Linearization and a superposition principle for deterministic and stochastic nonlinear Fokker-Planck-Kolmogorov equations
Abstract
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.
Keywords: Nonlinear Fokker-Planck equation, McKean-Vlasov stochastic differential equation, diffusion process, superposition principle
2010 MSC: 60J60, 58J65
1 Introduction
In this work we are concerned with nonlinear Fokker-Planck-Kolmogorov equations (FPK-equations) on , both deterministic
| (NLFPK) |
and perturbed by a first-order stochastic term driven by a finite-dimensional Wiener process
| (SNLFPK) |
with solutions being continuous curves of subprobability measures . Here, denotes the formal dual of a second-order differential operator acting on sufficiently smooth functions via
| (1) |
with coefficients and depending on and (in general non-locally) on the solution . These equations are to be understood in distributional sense, see Definition 3.1 and 4.1. The nonlinearity arises from the dependence of and 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 , the space of Borel probability measures on , of type
| (-CE) |
in the sense of distributions, with the linear operator acting on sufficiently smooth real functions on via the gradient operator on as
Precise information on this operator and equation (-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 to (-CE) is a superposition of solutions to (NLFPK), i.e. (denoting by the canonical projection )
| (2) |
for some probability measure 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 in of type
| (-FPK) |
where, roughly,
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 . The second main result of this work, Theorem 4.8, is then the stochastic generalization of the deterministic case: For any solution to (-FPK), there exists a solution process to (SNLFPK) on some probability space such that has distribution . 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 be an inhomogeneous vector field and consider the (nonlinear) ODE
| (ODE) |
and the linear continuity equation for curves of Borel (probability) measures on
| (CE) |
understood in distributional sense. In the seminal paper [1], L. Ambrosio showed the following: Any (probability) solution to (CE) with an appropriate global integrability condition is a superposition of solution curves to (ODE), i.e. there exists a (probability) measure on the space of continuous paths with values in the state space of (ODE), , which is concentrated on solutions to (ODE) such that
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 . Likewise, (CE) is replaced by (-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 (-CE), we linearize the equation.
Concerning the stochastic case, consider a stochastic differential equation on
| (SDE) |
By Itô’s formula, the one-dimensional marginals of any (weak) martingale solution solve the corresponding linear FPK-equation
| (FPK) |
where is a linear second-order diffusion operator with coefficients and . 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 to (FPK), there exists a (weak) martingale solution to (SDE) with one-dimensional marginals , 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 by the stochastic equation for measures (SNLFPK) and the corresponding second-order equation for measures (FPK) by (-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 and the corresponding continuity equation (for the deterministic case) and for martingale solutions and FPK-equations on (for the stochastic case) by Ambrosio and Trevisan ([2], [20]). The key technique is to transfer (-CE) and (NLFPK) (and, similarly, (-FPK) and (SNLFPK) for the stochastic case) to suitable equations on via a homeomorphism between and (replaced by 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
| (DDSDE) |
see the classical papers [15, 10, 18] as well as the more recent works [12, 11, 8]. Here, denotes the distribution of and is not to be confused with the operators and from above. As in the non-distribution 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 to (DDSDE) provides a solution to a nonlinear FPK-equation of type (NLFPK) via 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 (-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 , involving a geometric interpretation of the latter. On the other hand, it is our future plan to further study the geometry of 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 (-CE) and (-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 (-CE) and (-FPK) concentrated on probability measures to a measure on the space of continuous probability measure-valued paths , for technical reasons we more generally develop our results for vaguely continuous subprobability solutions (i.e. ). 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 and in Section 2, Section 3 contains the deterministic case, i.e. the superposition principle between solutions to (-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.
2 Notation and Preliminaries
We introduce notation and repeat basic facts on spaces and topologies of measures.
