aa r X i v : . [ m a t h . P R ] A p r ON A CLASS OF MARTINGALE PROBLEMS ON BANACHSPACES
MARKUS C. KUNZE
Abstract.
We introduce the local martingale problem associated to semi-linear stochastic evolution equations driven by a cylindrical Wiener processand establish a one-to-one correspondence between solutions of the martingaleproblem and (analytically) weak solutions of the stochastic equation. We alsoprove that the solutions of well-posed equations are strong Markov processes.We apply our results to semilinear stochastic equations with additive noisewhere the semilinear term is merely measurable and to stochastic reaction-diffusion equations with H¨older continuous multiplicative noise. Introduction
One of the most important tools in the study of stochastic differential equationsis the theory of associated martingale problems of Stroock and Varadhan [38]. Atthe heart of their approach is the equivalence between solutions of stochastic differ-ential equations (i.e. stochastic processes) and solutions of the associated martingaleproblem (i.e. probability measures on a function space).This equivalence is helpful in several ways. First, it can be used to prove existenceof solutions to stochastic differential equations by means of approximation andtightness arguments. Second, it plays an important role in proving uniqueness ofsolutions using techniques from semigroup theory or partial differential equations.Last but not least, the approach of Stroock and Varadhan yields, given existenceand uniqueness of solutions, the strong Markov property of the solutions. Thisplays an important role in the study of further properties of the solutions, e.g. theirasymptotic behavior.In this article, we set up a theory of (local) martingale problems for stochasticevolution equations(1.1) dX ( t ) = (cid:2) AX ( t ) + F ( X ( t )) (cid:3) dt + G ( X ( t )) dW H ( t ) , on a separable Banach space E . Here, A is the generator of a strongly continuoussemigroup S on E , W H is an H -cylindrical Wiener process where H is a separableHilbert space and the nonlinearities F : E → E and G : E → L ( H, E ) satisfy suit-able measurability and (local) boundedness assumptions. In fact, we shall considera slightly more general situation and allow the nonlinearities to take values in alarger Banach space ˜ E , resp. L ( H, ˜ E ). We will make our assumptions precise inSection 3.Martingale problems for equations of this form on 2-smoothable Banach spaceswere studied by Ondrej´at [34]. The usual solution concept for equations of theform (1.1) is that of a mild solution which involves a stochastic convolution term.We note that to assure that this term is well-defined, one has to impose additionalassumptions on the Banach space (typically geometric assumptions such as the Mathematics Subject Classification.
Key words and phrases. local Martingale problem, strong Markov property, stochastic partialdifferential equations.The author was supported by VICI subsidy 639.033.604 in the ‘Vernieuwingsimpuls’ programof the Netherlands Organization for Scientific Research (NWO).
UMD property or 2-smoothability) and/or the coefficients. This poses problemswhen extending the theory to general Banach spaces. Here, we overcome theseproblems by basing our theory on (analytically) weak solutions rather than on mildsolutions.Our approach does not only allow us to consider general Banach spaces, it alsoallows us to work without additional technical assumptions (such as the J-propertyin [34]) to ensure stochastic integrability of the occurring processes and to imposeonly minimal assumptions on the coefficients.Under these minimal assumptions, we introduce the local martingale problemassociated to equation (1.1) in Section 3 and establish a one-to-one correspondencebetween solutions of the local martingale problem and solutions of the stochasticevolution equation in Theorem 3.6. In Theorem 4.2 we prove, given existence anduniqueness of solutions, the strong Markov property for solutions