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

Let the coefficients $a_{ij}$ and $b_i$, $i,j \leq d$, 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.


Introduction
In this paper we are concerned with linear Fokker-Planck-Kolmogorov equations of the form which are second order parabolic equations for measures. Here a ij , b i : [0, T ] × R d → R are given coefficients with suitable measurability-and regularity-assumptions imposed below. For s ∈ [0, T ] and a Borel probability measure ν on R d , we consider the Cauchy problem of (1) with initial condition µ s = ν. Our notion of a solution to such a Cauchy problem is that of narrowly continuous probability curves, i.e. a family of Borel probability measures (µ t ) t∈[s,T ] such that [s, T ] ∋ t → µ t is weakly continuous and holds for all smooth, compactly supported f : R d → R. A shorthand notation for equation (1) is In particular, in this work Stroock and Varadhan prove the following: Given continuous and bounded coefficients a ij and b i , assume there exists at least one continuous solution to the martingale problem with start in x ∈ R d at time s ≥ 0 for every pair (s, x). Then there exists a strong Markovian selection of such solutions. More precisely, one can select a solution to the martingale problem P s,x for every initial condition (s, x) such that the family (P s,x ) (s,x)∈R+×R d is a strong Markov process on the space of continuous functions C(R + , R d ). Such a consideration goes back to an earlier work of Krylov from 1973 (c.f. [3]). It is also proven that such a selection is unique if and only if the martingale problem is well-posed.
The aim of this paper is to prove similar results for the Fokker-Planck-Kolmogorov equation (1). Our first main result (see Theorem 3.2) is the following: Assume all coefficients a ij and b i are Borel measurable, globally bounded and continuous in the spatial variable. Assuming that the Cauchy problem for equation (1) has at least one narrowly continuous probability solution for every initial condition (s, ν), we prove the existence of a family of solutions (µ s,ν ) s,ν such that and all probability measures ν. We regard (2) as a flow property for solutions to (1). Moreover, in Theorem 3.16 we show: There exists exactly one such flow if and only if the Fokker-Planck-Kolmogorov equation is well-posed among narrowly continuous probability solutions.
The structure of this paper is as follows: In Section 2 we introduce notation, present the exact notion of solution to the Cauchy problem of equation (1) and state the assumptions on the coefficients a ij and b i . In the third section, we present our two main results, which are Theorem 3.2 and 3.16. We set up all necessary notions and tools for its proofs. In particular, this includes the aforementioned superposition principle by Figalli and Trevisan. Afterwards we prove both main theorems.
It would be interesting to generalize our results to Fokker-Planck-Kolmogorov equations on infinite dimensional state spaces, e.g. replacing R d by a separable Hilbert space H. The techniques we developed within the proof of Theorem 3.2 seem promising for this more general case as well. In order to widen the spectrum of possible applications, it is also desirable to establish our main results under more general assumptions on the coefficients a ij and b i . This could be a direction of further research on this topic.

Notation
Let us introduce basic notation, which we will frequently use in the sequel. For a metric space X, P(X) denotes the set of all Borel probability measures on X. If X = R d , we will simply write P := P(R d ). x ∈ Ω. The spatial derivative in the i-th euclidean direction for such a function is denoted by ∂ i f , i ≤ d, and the derivative w.r.t. the time-variable by ∂ t f . As usual, for two topological spaces X and Y , C(X, Y ) denotes the set of all continuous functions f : X → Y . For a time interval I ⊆ R + , a Borel curve of Borel (probability) measures on R d is a family (µ t ) t∈I such that t → µ t (A) is Borel measurable for every A ∈ B(R d ). We call such a Borel curve narrowly continuous, if t → f dµ t is continuous for all f ∈ C b (R d ), i.e. in other words, if the map t → µ t is continuous w.r.t. the topology of weak convergence of measures on P. The one-dimensional Lebesgue measure on R or on an interval I ⊆ R is denoted by dt. The set of all symmetric, non-negative definit d × d-matrices with real entries is denoted by S + (R d ).