Notation
For a measure space and a measurable function , we set whenever the integral is well-defined. For , we denote by the Dirac measure in , i.e. if and only if and else. For a topological space with Borel -algebra we denote the set of continuous bounded functions by , the set of Borel probability measures on by and write . If , we let denote the trace of on . For , a family of finite Borel measures on is a Borel curve, if is Borel measurable for each . A set of functions is called measure-determining, if for each implies for any two finite Borel measures on .
For , the usual inner product is denoted by and, with slight abuse of notation, we also denote by the inner product in (the Hilbert space of square-summable real-valued sequences ). For , we set . If has first- and second-order partial derivatives, we denote them by and for .
We use notation for function spaces as follows. For , denotes the subset of functions in with continuous, bounded partial derivatives up to order , with the usual norm for . Likewise, denotes the subset of all such with compact support; for , we write instead. For , and a measure on , we denote by the space of Borel functions such that
where denotes the standard Euclidean norm on . For , denotes the usual inner product on the Hilbert space . For and a topological space , we write for the set of continuous functions . By we denote the space of symmetric, positive-semidefinite -matrices with real entries.
Basic properties of spaces of measures
Probability measures
For a topological space , we endow with the topology of weak convergence of measures, i.e. the initial topology of the maps , . If is Polish, then so is .
Subprobability measures
By we denote the set of all Borel subprobability measures on , i.e. if and only if is a non-negative measure on with . Throughout, we endow with the vague topology, i.e. the initial topology of the maps , . Hence, a sequence converges to in if and only if for each . Its Borel -algebra is denoted by . In particular, , the set of Borel probability measures on , is a topological space with the weak topology of probability measures on . The Riesz-Markov representation theorem yields that with the vague topology coincides with the positive half of the closed unit ball of the dual space of with the weak*-topology. Hence with the vague topology is compact. It is also Polish and is vaguely lower semicontinuous, see [13, Ch.4.1]. In particular,
Recall that . Hence, in the sequel we may consider measures as elements in with mass on .
In contrast to weak convergence in , vague convergence in 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 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.
3 Superposition Principle for deterministic nonlinear Fokker-Planck-Kolmogorov Equations
Fix throughout, let each component of the coefficients
be -measurable and consider the operator as in (1).
Definition 3.1.
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.
Remark 3.2.
-
(i)
Any subprobability solution with is a probability solution. Indeed, to prove this it suffices to show for each . Since fulfills (4), it suffices to choose a sequence , , from with the following properties: pointwise such that , pointwise with all first and second order derivatives bounded by some uniformly in and . Considering (4) for the limit , we obtain, by (3) and dominated convergence, for each
and hence the claim.
-
(ii)
By the above argument, one shows that for any subprobability solution, (4) holds for each .
Geometric approach to
For our goals, it is preferable to consider as a manifold-like space. We refer the reader to the appendix in [17], where for the space of probability measures the tangent spaces and a suitable test function class ,
| (5) |
have been introduced. Further, based on these choices, a natural pointwise definition of the gradient as a section in the tangent bundle
for as above is given by
which is shown to be independent of the representation of in terms of and . The setting in the present paper is nearly identical, but we consider the manifold-like space with the vague topology instead of with the weak topology as in [17], because is embedded in in the following sense. Let
| (6) |
be dense in with respect to such that no is constantly . Clearly, any such set of functions is dense in with respect to uniform convergence and measure-determining. Such sets of functions are sufficiently extensive to characterize the topology of as well as solutions to (NLFPK):
Lemma 3.3.
Proof.
Considering as a (infinite-dimensional) manifold-like topological space, any set of functions as above provides a global chart (i.e., an atlas consisting of a single chart) for , as it yields an embedding (cf. Lemma 3.4).
Consider as a Polish space with the topology of pointwise convergence and the range of as introduced below with its subspace topology. We write for the set of all elements in with values in . For , we denote by the canonical projection on
and, likewise, by the projection on . Subsequently, without further mentioning, we consider the spaces and with -algebras
respectively. These algebras coincide with the Borel -algebras with respect to the topology of uniform convergence (because both and are Polish). Also, consider with the natural subspace -algebra of . We refer to these -algebras as the canonical -algebras on the respective spaces and denote the set of probability measures on the respective -algebras by and .
