to (16) in the following sense: Let L be the operator acting on functions with L as in (1) and L as in (10). Consider the continuity equation for weakly continuous curves of Borel probability measures on R d × P. The exact notion of solution can be found in [17], where also the following observation is made: A pair (µ t , ν t ) t≤T solves (16) if and only if Λ t := ν t × δ µt solves (17). Using our main result, we prove the following conjecture posed in Remark 4.4. of [17].
Proof. By Corollary (3.11), the unique solution to (SP-CE) with initial condition δμ is (δ µt ) t≤T . Let (Λ (1) t ) t≤T and (Λ (2) t ) t≤T be two solutions to (17) with initial conditionν × δμ. It is straightforward to check that the curves of second marginals (Λ 2 ) t≤T are probability solutions to (SP-CE) with initial condition δμ (where we denote by Π 2 the projection from R d × P onto the second coordinate). Hence, for each t ∈ [0, T ] It is immediate to show that each curve γ (i) solves the second equation of (16) with fixed µ t and initial conditionν. Hence, γ Hence, the unique solution to (17) with initial conditionν × δμ is given by (ν t × δ µt ) t≤T .

Superposition Principle for stochastic nonlinear Fokker-Planck-Kolmogorov Equations
We make use of the following notation specific to the stochastic case.
For two real-valued n × n matrices A, B we write A:B = n k,l=1 A kl B kl . We use the same notation for A = (A kl ) k,l≥1 and B = (B kl ) k,l≥1 , if either A or B contain only finitely many non-trivial entries. For the Hilbert space ℓ 2 with topology induced by the usual inner product ·, · ℓ 2 and norm || · || ℓ 2 , we denote the space of continuous ℓ 2 -valued functions on [0, T ] by C T ℓ 2 . On ℓ 2 and C T ℓ 2 , we unambiguously use the same notation e t , p i and π n as on R ∞ and C T R ∞ in the previous section. Reminiscent to the previous section, we set B(C T ℓ 2 ) = σ(e t , t ∈ [0, T ]) and denote the set of probability measures on this space by P(C T ℓ 2 ). For σ-algebras A 1 , A 2 , we denote by A 1 A 2 the σ-algebra generated by A 1 and A 2 .
We call a filtered probability space (Ω, F , (F t ) t≤T , P) complete, provided both F and F 0 contain all subsets of P-negligible sets N ∈ F (i.e. P(N ) = 0). This notion does not require (F t ) t≤T to be right-continuous. A real-valued Wiener process W = (W t ) t≤T on such a probability space is called an F t -Wiener process, if W t is F t -adapted and W u − W t is independent of F t for each 0 ≤ t ≤ u ≤ T . Pathwise properties of stochastic processes such as continuity are to be understood up to a negligible set with respect to the underlying measure.
As in the previous section, we consider SP as a compact Polish space with the vague topology. Let d 1 ≥ 1 and consider product-measurable coefficients on t, µ, x)) i,j≤d ∈ R d×d1 such that σ is bounded, and let L be as before, i.e.
In contrast to the deterministic framework of the previous section, here we consider nonlinear stochastic FPK-equations of type (SNLFPK) on [0, T ], to be understood in distributional sense as follows. With slight abuse of notation, for σ ∈ R d×d1 and x ∈ R d , we write σ · x = ( d i=1 σ ik x i ) k≤d1 , which is consistent with the standard inner product notation σ · x in the case d 1 = 1. Definition 4.1. (i) A pair (µ, W ) consisting of an F t -adapted vaguely continuous SP-valued stochastic process µ = (µ t ) t≤T and an F t -adapted, d 1 -dimensional Wiener process W = (W t ) t≤T on a complete probability space (Ω, F , (F t ) t≤T , P) is a subprobability solution to (SNLFPK), provided for each i, j ≤ d, k ≤ d 1 , and A probability solution to (SNLFPK) is a pair as above such that µ is a P-valued process (µ t ) t≤T with weakly continuous paths.
is separable with respect to uniform convergence and since the paths t → µ t (ω) are vaguely continuous, the exceptional sets in the above definition can be chosen independently of ϕ and t.