Basic Setting
Let T > 0, which we regard as fixed throughout this paper, and let .., d} fulfill the following assumptions: a ij and b i are Borel measurable, continuous in x ∈ R d and globally bounded such that A(t, x) := (a ij (t, x)) i,j≤d ∈ S + (R d ) for every (t, x) ∈ [0, T ]×R d . These assumptions will be in force throughout the entire paper.
x) for every f : I × R d → R with at least two spatial derivatives. We always assume summation over repeated indices.
sometimes shortly written as is a narrowly continuous Borel curve of probability measures µ = (µ t ) t∈[s,T ] ∈ C [s, T ], P such that for all f ∈ C ∞ 0 (R d ) and all t ∈ [s, T ] . In particular µ s = ν.
Definition 2.2. The set of all narrowly continuous probability solutions to the above FPK-eq. with initial condition (s, ν) is denoted by F P (L, s, ν). Since we fix L = L A,b throughout, no confusion will occur when we write F P (s, ν) instead of F P (L, s, ν).

The Main Results
We say that the family (γ s,ν ) (s,ν)∈[0,T ]×P has the flow property, if for every 0 ≤ s ≤ r ≤ t ≤ T and ν ∈ P we have Our first main result is the following For the proof, we need several preparations.
For 0 ≤ s ≤ T < +∞, we set Q T s := [s, T ] ∩ Q. Let P Q T s := (γ q ) q∈Q T s |γ q ∈ P be endowed with the product topology of weak convergence of probability measures. Note that this space is Polish, since the topology of weak convergence on P is metrizable by the Prohorov metric. Moreover, the space C([s, T ], R d ) will always be endowed with the norm of uniform convergence.
is such a measure-determining family.
Remark 3.6. Note that for 0 ≤ s ≤ r ≤ T , the sequence (m r l ) l∈N0 is a subsequence of (m s l ) l∈N0 . This will be important when we verify the flow-property in the proof of Theorem 3.2.
i.e. it is simply the projection on all coordinates q ∈ Q T s .
Remark 3.8. In the sequel we will always consider C [s, T ], P with the product topology of weak convergence of probability measures and not, as common, with the topoloy of uniform convergence. Then J s as in Definition 3.7 is clearly continuous for every s ∈ [0, T ]. However, note that C [s, T ], P endowed with the product topology of weak convergence of measures is not metrizable.
The next remark points out an important technique of the proof of Theorem 3.2.
s . The advantage of this consideration is that P Q T s is, in contrast to C [s, T ], P , metrizable and hence continuity of a map f : P Q T s → R is equivalent to sequential continuity.
Before we turn to the proof of Theorem 3.2, we need to introduce a few additional important results about Fokker-Planck-Kolmogorov equations and their connection to the corresponding martingale problem. We start by recalling the definition of a solution to the martingale problem associated to the given coefficients a ij and b i . Definition 3.10. Let L = L A,b denote the differential operator associated to the given coefficients a ij and b i from above. A (continuous) solution to the martingale problem associated to L with initial  Proof. Obviously (P • π −1 t ) t∈[s,T ] is a Borel curve of probability measures. Due to the continuity of the canoncial projections π t for t ∈ [s, T ], (P • π −1 t ) t∈[s,T ] is clearly narrowly continuous. Using the martingale property of Definition 3.10 (ii), we obtain by integration with P , Fubini's theorem, a change of variables for image measures and (i) of the previous definition: The following theorem by Trevisan gives sort of an inverse of the above proposition. We point out that Trevisan's result (c.f. Theorem 2.5 in [5]), which is an extension of an earlier work by Figalli (c.f. [2]), does not require any continuity or boundedness of the coefficients (instead of the latter, in [5] global integrability of a ij and b i against dγ t (x)dt over [s, T ] × R d is required for any solution γ).
there exists a constant c f ≥ 0, which does not depend on P ∈ M, such that f a (π t ) + c f t is a non-negative submartingale w.r.t. to the natural filtration on C [s, T ], R d for every P ∈ M and every a ∈ R d .
Proof. It suffices to note that the proof of Theorem 1.4.6. in [4] still holds when one replaces C R + , R d by C [s, T ], R d for arbitrary 0 ≤ s ≤ T < ∞.