Lemma 3.4.
Let be a set of functions as in (6).
-
(i)
The map , depending on ,
(8) is a homeomorphism between and its range (hence, formally, a global chart for ). In particular, is compact. Moreover, if is another set as in (6) with corresponding chart , then for a unique homeomorphism on .
-
(ii)
The map
is measurable and one-to-one with measurable inverse . Further, is a measurable set, i.e. .
Proof.
-
(i)
The continuity of is obvious by definition of the vague topology on and since . Since is compact with respect to the vague topology, compactness of follows. is measure-determining on , which implies that is one-to-one. Since by definition
continuity of follows from Lemma 3.3 (i). The final assertion follows, since for as in the assertion, with , is a homeomorphism.
-
(ii)
Since is one-to-one and measurable, so is . Clearly, is the range of and hence is a bijection between standard Borel spaces (the latter, because and with the respective topologies are Polish). This yields the measurability of . Finally, closedness of implies that is a measurable set, because carries the subspace topology inherited from .
∎
By part (i) of the previous lemma it is justified to fix a set for the remainder of the section. In order to switch between test functions on and in an equivalent way, we slightly deviate from the test function class presented in [17] (see (5)) and, instead, consider
where the restriction is made for consistency with the stochastic case later on only. We summarize our geometric interpretation of , which is of course still a close adaption of the ideas presented in [17]:
For the manifold-like space , we consider smooth test functions , with being fixed as in (6). For each , we have the tangent space and the gradient
for as a section in the tangent bundle , which is independent of the representation of . Adding to the approach of as a manifold-like space, the global chart as in (8) embeds into . However, we do not rigorously treat as a (Fréchet-)manifold and consider the embedding merely as a tool to transfer (NLFPK) and its corresponding continuity equation to equivalent equations over , as outlined below.
The continuity equation (-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 . More precisely, in analogy to the derivation in [17], it is readily seen that any subprobability solution to (NLFPK) induces a curve of elements in , , , with
| (9) |
for each and . Here, we set (similarly for ), and abbreviated
We rewrite (9) in distributional form in duality with as
Setting
| (10) |
this is just the linear continuity equation (-CE). The term has rigorous meaning only, if has sufficiently regular components in order to put the derivative on via integration by parts, which we do not assume at any point.
Considering as a manifold-like space, one may formally regard to as a time-dependent section in the tangent bundle .
More generally, we introduce the following notion of solution to (-CE) (see [17]):
Definition 3.5.
The choice of as in (6) implies that any solution in the above sense fulfills (12) even for each , i.e. for the larger class of test functions considered in [17] (upon extending their domain from to ). In particular, this notion of solution is independent of . The main result of this chapter, Theorem 3.7, states that any solution to (-CE) as in Definition 3.5 arises as a superposition of solutions to (NLFPK). Note that for , uniqueness of solutions to (-CE) with implies uniqueness of subprobability solutions to (NLFPK) with .
Transferring (NLFPK) and (-CE) to
We use the global chart and the map of Lemma 3.4 to reformulate both (NLFPK) and (-CE) on . Define a Borel vector field component-wise as follows. For , consider the Borel set ,
and define via
Now define via
which is Borel measurable by Lemma 3.4. Next, consider the differential equation on
| (-ODE) |
which turns out to be the suitable analogue to (NLFPK) on . Analogously, the corresponding continuity equation for curves of Borel probability measures on , i.e.
| (-CE) |
with as introduced below, is the natural analogue of the linear continuity equation (-CE). Roughly, these analogies are to be understood in the sense that solutions to (NLFPK) and (-CE) can be transferred to solutions to (-ODE) and (-CE), respectively, via the chart . We refer to the proof of the main result below for more details. Let
denote the canonical projection to the -th component, set and
By we denote the gradient-type operator on , acting on via
| (13) |
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.