(ii) The first integral on the right-hand side of (19) is a pathwise (that is, for individual fixed ω ∈ Ω) integral with respect to the finite measure µ s (ω)ds on [0, T ] × R d . The second integral is a stochastic integral, which is defined, since the integrand is R d1 -valued, bounded, product-measurable and F t -adapted (Thm. 3.8 [7]). More precisely, where σ α = (σ iα ) i≤d denotes the α-th column of σ and the components W α , α ≤ d 1 , of W are real, independent Wiener processes.
By the global integrability assumption (18) and since σ is bounded, we obtain (in analogy to Remark 3.2) the following conservation of mass, which we use to prove the final assertion of the main result Theorem 4.8. Lemma 4.3. Let (µ t ) t≤T be a subprobability solution to (SNLFPK). If µ 0 ∈ P P-a.s., then the paths of t → µ t are P-valued P-a.s. and, hence, in particular weakly continuous.
Since the stochastic integral is continuous in t, a classical diagonal argument yields that there exists a subsequence (k l ) l≥1 along which (20) holds for all t ∈ [0, T ] on a set of full P-measure, independent of t. Let ω ′ ∈ Ω be from this set such that also µ 0 (ω ′ ) ∈ P and (19) holds for each t and ϕ. Note that the set of all such ω ′ has full P-measure. Then, similar to the reasoning in Remark 3.2 and by using (20), considering (19) for such ω ′ with ϕ k l in place of ϕ for the limit l −→ +∞, we obtain and hence the result.
Note that the above proof can be adjusted to extend (19) to each ϕ ∈ C 2 b (R d ).
Embedding SP into ℓ 2 In comparison with the deterministic case, we still consider SP as a manifold-like space with tangent spaces T µ SP = L 2 (R d , R d ; µ) as before. However, instead of embedding into R ∞ by G as in the previous section, now we need a global chart H : SP → ℓ 2 in order to handle the stochastic integral term later on. To this end, we replace the set of functions G = {g i , i ≥ 1} of the deterministic case by and consider the map The following lemma collects useful properties of H and H, which are in the spirit of Lemma 3.3 and 3.4. We point out that we could have used the function class H instead of G already in Section 3, but we decided to pass from G to H at this point in order to stress the technical adjustments necessary due to the stochastic case. (ii) H is a homeomorphism between SP and its range H(SP) ⊆ ℓ 2 , endowed with the ℓ 2 -subspace topology. In particular, H(SP) ⊆ ℓ 2 is compact.
Proof. (i) The first claim is obvious, since G is measure-determining. Concerning the second claim, note that it is clearly sufficient to have (19) g i are dense in the unit ball of C 2 c (R d ) with respect to || · || C 2 b , it is sufficient to have (19) for each such normalized function. Indeed, if ϕ k −→ k→∞ ϕ uniformly up to second-order partial derivatives, then by Itô-isometry which converges to 0 as k −→ ∞ due to the boundedness of σ. Hence, along a subsequence (k l ) l≥1 , we have a.s.
The a.s.-convergence of all other terms in (19) is clear. Therefore, it is sufficient to require (19) for a dense subset of the unit ball of C 2 c (R d ). Clearly, this yields at once that it is sufficient to have (19) for each h i ∈ H.
(ii) By definition, H maps into ℓ 2 . Since H is measure-determining, H is one-to-one, hence bijective onto its range. If µ n −→ n→∞ µ vaguely in SP, clearly H(µ n ) converges to H(µ) in the product topology. Since for any i ≥ 1 sup the convergence holds in ℓ 2 as well, which implies continuity of H. In particular, H(SP) ⊆ ℓ 2 is compact. Conversely, if H(µ n ) converges in ℓ 2 to some z = (z i ) i≥1 , then, by closedness of H(SP) ⊆ ℓ 2 , we have z = H(µ) for a unique element µ ∈ SP and µ n −→ n→∞ µ vaguely. Indeed, the latter follows as in Lemma 3.4 (i).