of (1.1), usingsome abstract results about local martingale problems presented in Section 2.Thus, Sections 2 – 4 contain the abstract theory of martingale problems onBanach spaces. In Sections 5 and 6 we discuss related results, which we believe arehelpful to apply the theory.In Section 5 we extend the Yamada-Watanabe theory [39] to the setting of Ba-nach spaces and prove that pathwise uniqueness implies uniqueness in law (this isthe uniqueness concept used in the abstract theory above) and strong existence ofsolutions. As in finite dimensions, pathwise uniqueness can be much easier veri-fied than uniqueness in law in certain situations, in particular for equations with(locally) Lipschitz continuous coefficients.In Section 6 we show that (analytically) weak and mild solutions coincide if eitherthe coefficient G is constant, i.e. in equations with additive noise, or if the Banachspace E is a UMD space. Working with mild solutions is especially helpful to proveexistence of solutions, as the standard approach via approximation and tightnessoften uses the factorization method of [7] as a tool, which, in turn, requires aBanach space valued stochastic integral. Here, we use the Banach space valuedWiener integral, see [32], in the case of constant G and the theory of integrationin UMD Banach spaces [30] in the second case. Note that this is the only sectionwhere we make use of a stochastic integral, all our abstract results do not dependon geometric assumptions on E .Let us close this introduction by discussing applications of our theory to concretestochastic evolution equations. Techniques inspired by martingale problems can befound frequently in the literature on infinite dimensional stochastic equations eventhough, more often than not, a martingale problem is not used directly. This is mostapparent in the term martingale solution which in infinite dimensions does not referto solutions of the martingale problem but is used synonymously for stochasticallyweak solutions (thus for stochastic processes). Such solutions were constructed, forexample, in [6, 12, 2, 41]. Concerning uniqueness, several authors [6, 11, 40] haveproved uniqueness in law for certain equations by using partial differential equationson Hilbert spaces.Naturally, the results contained in this article can be used to prove, given well-posedness, the strong Markov property for solutions of stochastic evolution equa-tions in arbitrary separable Banach spaces. However, the results obtained here canalso be used to establish well-posedness of a given equation. Naturally, the proofof well-posedness of a stochastic evolution equation requires additional argumentswhich depend on the equation in question. Thus, the full proofs of our applicationsto stochastic evolution equations will be given elsewhere [20, 19]. We will, however,give a rough sketch in Section 7 and discuss how the results of this article enter thearguments. N A CLASS OF MARTINGALE PROBLEMS ON BANACH SPACES 3 Markov processes and local Martingale Problems
In this section (
E, d ) is a complete, separable metric space. We denote theBorel σ -algebra of E by B ( E ). The spaces of scalar-valued measurable, boundedmeasurable, continuous and bounded continuous functions will be denoted by B ( E ) ,B b ( E ) , C ( E ) and C b ( E ) respectively. P ( E ) denotes the set of all probabilitymeasures on ( E, B ( E )). For x ∈ E , the Dirac measure in x is denoted by δ x .By C ([0 , ∞ ); E ) we denote the space of all continuous, E -valued functions. Theelements of C ([0 , ∞ ); E ) will be denoted by bold lower case letters: x , y , z . Endowedwith the metric δδδ , defined by δδδ ( x , y ) := ∞ X k =1 − k sup t ∈ [0 ,k ] d ( x t , y t ) ∧ ,C ([0 , ∞ ); E ) is a complete, separable metric space in its own right. We denoteits Borel σ -algebra by B . It is well-known that B = σ ( x s : s ≥ x s with the E -valued map x x s . We shall do so in what follows without further notice. Thefiltration generated by these ‘coordinate mappings’ is denoted by B := ( B t ) t ≥ , i.e. B t := σ ( x s : s ≤ t ).The space P ( C ([0 , ∞ ); E )) of probability measures on the Borel σ -algebra of C ([0 , ∞ ); E ) will be topologized by the weak topology , i.e. the coarsest topology forwhich for all bounded continuous function Φ on C ([0 , ∞ ); E ) the map P R Φ d P is continuous. It is well known that this topology is metrizable through a complete,separable metric, see [36, Section II.6], i.e. P ( C ([0 , ∞ ); E )) is a Polish space.A probability measure P on ( C ([0 , ∞ ); E ) , B ) is called a Markov measure if thecoordinate process ( x t ) t ≥ defined on ( C ([0 , ∞ ); E ) , B , P ) is a Markov process withrespect to B , i.e. for all f ∈ B b ( E ) and s, t ≥ E (cid:2) f ( x t + s ) (cid:12)(cid:12) B t (cid:3) = E (cid:2) f ( x t + s ) (cid:12)(cid:12) x t (cid:3) P − a.e., where E denotes (conditional) expectation with respect to P . If this equation alsoholds whenever t is replaced with a B -stopping time τ which is almost surely finite,i.e. the coordinate process is a strong Markov process with respect to B , then P iscalled a strong Markov measure . Here, as usual, B τ is the σ -algebra B τ := { A ∈ B : A ∩ { τ ≤ t } ∈ B t for all t ≥ } . A transition semigroup is a family T := ( T ( t )) t ≥ of positive contractions on B b ( E ) such that(1) T is a semigroup, i.e. T (0) = I and T ( t + s ) = T ( t ) T ( s ) for all t, s ≥ T ( t ) is associated with a Markovian kernel , i.e a map p t : E × B ( E ) → [0 ,
1] such that (i) p t ( x, · ) ∈ P ( E ) for all x ∈ E and (ii) p t ( · , A ) ∈ B b ( E ) for all A ∈ B ( E ). That T ( t ) is associated with p t meansthat T ( t ) f ( x ) = R E f ( y ) p t ( x, dy ) for all f ∈ B b ( E ).The kernels p t themselves are referred to as transition functions or transition proba-bilities . The semigroup property above is equivalent with the Chapman-Kolmogorovequations .A probability measure P on C ([0 , ∞ ); E ) is called Markov measure with transitionsemigroup T if for all f ∈ B b ( E ) and s, t ≥ E (cid:2) f ( x t + s ) (cid:12)(cid:12) B t (cid:3) = E (cid:2) f ( x t + s ) (cid:12)(cid:12) x t (cid:3) = (cid:2) T ( s ) f (cid:3) ( x t ) P − a.e. If this equation also holds whenever t is replaced with an P -a.s. finite B -stoppingtime τ , then P is called a strong Markov measure with transition semigroup T .The connection between martingale problems and Markovian measures is wellestablished, see [9, Chapter 4]. However, if we want to treat stochastic evolution MARKUS C. KUNZE equations on Banach spaces, we have to consider local martingale problems ratherthan martingale problems.
Definition 2.1. An admissible operator is a map L , defined on a subset D ( L ) ⊂ C ( E ) and taking values in B ( E ) such that for all f ∈ D ( L ) the function L f isbounded on compact subsets of E .Given an admissible operator L , a probability measure P on C ([0 , ∞ ); E ) is saidto solve the local martingale problem for L if for every f ∈ D ( L ) the process M f defined by (cid:2) M f ( x ) (cid:3) ( t ) := f ( x t ) − f ( x ) − Z t L f ( x s ) ds is a local martingale under P . This of course means that there exists a sequence τ n , which may depend on f , of B -stopping times with τ n ↑ ∞ P -almost surely suchthat the stopped processes M fτ n , defined by M fτ n ( t ) := M f ( t ∧ τ n ), are martingalesfor all n ∈ N .If an initial distribution µ ∈ P ( E ) is specified, we say that P is a solution to thelocal martingale problem for ( L , µ ) to indicate that in addition to being a solutionto the local martingale problem for L , the measure P satisfies P ( x ∈ Γ) = µ (Γ)for all Γ ∈ B ( E ), i.e. under P the random variable x has distribution µ .We note that by the continuity of t x t and since L f is bounded on compactsubsets of E , the process M f is well-defined. In fact, since f is a continuous function,it follows that M f is a continuous process.The proofs of our results in Section 4 are based on the following theorem. Theorem 2.2.