3.1 Main Result: Deterministic case
The following theorem is the main result for the deterministic case.
Theorem 3.7.
Let be Borel coefficients on . For any weakly continuous solution to (-CE) in the sense of Definition 3.5, there exists a probability measure , which is concentrated on vaguely continuous subprobability solutions to (NLFPK) such that
Moreover, if , then is concentrated on weakly continuous probability solutions to (NLFPK).
The proof relies on a superposition principle for measure-valued solution curves of continuity equations on and its corresponding differential equation, which we recall in Proposition 3.8 below. More precisely, we proceed in three steps. First, we transfer to a solution to (-CE). Then, by Proposition 3.8 below we obtain a measure with , which is concentrated on solution curves to (-ODE). Finally, we transfer back to a measures with the desired properties. Below, we denote by the set of test functions of same type as in , but with in place of .
Proposition 3.8.
We proceed to the proof of the main result.
Proof of Theorem 3.7: Let be a weakly continuous solution to (-CE) as in Definition 3.5.
Step 1: From (-CE) to (-CE): Set
with as in Lemma 3.4, which corresponds to the fixed set of functions . Since is continuous, is a weakly continuous curve of Borel subprobability measures on . We show that solves (-CE). Indeed, the integrability condition (14) is fulfilled, since fulfills Definition 3.5. Further, since solves (-CE), we have for any and
| (15) |
and hence, abbreviating by and setting for as above, we have
and, furthermore, for each
Comparing with (15), it follows that is a solution to (-CE) as claimed, because was arbitrary and hence as above is arbitrary in . By standard approximation, one extends the above equation to test functions from .
Step 2: From (-CE) to (-ODE): Proposition 3.8 implies the existence of a measure such that
-
(i)
for each
-
(ii)
is concentrated on solution paths of (-ODE).
Step 3: From (-ODE) to (NLFPK): We show that the measure , with as in Lemma 3.4 fulfills all desired properties. Indeed, since
for each we deduce that is concentrated on . By Lemma 3.4, is closed. Since by construction is concentrated on continuous curves in , is concentrated on . Further, is a measurable set and is measurable by Lemma 3.4. Therefore, we may define via
It remains to verify for all and that is concentrated on subprobability solutions to (NLFPK). Concerning the first matter, we have
and
Since and coincide as measurable maps on and it was shown above that is concentrated on , we obtain
Concerning the second aspect, note that by definition of and and by the equality , (11) for implies that is concentrated on vaguely continuous curves in with the global integrability property (3) such that is a solution to (-ODE). Each such curve is a subprobability solution to (NLFPK). Indeed, due to -a.s., we have
and Lemma 3.3 (ii) applies.
It remains to prove the additional assertion about probability solutions. To this end, assume is concentrated on . Then, and hence the claim follows by Remark 3.2. ∎
The final assertion of the theorem in particular implies: If for a weakly continuous solution to (-CE), then for each . 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 to (-CE) with , we cannot prove that in Theorem 3.7 is concentrated on (in fact, not even could be shown). Indeed, inspecting the proof above, one may only prove that is concentrated on for each . But since is not closed, curves in the support of may be proper subprobability-valued at single times. The deeper reason for this is that the range of as in 3.3 as a map on with the weak topology is not closed in . It seems that one cannot resolve this issue by simply changing the function set , 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 with the vague topology is compact and the vague test function class 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 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 , the vague closure of , by the reasons outlined above. In order to replace by some larger space of measures , it seems indispensable that Lemma 3.4 remains true, i.e. that the range of under a suitable homeomorphism is closed in .
3.2 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 (-CE).
Corollary 3.10.
Proof.
Corollary 3.11.
Proof.