For consistency of notation, below we denote the test function class of the manifold-like space SP by F C 2 b (H) to stress that the base functions g i are now replaced by h i ∈ H. However, the class of test functions remains unchanged, because the transition from g i to h i can be incorporated in the choice of f .

Linearization of (SNLFPK)
As in the deterministic case, also for the stochastic nonlinear equation (SNLFPK) one can consider an associated linear equation for curves in P(SP). To the best of our knowledge, such a linearization for stochastic FPK-equations has not yet been considered in the literature. Of course, the basic idea stems from the deterministic case [17] discussed in the previous section. From Itô's formula one expects this linearized equation to be of second-order.
with the martingale M F given as Since M F 0 = 0 P-a.s., integrating with respect to P and defining the curve of measures in P(SP) As for the first-order term, which is interpreted as the pairing of the gradient ∇ SP F with the inhomogeneous vector field b + a∇ in the tangent bundle T SP, also the second-order term allows for a geometric interpretation: Recall that for a smooth, real function F on a Riemannian manifold M with tangent bundle T M , the Hessian Hess(F ) p at p ∈ M is a bilinear form on T p M with denotes the change of the vector field ∇F in direction η p at p. Recall that we consider SP as a manifoldlike space with gradient ∇ SP and that hence the reasonable notion of the Levi-Civita connection∇ L,SP on SP for σ ∈ T µ SP = L 2 (R d , R d ; µ), Y ∈ T SP at µ is given by For the representation of Hess(F ) for a test function F ∈ F C 2 b (H), we need to set Y = ∇ SP F . In this case, we can indeed make sense of The linear combination has to be understood in an x-wise sense with coefficient functions ∇h k , which are independent of the variable of interest µ. Denoting F k (µ) := (∂ k f ) µ(h 1 ), . . . , µ(h n ) , we then define Consequently, we have a reasonable notion of the Levi-Civita connection on SP at µ for σ ∈ T µ SP and ∇ SP F for F ∈ F C 2 b (H) as The section (∇ SP ) 2 F in T SP * ⊗ T SP * (and hence ∇ L,SP σ ∇ SP F and Hess(F ) below) is independent of the particular representation of F in (24). Indeed, we have (c.f. Appendix A [17]) for Since the gradient ∇ SP F is independent of the particular representation of F and σ ∈ L 2 (R d , R d ; µ) is arbitrary, also (∇ SP ) 2 F is independent of the representation of F .
by Itô's formula, the corresponding linear second-order equation for measures in distributional form is where Hess(f ) denotes the usual Euclidean Hessian matrix of f ∈ C 2 (R d ). In this spirit, it seems natural to consider (SNLFPK) as a stochastic equation with state space SP instead of R d as for (28) and (SP-CE) as the corresponding linear Fokker-Planck-type equation on SP.
By the above derivation, for any subprobability solution process (µ t ) t≤T to (SNLFPK) the curve (Γ t ) t≤T , Γ t := P • µ −1 t in P(SP) solves (SP-CE) in the sense of the following definition.
is fulfilled and for each F ∈ F C 2 b (H), 27 holds for each t ∈ [0, T ].

Transferring (SNLFPK) and (SP-FPK) to ℓ 2
Reminiscent to the deterministic case, we use the global chart H : SP → ℓ 2 to introduce auxiliary equations on ℓ 2 and the space of measures on ℓ 2 , respectively, as follows. Again, we use the notation For i, j ≥ 1, α ≤ d 1 , define the measurable coefficients B i for (t, µ) such that µ ∈ A t , and Σ α i and A ij on [0, T ] × SP by where C = C(a, b, d) is a finite constant independent of t, z and i ≥ 1. A similar argument is valid for eachΣ α . EachB i andΣ α i is product-measurable with respect to the ℓ 2 -topology due to the measurability of B and Σ α . Reminiscent to (R ∞ -CE) in the previous section, we associate to (SP-FPK) the FPK-equation on which we understand in the sense of the following definition, with∇ as in (13). Subsequently, we denote by F C 2 b (ℓ 2 ) the set of all mapsF : ℓ 2 → R of typeF = f • π n for n ≥ 1 and f ∈ C 2 b (R n ). Also, set Consequently, both summands in (31) contain only finitely many non-trivial summands.