Let L be admissible. Suppose that for every µ ∈ P ( E ) anytwo solutions P , Q of the local martingale problem for ( L , µ ) have the same one-dimensional distributions, i.e. for all t ≥ we have P ( x t ∈ Γ) = Q ( x t ∈ Γ) ∀ Γ ∈ B ( E ) . Then (1)
Every solution of the local martingale problem for L is a strong Markovmeasure. (2) For every µ ∈ P ( E ) , there is at most one solution to the local martingaleproblem for ( L , µ ) .If in addition to the uniqueness assumption above for every x ∈ E there exists asolution P x to the local martingale problem for ( L , δ x ) and if the map x P x ( B ) is Borel measurable for all B ∈ B , then (3) For every µ ∈ P ( E ) , there exists a solution P µ of the local martingaleproblem for ( L , µ ) . (4) Define the operator T ( t ) by T ( t ) f ( x ) := R f ( x t ) d P x for f ∈ B b ( E ) . Thenevery solution P of the local martingale problem for L is a strong Markovmeasure with transition semigroup T := ( T ( t )) t ≥ .Proof. This Theorem is a generalization of [9, Theorem 4.4.2] to local martingaleproblems. Hence, we have the added difficulty that in the definition of “solution ofthe local martingale problem” a sequence of stopping times appears. We only givethe proof of statement (1), the other statements are derived following the proofs ofthe corresponding statements in [9, Theorem 4.4.2] with similar changes due to thepresence of stopping times.Let P be a solution of the local martingale problem for ( L , µ ). We denote(conditional) expectation with respect to P by E . Let ρ be a stopping time with ρ < ∞ almost surely and define the mappings Θ ρ and Ψ ρ : C ([0 , ∞ ); E ) → C ([0 , ∞ ); E )by (Θ ρ x )( t ) := x ( t + ρ ( x )) and (Ψ ρ x )( t ) := x (( t − ρ ( x )) + ) . N A CLASS OF MARTINGALE PROBLEMS ON BANACH SPACES 5
Then Θ ρ and Ψ ρ are measurable mappings with Ψ ρ Θ ρ x = x for all x ∈ C ([0 , ∞ ); E ).Now fix A ∈ B ρ with P ( A ) > P , P on C ([0 , ∞ ); E )by P ( B ) := E (cid:2) A E [ Θ − ρ B | B ρ ] (cid:3) P ( A ) and P ( B ) := E (cid:2) A E [ Θ − ρ B | x ( ρ )] (cid:3) P ( A ) . We note that under P and P the distribution of x (0) are identical, namely forΓ ∈ B ( E ) we have P ( x (0) ∈ Γ) = P ( x (0) ∈ Γ) = P ( x ( ρ ) ∈ Γ | A ) . Hence, if we prove that P and P solve the local martingale problem associatedwith L , we can conclude from our assumption that P and P have the same one-dimensional distributions. This will then imply that for t > ∈ B ( E ), wehave P ( x ( t ) ∈ Γ) = P ( A ) − E (cid:2) A E [ x ( t + ρ ) ∈ Γ | B ρ ] (cid:3) = P ( x ( t ) ∈ Γ) = P ( A ) − E (cid:2) A E [ x ( t + ρ ) ∈ Γ | x ( ρ )] (cid:3) . Multiplying with P ( A ) and observing that A with P ( A ) > E [ x ( t + ρ ) ∈ Γ | B ρ ] = E [ x ( t + ρ ) ∈ Γ | x ( ρ )]. Since t, ρ and Γ were arbitrary, thisproves that ( x ( t )) t ≥ is a strong Markov process under P .It remains to prove that P and P solve the local martingale problem associatedwith L . Fix f ∈ D ( L ). Since P solves the local martingale problem, there existsa sequence τ n of stopping times with τ n → ∞ almost everywhere with respect to P such that M fτ n is a martingale under P . We put σ n := τ n ◦ Ψ ρ . Note that { σ n ≤ t } = Ψ − ρ { τ n ≤ t } ∈ B t , since τ n is a stopping time and since Ψ − ρ A ∈ B t for all A ∈ B t , as is easy to see. Hence σ n is a stopping time. Since Ψ ρ Θ ρ x = x , itfollows from the definition of P and P that σ n ↑ ∞ almost surely with respect to P and P .Now fix t > s and C ∈ B s and observe that ξ ( x ) := h(cid:0) M fσ n ( t ) − M fσ n ( s ) (cid:1) C (cid:3) (Θ ρ x ) = h(cid:0) M fτ n ( t + ρ ) − M fτ n ( s + ρ ) (cid:1) Θ − ρ C (cid:3) ( x )where Θ − ρ C ∈ B s + ρ . Since M fτ n is a continuous P -martingale, it follows from theoptional sampling theorem that E [ ξ | B ρ ] = 0, and hence, since σ ( x ( ρ )) ⊂ B ρ , also E [ ξ | x ( ρ )] = 0. Recalling the definition of P and P , we see that that M fσ n is amartingale under P and P . (cid:3) Definition 2.3.