Let and be weakly continuous solutions to (-CE) with for . By Theorem 3.7, there exist probability measures , , concentrated on subprobability solutions to (NLFPK) with initial condition such that for each and . By assumption, we obtain for a unique element and thus also . If , then by Remark 3.2, which gives the second assertion. ∎
3.2.1 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
| (16) |
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 to the first equation in the nonlinearity spot of . For an initial condition , (16) is said to have a unique solution, if there exists a unique probability solution to the first equation in the sense of Definition 3.1 with and a unique weakly continuous curve , which solves the second equation with fixed coefficient with (we refer to [17] for more details). The authors associate a linear continuity equation on to (16) in the following sense: Let be the operator acting on functions
via
with as in (1) and as in (10). Consider the continuity equation
| (17) |
for weakly continuous curves of Borel probability measures on . The exact notion of solution can be found in [17], where also the following observation is made: A pair solves (16) if and only if solves (17). Using our main result, we prove the following conjecture posed in Remark 4.4. of [17].
Proposition 3.12.
Proof.
By Corollary (3.11), the unique solution to (-CE) with initial condition is . Let and be two solutions to (17) with initial condition . It is straightforward to check that the curves of second marginals and are probability solutions to (-CE) with initial condition (where we denote by the projection from onto the second coordinate). Hence, for each
Consequently, is of product type, i.e. for weakly continuous curves , . It is immediate to show that each curve solves the second equation of (16) with fixed and initial condition . Hence, for each and , which implies . Hence, the unique solution to (17) with initial condition is given by . ∎
4 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 matrices we write :. We use the same notation for and , if either or contain only finitely many non-trivial entries.
For the Hilbert space with topology induced by the usual inner product and norm , we denote the space of continuous -valued functions on by . On and , we unambiguously use the same notation and as on and in the previous section. Reminiscent to the previous section, we set and denote the set of probability measures on this space by . For -algebras , , we denote by the -algebra generated by and .
We call a filtered probability space complete, provided both and contain all subsets of -negligible sets (i.e. ). This notion does not require to be right-continuous. A real-valued Wiener process on such a probability space is called an -Wiener process, if is -adapted and is independent of for each . 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 as a compact Polish space with the vague topology. Let and consider product-measurable coefficients on
such that is bounded, and let be as before, i.e.
for and .
In contrast to the deterministic framework of the previous section, here we consider nonlinear stochastic FPK-equations of type (SNLFPK) on , to be understood in distributional sense as follows. With slight abuse of notation, for and , we write , which is consistent with the standard inner product notation in the case .
Definition 4.1.
-
(i)
A pair consisting of an -adapted vaguely continuous -valued stochastic process and an -adapted, -dimensional Wiener process on a complete probability space is a subprobability solution to (SNLFPK), provided
(18) for each , and
(19) holds -a.s. for each and .
-
(ii)
A probability solution to (SNLFPK) is a pair as above such that is a -valued process with weakly continuous paths.
Remark 4.2.
-
(i)
Since is separable with respect to uniform convergence and since the paths are vaguely continuous, the exceptional sets in the above definition can be chosen independently of and .
-
(ii)
The first integral on the right-hand side of (19) is a pathwise (that is, for individual fixed ) integral with respect to the finite measure on . The second integral is a stochastic integral, which is defined, since the integrand
is -valued, bounded, product-measurable and -adapted (Thm. 3.8 [7]). More precisely,
where denotes the -th column of and the components , , of are real, independent Wiener processes.
By the global integrability assumption (18) and since 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 be a subprobability solution to (SNLFPK). If -a.s., then the paths of are -valued -a.s. and, hence, in particular weakly continuous.
Proof.
Let approximate the constant function as in Remark 3.2. Then, by Itô-isometry, for each , there exists a subsequence such that
| (20) |
Since the stochastic integral is continuous in , a classical diagonal argument yields that there exists a subsequence along which (20) holds for all on a set of full -measure, independent of . Let be from this set such that also and (19) holds for each and . Note that the set of all such has full -measure. Then, similar to the reasoning in Remark 3.2 and by using (20), considering (19) for such with in place of for the limit , we obtain
and hence the result. ∎
Note that the above proof can be adjusted to extend (19) to each .