Definition 4.7. A weakly continuous curve (Γ t ) t≤T ⊆ P(ℓ 2 ) is a solution to (ℓ 2 -FPK), if it fulfills the integrability condition and for anyF holds for each t ≤ T .

Main Result: Stochastic case
The main result of this section is the following superposition principle for solutions to (SNLFPK) and (SP-FPK), which generalizes Theorem 3.7 to stochastically perturbed equations.
As in the proof of 3.7, we proceed in three steps. Since parts of the proof are technically more involved than in the deterministic case, we first present the ingredients of each step and afterwards state the proof of Theorem 4.8 as a corollary.
Step 2: From (ℓ 2 -FPK) to the martingale problem (ℓ 2 -MGP): We introduce a martingale problem on ℓ 2 , which is related to (ℓ 2 -FPK) in the sense of Remark 4.11 below and is, roughly speaking, the stochastic analogue to (R ∞ -ODE) from the previous section. Recall the notation e t for the projection e t : C T ℓ 2 → ℓ 2 , e t : γ → γ t for t ≤ T . andF
In view of Proposition 4.13 below, we extend the coefficientsB i ,Σ α i (and hence alsoĀ ij ) to R ∞ viaB i := 0 =:Σ α i on [0, T ] × R ∞ \ℓ 2 . We still use the notationB,Σ α andĀ and note that they are B([0, T ]) ⊗ B(R ∞ )/B(R ∞ )-measurable due to Remark 4.12 below. Due to the same remark, we may regard any solution (Γ t ) t≤T to (ℓ 2 -FPK) as a solution to a FPK-equation on R ∞ by considering (ℓ 2 -FPK) with the extended coefficients and test functionsF ∈ F C 2 b (ℓ 2 ) extended to R ∞ by considering π n on R ∞ instead of ℓ 2 . Similarly, the formulation of the martingale problem (ℓ 2 -MGP) as in Definition 4.10 extends to R ∞ in the sense that a measureQ ∈ P(C T R ∞ ) is understood as a solution, provided the process (33) is aQ-martingale on C T R ∞ with respect to the natural filtration for eachF = f • π n : R ∞ → R as above.
We shall need the following superposition principle from [20], which lifts a solution to a FPKequation on R ∞ to a solution of the associated martingale problem. Note that in [20], the author assumes an integrability condition of order p > 1 instead of p = 1 as in (30) in order to essentially reduce the proof to the corresponding finite-dimensional result, see [20,Thm.2.14], which requires such a higher order integrability. However, since the latter result was extended to the case of an L 1 -integrability condition by the same author [21,Thm.2.5], it is easy to see that also the infinitedimensional result [20, Thm.7.1] holds for solutions with L 1 -integrability as in (30). [Superposition principle on R ∞ , Thm.7.1. [20] For any weakly continuous solution (Γ t ) t≤T ⊆ P(R ∞ ) to the R ∞ -extended version of (ℓ 2 -FPK), there existsQ ∈ P(C T R ∞ ), which solves the R ∞ -extended version of (ℓ 2 -MGP) such thatQ • e −1 t =Γ t for each t ∈ [0, T ]. Moreover, we have the following consequence for the solutions we are interested in. Note that paths t → z t ∈ H(SP) are continuous with respect to the product topology if and only if they are ℓ 2 -continuous. Hence, we may use the notation C T H(SP) unambiguously and consider it as a subset of either C T R ∞ or C t ℓ 2 . Since H(SP) ⊆ ℓ 2 is closed even with respect to the product topology, C T H(SP) belongs to B(C T ℓ 2 ) and B(C T R ∞ ).