Let L be an admissible operator. We say that the local martingaleproblem for L is well-posed if for every x ∈ E , there exists a unique solution P x ofthe local martingale problem for ( L , δ x ).We say that the martingale problem for L is completely well-posed , if (i) forevery µ ∈ P ( E ) there exists a unique solution P µ of the local martingale problemfor ( L , µ ) and (ii) the map x P x ( B ) is measurable for every B ∈ B .In the case of uniqueness, we will use the notation P x resp. P µ for the solutionof the local martingale problem for ( L , δ x ), resp. ( L , µ ).In Theorem 4.2, we will prove that if the martingale problem for L is well-posed, then it is already completely well-posed. Thus, we obtain the measurabilityof the map x P x and existence and uniqueness of solutions for arbitrary initialdistributions µ for free.We note that by (2) of Theorem 2.2, the uniqueness assumption in the definitionof ‘completely well-posed’ can be weakened to uniqueness of the one-dimensionalmarginals. Similarly, by (3) of Theorem 2.2, in the definition of ‘completely well-posed’ it suffices to assume existence of solutions only for degenerate initial distri-butions δ x , for all x ∈ E . MARKUS C. KUNZE
By part (4) of Theorem 2.2, if the local martingale problem for L is completelywell-posed, then there exists a transition semigroup T such that every solution P µ is a strong Markov measure with transition semigroup T . This semigroup T isuniquely determined by L and will be called the associated semigroup .3. Stochastic differential equations and the associated localmartingale problem
We now turn our attention to the stochastic evolution equation (1.1). In orderto stress the dependence on the coefficients, we will also refer to equation (1.1) asequation [
A, F, G ]. The following are our standing hypotheses on the coefficientsand will be assumed in the rest of this paper.
Hypothesis . ˜ E is a separable Banach space and A generates a strongly continuoussemigroup S := ( S ( t )) t ≥ = ( S t ) t ≥ on ˜ E . H is a separable Hilbert space and W H is an H -cylindrical Wiener process. E is a separable Banach space such that D ( A ) ⊂ E ⊂ ˜ E with continuous and dense embeddings. Throughout, all Banachspaces are real. Furthermore,(1) F : E → ˜ E is strongly measurable and bounded on bounded subsets of E ;(2) G : E → L ( H, ˜ E ) is H -strongly measurable, i.e. Gh : E → ˜ E is stronglymeasurable for all h ∈ H , and G is bounded on bounded subsets of E . Example . Let us describe typical examples in which Hypothesis 3.1 is satisfied.In the easiest example, ˜ E = E and A is the generator of a strongly continuous S on ˜ E . In applications, A is typically a differential operator and ˜ E is an L p -space. Inthat situation, it is also possible to replace E with a suitable Sobolev space or a spaceof continuous functions. To model equations driven by (additive or multiplicative)white noise, it is often useful to replace ˜ E with a suitable extrapolation space, see,for example, [31].In these situations, the semigroup S typically maps ˜ E into E and restricts toa strongly continuous semigroup on E . Moreover, one has some control over thenorms k S ( t ) k L ( ˜ E,E ) at t = 0. It should be noted, that we assume none of this inHypothesis 3.1. However, later on (in Hypothesis 6.5) we will make precisely theseassumptions.Before defining what we mean by ‘a solution’ of equation [ A, F, G ], let us recallthe notion of an H -cylindrical Wiener process. Let (Ω , Σ , F , P ) be a stochastic basis,i.e. a probability space (Ω , Σ , P ) together with a filtration F = ( F t ) t ≥ . We say thatthe usual conditions are satisfied if F contains all P -null sets and the filtration isright continuous.An H -cylindrical Wiener process (with respect to F ) is a bounded linear operator W H from L (0 , ∞ ; H ) to L (Ω , Σ , P ) with the following properties:(1) for all f ∈ L (0 , ∞ ; H ) the random variable W H ( f ) is centered Gaussian.(2) for all t ≥ f ∈ L (0 , ∞ ; H ) with support in [0 , t ], the random variable W H ( f ) is F t -measurable.(3) for all t ≥ f ∈ L (0 , ∞ ; H ) with support in [ t, ∞ ), the randomvariable W H ( f ) is independent of F t .(4) for all f , f ∈ L (0 , ∞ ; H ) we have E ( W H ( f ) W H ( f )) = [ f , f ] L (0 , ∞ ; H ) .We shall write W H ( t ) h := W H ( (0 ,t ] ⊗ h ) , t > , h ∈ H. It is easy to see that for h ∈ H the process W H h := ( W H ( t ) h ) t ≥ is a real-valuedBrownian motion (which is standard if k h k H = 1).We now define the concept of a weak solution. The relation of weak solutionwith other solution concepts will be discussed in Section 6. N A CLASS OF MARTINGALE PROBLEMS ON BANACH SPACES 7
Definition 3.3.