Embedding into
In comparison with the deterministic case, we still consider as a manifold-like space with tangent spaces as before. However, instead of embedding into by as in the previous section, now we need a global chart
in order to handle the stochastic integral term later on. To this end, we replace the set of functions of the deterministic case by
| (21) |
and consider the map
The following lemma collects useful properties of and , which are in the spirit of Lemma 3.3 and 3.4. We point out that we could have used the function class instead of already in Section 3, but we decided to pass from to at this point in order to stress the technical adjustments necessary due to the stochastic case.
Lemma 4.4.
Proof.
-
(i)
The first claim is obvious, since is measure-determining. Concerning the second claim, note that it is clearly sufficient to have (19) for each with . Since the functions are dense in the unit ball of with respect to , it is sufficient to have (19) for each such normalized function. Indeed, if uniformly up to second-order partial derivatives, then by Itô-isometry
which converges to as due to the boundedness of . Hence, along a subsequence , 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 . Clearly, this yields at once that it is sufficient to have (19) for each .
-
(ii)
By definition, maps into . Since is measure-determining, is one-to-one, hence bijective onto its range. If vaguely in , clearly converges to in the product topology. Since for any
the convergence holds in as well, which implies continuity of . In particular, is compact. Conversely, if converges in to some , then, by closedness of , we have for a unique element and 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 by to stress that the base functions are now replaced by . However, the class of test functions remains unchanged, because the transition from to can be incorporated in the choice of .
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 . 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.
Let be a subprobability solution to (SNLFPK) (with underlying measure ) and choose any from . Again, we abbreviate and similarly for and . By Itô’s formula, we have -a.s.
with the martingale given as
Since -a.s., integrating with respect to and defining the curve of measures in
yields
| (22) |
As for the first-order term, which is interpreted as the pairing of the gradient with the inhomogeneous vector field in the tangent bundle , also the second-order term allows for a geometric interpretation: Recall that for a smooth, real function on a Riemannian manifold with tangent bundle , the Hessian at is a bilinear form on with
| (23) |
where denotes the Levi-Civita-connection on , the unique affine connection compatible with the metric tensor on and denotes the usual gradient on . Intuitively, denotes the change of the vector field in direction at . Recall that we consider as a manifold-like space with gradient and that hence the reasonable notion of the Levi-Civita connection on for at is given by
whenver is defined in . For the representation of for a test function , we need to set . In this case, we can indeed make sense of
because the gradient
is a linear combinations of the ”-like” functions . The linear combination has to be understood in an -wise sense with coefficient functions , which are independent of the variable of interest . Denoting , we then define
| (24) |
Consequently, we have a reasonable notion of the Levi-Civita connection on at for and for as
| (25) |
The section in (and hence and below) is independent of the particular representation of in (24). Indeed, we have (c.f. Appendix A [17]) for
the following pointwise (in ) equality for each
Since the gradient is independent of the particular representation of and is arbitrary, also is independent of the representation of .
Considering (23), we then set for and
| (26) |
which is a (symmetric) bilinear form on and rewrite (4) as
| (27) |
(with and similarly for and ). Introducing the second-order operator , acting on via
we arrive at the distributional formulation of (-FPK)
as in the introduction.
Remark 4.5.
Equation (-CE) is the natural analogue to second-order FPK-equations over Euclidean spaces. Indeed, for a stochastic equation on
| (28) |
by Itô’s formula, the corresponding linear second-order equation for measures in distributional form is
with
where denotes the usual Euclidean Hessian matrix of . In this spirit, it seems natural to consider (SNLFPK) as a stochastic equation with state space instead of as for (28) and (-CE) as the corresponding linear Fokker-Planck-type equation on .