Lemma 4.14. If in the situation of the previous proposition eachΓ t is concentrated on the Borel set H(SP) ⊆ R ∞ , thenQ is concentrated on continuous curves in H(SP). In particular, in this caseQ may be regarded as an element of P(C T ℓ 2 ) and a solution to the martingale problem (ℓ 2 -MGP) as in with mass on C T H(SP). It is clear that this measure fulfills Definition 4.10.
Hence, subsequently we may regard toQ as in Proposition 4.13 as a solution to (ℓ 2 -MGP) on either R ∞ or ℓ 2 without differing the notation. Recall the notation p i : ℓ 2 → R, p i (z) = z i . Proposition 4.16. Let (Γ t ) t≤T be a weakly continuous solution to (ℓ 2 -FPK) such thatΓ t (H(SP)) = 1 for each t ∈ [0, T ]. Then, there exists a solutionQ ∈ P(C T ℓ 2 ) to the martingale problem (ℓ 2 -MGP) such thatQ is concentrated on C T H(SP) withQ • e −1 t =Γ t for each t ∈ [0, T ]. Further, the results of Lemma 4.15 apply toQ.
Step 3: From (ℓ 2 -MGP) to (SNLFPK): For a given solutionQ ∈ P(C T ℓ 2 ) to (ℓ 2 -MGP), set C := B(C T ℓ 2 ) NQ and C t := σ(e s , s ≤ t) NQ for t ≤ T , where NQ denotes the collection of all subsets of sets N ∈ B(C T ℓ 2 ) withQ(N ) = 0. Of course, C and C t depend onQ, but we suppress this dependence in the notation. Without further mentioning, we understand suchQ as extended to C in the canonical way. Then, (C T ℓ 2 , C, (C t ) t≤T ,Q) is a complete filtered probability space. Clearly, (t, γ) →Σ(t, e t (γ)) is C t -progressively measurable from [0, T ] × C T ℓ 2 to L(R d1 , ℓ 2 ), the space of bounded linear operators from R d1 to ℓ 2 .
Remark 4.17. We extend (C T ℓ 2 , C, (C t ) t≤T ,Q) as follows. Let (Ω ′ , F ′′ , (F ′′ t ) t≤T , P ) be a complete filtered probability space with a real-valued F ′′ t -Wiener process β on it, define let F and F t be the P ′ -completion of F ′ and F ′ t , respectively, and denote the canonical extension of P ′ to F by P. Further, we denote the Wiener process β on the i-th copy of Ω ′ by β i and extend each β i to Ω by β i (ω) := β i (ω i ) for ω = γ × (ω i ) i≥1 ∈ Ω. Similarly, we extend each projection e t from C T ℓ 2 to Ω via e t (ω) := e t (γ) for ω as above, but keep the same notation for this extended process. Obviously, (e t ) t≤t is a continuous, F t -adapted process on Ω and each β i is an F t -Wiener process on Ω under P. Moreover, (e t ) t≤T and (β i ) i≥1 are independent on Ω with respect to P by construction. Further, it is clear that the process M i as in (34) is a P-martingale with respect to F t for each i ≥ 1 with covariation as in (35) and that (t, γ) →Σ(t, e t (γ)) ∈ L(R d1 , ℓ 2 ) is F t -progressively measurable on [0, T ] × Ω.
Finally, we need the following result, which is a special case of Theorem 2, [16].
Proposition 4.18. LetQ ∈ P(C T ℓ 2 ) be a solution to the martingale problem (ℓ 2 -MGP). Then, there exists a complete filtered probability space with an adapted d 1 -dimensional Wiener process W = (W α ) α≤d1 and an ℓ 2 -valued adapted continuous process X = (X t ) t≤T such that the law of X on C T ℓ 2 isQ and for i ≥ 1 and t ∈ [0, T ], we have a.s. These choices fulfill all requirements of [16]. In this case, the ℓ 2 -valued process X is given by X t = e t on Ω. Since all terms in (37) are continuous in t, the exceptional set may indeed by chosen independently of t ∈ [0, T ] and i ≥ 1.
The proof of Theorem 4.8 now follows from the above three step-scheme as follows.