A tuple (cid:0) (Ω , Σ , F , P ) , W H , X (cid:1) , where (Ω , Σ , F , P ) is stochastic basissatisfying the usual conditions, W H is an H-cylindrical Wiener process with respectto F and X = ( X t ) t ≥ is a continuous, F -progressive, E -valued process is called weak solution of (1.1) if for all x ∗ ∈ D ( A ∗ ) ⊂ ˜ E ∗ and t ≥ h X t , x ∗ i = h X , x ∗ i + Z t h X s , A ∗ x ∗ i ds + Z t h F ( X s ) , x ∗ i ds + Z t G ( X s ) ∗ x ∗ dW H ( s ) , P -a.e. Remark . Weak solutions are weak both in the analytic sense, i.e. we require (3.1)to hold only if tested against functionals x ∗ ∈ D ( A ∗ ) and in the probabilistic sense,i.e. the stochastic basis and the cylindrical Wiener process are part of the solution.More appropriately, we should speak of ‘analytically weak and stochastically weaksolution’ or ‘weak martingale solution’. However, to shorten notation, we havesettled on the term ‘weak solution’.By the continuity of the paths and our assumptions in Hypothesis 3.1, theLebesgue-integral in (3.1) is well defined. The stochastic integral in equation (3.1) isan integral of an H ≃ H ∗ -valued stochastic processes with respect to a cylindricalWiener process. It is well known how to construct such an integral for progres-sive H -valued processes Φ such that Φ ∈ L (0 , T ; H ) almost surely for all T > h k ) is a (finite or countably infinite) orthonormal basis of the separableHilbert space H and we define β k ( s ) := W H ( s ) h k , then Z t Φ( s ) dW H ( s ) := X k Z t [Φ( s ) , h k ] H dβ k ( s ) . The integral process I ( t ) := R t Φ( s ) dW H ( s ) is a real-valued, continuous, local mar-tingale with with quadratic variation J I K t = R t k Φ( s ) k H ds . We also note that foran F -stopping time τ we have almost surely I ( t ∧ τ ) = R t [0 ,τ ] ( s )Φ( s ) dW H ( s ) forall t ≥ X is a weak solutionof (1.1), meaning that X is a continuous, progressive, E -valued process, definedon a stochastic basis (Ω , Σ , P , F ), satisfying the usual conditions, on which an H -cylindrical Wiener process W H with respect to F is defined such that the tupel((Ω , Σ , F , P ) , W H , X ) is a weak solution of (1.1). In this case, unless stated other-wise, P will denote the measure on the probability space and W H the H -cylindricalWiener process. These remarks apply, mutatis mutandis, also for the other solutionconcepts that we will introduce. Remark . We note that the exceptional set in (3.1) which initially depends on x ∗ and t may be chosen independently of t , since the deterministic integrals as wellas the stochastic integral in (3.1) are pathwise continuous in t .We now establish a one-to-one correspondence between weak solutions of equa-tion [ A, F, G ] and solutions of the local martingale problem for an (admissible)operator L [ A,F,G ] which we call the associated local martingale problem .The operator L [ A,F,G ] is defined as follows.By D we denote the vector space of all functions f : E → R of the form f ( x ) = ϕ ( h x, x ∗ i , . . . , h x, x ∗ n i )where n ∈ N , ϕ ∈ C ( R n ) and x ∗ , . . . , x ∗ n ∈ D ( A ∗ ). MARKUS C. KUNZE