We now state a crucial compactness result for the set of solutions to the martingale problem associated to L with initial condition (s, ν). Essentially, this result is formulated as part of Lemma 12.2.1 in [4]. However, as this lemma only covers the compactness for deterministic initial conditionsi.e. ν = δ x for x ∈ R d -in the case of time-independent coefficients, for the convenience of the reader we decided to give a proof for the more general version, which we shall need below. Proof. Using Proposition 3.13, we first show that M P (s, ν) is precompact. Indeed, since P • π −1 s = ν for all P ∈ M P (s, ν) and every Borel probability meaure on R d is tight, we obtain (i) of Proposition 3.13. Concerning (ii), note that for non-negative f ∈ C ∞ 0 (R d ), due to the boundedness of a ij and b i and since f is compactly supported, there is a constant c f ≥ 0 such that for all u ∈ [s, T ]. Hence, [s, T ] ∋ t → t s L u f (π u )du + c f t is non-negative and increasing and thus for any P ∈ M P (s, ν) the process f (π t ) + c f t is a non-negative submartingale on [s, T ] w.r.t. the natural filtration on C([s, T ], R d ) under P . It is clear that the same constant c f works for all translates f a as well, since the boundedness of all coefficients and f ∈ C ∞ 0 (R d ) yields that (5) holds for every f a . Hence Proposition 3.13 applies and M P (s, ν) ⊆ P C([s, T ], R d ) is precompact.
It remains to show the closedness of M P (s, ν). Therefore, let {P n } n∈N ⊆ M P (s, ν) such that P n → n→∞ P weakly in P C([s, T ], R d ) . First of all it is obvious that P • π −1 s = ν, since the canoncial projection π s : C([s, T ], R d ) → R d is continuous and P n • π −1 s = ν for all n ∈ N. In order to prove P ∈ M P (s, ν), we show (s, T ) × R d and every continuous, bounded F r -measurable G r : C([s, T ], R d ) → R. But as P n ∈ M P (s, ν), this holds for every n ∈ N and since φ(z, π z ) − z s ∂ u φ(u, ·) • π u + L u φ(u, π u )du G r : C([s, T ], R d ) → R is bounded and continuous for every z ∈ [s, T ] (the latter due to a classical criterion for continuity of parameter-dependent integrals, using the continuity of the canonical projections π u plus the continuity in x ∈ R d and boundedness of a ij , b i and the boundedness of φ), the weak convergence P n → n→∞ P implies the desired equality. Therefore P ∈ M P (s, ν) and the proof is complete.
We now turn to the proof of Theorem 3.2.
Proof of Theorem 3.2: Let {f n } n∈N ⊆ C b (R d ) be a measure-determining family and η a fixed enumeration, as presented in Definition 3.5. Below we adopt all notations of Definition 3.5. Further fix (s, ν) ∈ [0, T ] × P. We define the following values, maps and sets, where for abbreviation we occasionally write µ instead of (µ t ) t∈[s,T ] . The map J s is as in Defnition 3.7. : We make the following observations: Since we assume F P (s, ν) = ∅ for all (s, ν) ∈ [0, T ] × P, we have J s F P (s, ν) = ∅. Therefore, and because each f n is bounded and for µ ∈ J s F P (s, ν) every marginal µ q is a probability measure, we have u m s 0 (s, ν) ∈ R. By the same argument we have u m s k+1 (s, ν) ∈ R for k ∈ N 0 , provided M m s k = ∅. Moreover, G s,ν m s 0 is continuous, because (µ q ) q∈Q T s → µ q m s 0 is clearly continuous from J s F P (s, ν) with the induced product topology of weak convergence to P with the topology of weak convergence and µ q m s  , ν), then, since {m s k |k ∈ N 0 } is an enumerating sequence of N × Q T s , we have f n dµ 1 q = f n dµ 2 q for all (n, q) ∈ N × Q T s . Since {f n } n∈N is measure-determining, we obtain µ 1 q = µ 2 q for all q ∈ Q T s , which implies k∈N0 M m s k (s, ν) ≤ 1. Even more, by definition of M m s 0 (s, ν), both (µ 1 q ) q∈Q T s and (µ 2 q ) q∈Q T s are elements of J s F P (s, ν) and thus, there is a unique element (μ t ) t∈[s,T ] ∈ F P (s, ν) such that µ 1 q =μ q = µ 2 q for all q ∈ Q T s . We conclude: There exists at most one element µ = (µ t ) t∈[s,T ] ∈ F P (s, ν) such that (µ q ) q∈Q T s ∈ J s F P (s, ν) .
We now show k∈N0 M m s k (s, ν) ≥ 1 :We start by showing that J s F P (s, ν) ⊆ P Q T s is compact.