Transferring (SNLFPK) and (-FPK) to
Reminiscent to the deterministic case, we use the global chart to introduce auxiliary equations on and the space of measures on , respectively, as follows. Again, we use the notation
For , , define the measurable coefficients for such that , and and on by
and set
Now, transferring to , define and on component-wise via
,
and are -valued, since for
where is a finite constant independent of and . A similar argument is valid for each . Each and is product-measurable with respect to the -topology due to the measurability of and . Reminiscent to (-CE) in the previous section, we associate to (-FPK) the FPK-equation on
| (-FPK) |
which we understand in the sense of the following definition, with as in (13). Subsequently, we denote by the set of all maps of type for and . Also, set
Consequently, both summands in (31) contain only finitely many non-trivial summands.
Definition 4.7.
A weakly continuous curve is a solution to (-FPK), if it fulfills the integrability condition
| (30) |
and for any , ,
| (31) |
holds for each .
4.1 Main Result: Stochastic case
The main result of this section is the following superposition principle for solutions to (SNLFPK) and (-FPK), which generalizes Theorem 3.7 to stochastically perturbed equations.
Theorem 4.8.
Let be bounded on . Let be a weakly continuous solution to (-FPK). Then, there exists a complete filtered probability space , an adapted -dimensional Wiener process and a -valued adapted vaguely continuous process such that solves (SNLFPK) and
holds for each .
Moreover, if is concentrated on , i.e. , then the paths are -valued for -a.e. and hence even weakly continuous.
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 1: From (-FPK) to (-FPK):
Proof.
Step 2: From (-FPK) to the martingale problem (-MGP): We introduce a martingale problem on , which is related to (-FPK) in the sense of Remark 4.11 below and is, roughly speaking, the stochastic analogue to (-ODE) from the previous section. Recall the notation for the projection , for .
Definition 4.10.
A measure is a solution to the -martingale problem (-MGP), provided
| (32) |
and
| (33) |
is a -martingale on with respect to the natural filtration on for any .
Remark 4.11.
In view of Proposition 4.13 below, we extend the coefficients (and hence also ) to via
We still use the notation , and and note that they are -measurable due to Remark 4.12 below. Due to the same remark, we may regard any solution to (-FPK) as a solution to a FPK-equation on by considering (-FPK) with the extended coefficients and test functions extended to by considering on instead of . Similarly, the formulation of the martingale problem (-MGP) as in Definition 4.10 extends to in the sense that a measure is understood as a solution, provided the process (33) is a -martingale on with respect to the natural filtration for each as above.
Remark 4.12.
We recall that and . In particular, any probability measure uniquely extends to a Borel probability measure on via , .
We shall need the following superposition principle from [20], which lifts a solution to a FPK-equation on to a solution of the associated martingale problem. Note that in [20], the author assumes an integrability condition of order instead of 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 -integrability condition by the same author [21, Thm.2.5], it is easy to see that also the infinite-dimensional result [20, Thm.7.1] holds for solutions with -integrability as in (30).
Proposition 4.13.
Moreover, we have the following consequence for the solutions we are interested in. Note that paths are continuous with respect to the product topology if and only if they are -continuous. Hence, we may use the notation unambiguously and consider it as a subset of either or . Since is closed even with respect to the product topology, belongs to and .
Lemma 4.14.
If in the situation of the previous proposition each is concentrated on the Borel set , then is concentrated on continuous curves in . In particular, in this case may be regarded as an element of and a solution to the martingale problem (-MGP) as in Definition 4.10.
Proof.
The closedness of yields
due to for each . By the observation above this lemma, it follows
and we can therefore consider as a probability measure on via
with mass on . It is clear that this measure fulfills Definition 4.10. ∎
Hence, subsequently we may regard to as in Proposition 4.13 as a solution to (-MGP) on either or without differing the notation. Recall the notation ,
Lemma 4.15.
Let be a solution to the martingale problem (-MGP) on . Then, for any , the process
| (34) |
is a real-valued, continuous -martingale on with respect to the canonical filtration. The covariation of and is -a.s. given by
| (35) |
Proof.
For , let , consider , and let
Note that on , independent of . For , introduce the stopping time with respect to the canonical filtration on . Clearly, pointwise. Consider such that for .