By Proposition 3.14, M P (s, ν) is a compact subset of P C([s, T ], R d ) with the topology of weak convergence. Now define the following map: We prove continuity of Λ by letting Q n → n→∞ Q in P C([s, T ], R d ) , i.e. GdQ n → n→∞ GdQ for all in C [s, T ], P (endowed with the product topology of weak convergence of measures) and therefore Λ is continuous. Hence Λ M P (s, ν) ⊆ C [s, T ], P is compact. By Proposition 3.11 (giving "⊆") and Theorem 3.12 (giving " ⊇ "), we clearly have and therefore F P (s, ν) ⊆ C [s, T ], P is compact.
By Remark 3.8 and the above, the map is continuous as well. Therefore and by (6) Let us recapitulate what we have achieved so far: For each initial condition (s, ν) ∈ [0, T ] × P, we have characterized a unique element µ s,ν = (µ s,ν t ) t∈[s,T ] ∈ F P (s, ν) in the sense that it is the unique element in F P (s, ν) such that (µ s,ν ) q∈Q T s ∈ k∈N0 M m s k (s, ν). We consider µ s,ν as an extremal element in F P (s, ν), because by construction it is "iteratively maximal" in the sense that (µ s,ν q ) q∈Q T s ∈ M m s k for every k ∈ N 0 .
To conclude the proof, it remains to show that the family (µ s,ν ) (s,ν)∈[0,T ]×P has the desired flow property, i.e. it fulfills (4) of Definition 3.1. In order to prove this, let us fix 0 ≤ s ≤ r ≤ T and ν ∈ P and let µ s,ν = (µ s,ν t ) t∈[s,T ] ∈ F P (s, ν) be the unique iteratively maximal solution for the initial condition (s, ν) as described in the previous passage. Consider the initial condition (r, µ s,ν r ) and let γ = (γ t ) t∈[r,T ] ∈ F P (r, µ s,ν r ) be the unique iteratively maximal solution for the initial condition (r, µ s,ν r ), i.e. in our notation γ t = µ r,µ s,ν r t . We need to show We proceed as follows: Define ζ = (ζ t ) t∈[s,T ] by which is a well-defined Borel curve of probability measures, since γ r = µ s,ν r and since both µ s,ν and γ are Borel curves. Clearly ζ ∈ F P (s, ν): Indeed, it is obvious that [s, T ] ∋ t → ζ t is narrowly continuous, ζ s = ν holds and for f ∈ C ∞ 0 (R d ) and t ∈ [s, T ] we have by the characterizing property of γ and hence equality in (9). If q m s 0 , q m s 1 ∈ ]r, T ], then m s 0 = m r 0 , m s 1 = m r 1 and both (µ s,ν q ) q∈Q T r and (γ q ) q∈Q T r are in M m r 0 (r, µ s,ν r ) and we also obtain (10). Hence, equality in (9) holds in any case. Iterating this procedure yields f n m s k dµ s,ν q m s k = f n m s k dζ q m s k for all k ∈ N 0 . As (m s k ) k∈N0 is the enumerating sequence of N×Q T s and {f n } n∈N is measure-determining on R d , this yields µ s,ν q = ζ q for all q ∈ Q T s , so in particular µ s,ν q = γ q for all q ∈ Q T r . Since both (γ q ) q∈Q T r and (µ s,ν q ) q∈Q T r belong to J r F P (r, µ s,ν r ) , we obtain (7). Remark 3.15. We point out that if we perform the procedure of the above proof for a different measure-determining family {g n } n∈N ⊆ C b (R d ) and/or with a different enumeration δ instead of η, we may obtain a different family of solutions with the flow property. This also becomes apparent in the next theorem and its proof. Above that, in principle, one could also consider a different dense, countable subset of [0, T ] instead of Q T 0 . This could also lead to a differnt solution family with the flow property.
The following theorem is an interesting consequence of the method we used to construct a flow of solutions within the proof of Theorem 3.2.
Theorem 3.16. Let all assumptions of Theorem 3.2 be in force. Then the following are equivalent: (i) The FPK-eq. is well-posed among narrowly continuous probability solutions, i.e. F P (s, ν) = 1 for all (s, ν) ∈ [0, T ] × P.
Proof. The implication (i) =⇒ (ii) follows immediately, because the existence of a flow follows by Theorem 3.2 and due to well-posedness, there can obviously not be two differing flows.