Since and for , we have
and, setting ,
Since the latter is a continuous -martingale for each , it follows that is a continuous local -martingale. Concerning (35), it suffices to prove that for any , , we have
| (36) |
with
Indeed, from here (35) follows by considering (36) for , localization of the local martingale and polarization for the quadratic (co-)variation. Concerning (36), it is standard (cf. [19, p.73,74]) to use Itô’s product rule to obtain that
is a continuous -martingale on , where we denote by the integrand of the integral term in the definition of . A straightforward calculation yields
which completes the proof. ∎
We summarize the results of this step in the following proposition.
Proposition 4.16.
Step 3: From (-MGP) to (SNLFPK): For a given solution to (-MGP), set
and
for , where denotes the collection of all subsets of sets with . Of course, and depend on , but we suppress this dependence in the notation. Without further mentioning, we understand such as extended to in the canonical way. Then, is a complete filtered probability space. Clearly, is -progressively measurable from to , the space of bounded linear operators from to .
Remark 4.17.
We extend as follows. Let be a complete filtered probability space with a real-valued -Wiener process on it, define
let and be the -completion of and , respectively, and denote the canonical extension of to by . Further, we denote the Wiener process on the -th copy of by and extend each to by for . Similarly, we extend each projection from to via for as above, but keep the same notation for this extended process. Obviously, is a continuous, -adapted process on and each is an -Wiener process on under . Moreover, and are independent on with respect to by construction. Further, it is clear that the process as in (34) is a -martingale with respect to for each with covariation as in (35) and that is -progressively measurable on .
Finally, we need the following result, which is a special case of Theorem 2, [16].
Proposition 4.18.
Let be a solution to the martingale problem (-MGP). Then, there exists a complete filtered probability space with an adapted -dimensional Wiener process and an -valued adapted continuous process such that the law of on is and for and , we have a.s.
| (37) |
and the exceptional set can be chosen independent of and .
To see this, consider Theorem 2 of [16] with , , , the processes given by as in (34) on the probability space of Remark 4.17 and
These choices fulfill all requirements of [16]. In this case, the -valued process is given by on . Since all terms in (37) are continuous in , the exceptional set may indeed by chosen independently of and .
The proof of Theorem 4.8 now follows from the above three step-scheme as follows.
Proof of Theorem 4.8: Let be a weakly continuous solution to (-FPK). By Lemma 4.9 of Step 1, the weakly continuous curve of Borel probability measures on
solves (-FPK) and each is concentrated on . By Proposition 4.16 of Step 2, there exists a solution to the martingale problem (-MGP), which is concentrated on such that
Further, Lemma 4.15 applies to . By Lemma 4.15 and Proposition 4.18 of Step 3, there is a -dimensional -adapted Wiener process and an -adapted process on some complete filtered probability space , which fulfill (37) and -a.s. such that is the law of .
Possibly redefining on a -negligible set (which preserves (37) and its adaptedness, the latter due to the completeness of the underlying filtered probability space), we may assume for some for each . The continuity of and implies vague continuity of
| (38) |
for each and -adaptedness of the -valued process . Considering (37), and the definition of and , we obtain, recalling for each ,
-a.s. for each . From here, it follows by Lemma 4.4 (i) that is a solution to (SNLFPK) as in Definition 4.1. Further,
It remains to prove the final assertion of the theorem. To this end, note that implies and hence -a.s. with as in (38). From here, the assertion follows by Lemma 4.3. ∎
Remark 4.19.
The particular type of noise we consider for (SNLFPK) was partially motivated by [8], where the natural connection of equations of type (SNLFPK) to interacting particle systems with common noise was investigated. Other types of noise terms may be treated in the future, including a possible extension to infinite-dimensional ones. In particular, Proposition 4.18 via [16, Thm.2] in the final step of the proof seems capable of such extensions, since the latter is an infinite-dimensional result.
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Marco Rehmeier Faculty of Mathematics, Bielefeld University, Universitätsstraße 25, 33615 Bielefeld, Germany
E-mail address: [email protected]