For f = ϕ ( h· , x ∗ i , . . . , h· , x ∗ n i ) ∈ D we put(3.2) L [ A,F,G ] f ( x ) := n X k =1 ∂ϕ∂u k ( h x, x ∗ i , . . . , h x, x ∗ n i ) · (cid:2) h x, A ∗ x ∗ k i + h F ( x ) , x ∗ k i (cid:3) + 12 n X k,l =1 [ G ( x ) ∗ x ∗ k , G ( x ) ∗ x ∗ l ] H ∂ ϕ∂u k ∂u l ( h x, x ∗ i , . . . , h x, x ∗ n i )The operator L [ A,F,G ] is defined by D ( L ) = D and L [ A,F,G ] f := L [ A,F,G ] f .Put D min := (cid:8) h· , x ∗ i j : x ∗ ∈ D ( A ∗ ) , j = 1 , (cid:9) . We will also use the operator L min[ A,F,G ] := L [ A,F,G ] | D min . We note that since F and G are bounded on boundedsubsets of E , the operators L [ A,F,G ] and L min[ A,F,G ] are admissible. We would like topoint out that the function L [ A,F,G ] f even if ϕ has compact support. This is thereason for considering local martingale problems, rather than martingale problems. Theorem 3.6.
Suppose that X is a weak solution of equation [ A, F, G ] . Then thelaw P of X solves the local martingale problem for L [ A,F,G ] .Conversely, if P solves the local martingale problem for L min[ A,F,G ] , then there existsa weak solution X of equation [ A, F, G ] with distribution P .Proof. First suppose that X is a weak solution of equation [ A, F, G ].Let f = ϕ ( h· , x ∗ i , . . . , h· , x ∗ n i ) ∈ D and define the R n -valued process ξ by ξ k ( t ) = h X ( t ) , x ∗ k i for all t ≥ k = 1 , . . . , n . We also define R n -valued processes V and M by V k ( t ) := Z t h X s , A ∗ x ∗ k i + h F ( X s ) , x ∗ k i ds , M k ( t ) := Z t G ( X s ) ∗ x ∗ k dW H ( s ) , for k = 1 , . . . , n . Note that, almost surely, V has continuous trajectories of locallybounded variation and that M is a continuous, local martingale. Since X is a weaksolution, it follows that ξ = ξ + M + V .Itˆo’s formula [9, Theorem 5.2.9] yields f ( X t ) − f ( X ) = ϕ ( ξ t ) − ϕ ( ξ )= n X k =1 Z t ∂ϕ∂u k ( ξ s ) dV k ( s ) + 12 n X k,l =1 Z t ∂ ϕ∂u k ∂u l ( ξ s ) d J M k , M l K s + n X k =1 Z t ∂ϕ∂u k ( ξ s ) dM k ( s )= Z t (cid:2) L [ A,F,G ] f (cid:3) ( X s ) ds + n X k =1 Z t ∂ϕ∂u k ( ξ s ) dM k ( s ) , for all t ≥
0. Here, we have used that J M k , M l K t = R t [ G ( X s ) ∗ x ∗ k , G ( X s ) ∗ x ∗ l ] H ds .It thus follows that f ( X t ) − f ( X ) − Z t [ L [ A,F,G ] f ]( X s ) ds is a continuous local martingale with respect to F . Passing to the range space C ([0 , ∞ ); E ), it follows that under the distribution P of X , the process M f is acontinuous local martingale with respect to B .We now prove the converse. First note that if x ∗ ∈ D ( A ∗ ), then for f ( x ) = h x, x ∗ i we have L [ A,F,G ] f ( x ) = h x, A ∗ x ∗ i + h F ( x ) , x ∗ i and for f ( x ) = h x, x ∗ i wehave L [ A,F,G ] f ( x ) = 2 h x, x ∗ i · (cid:2) h x, A ∗ x ∗ i + h F ( x ) , x ∗ i (cid:3) + k G ( x ) ∗ x ∗ k H . If P is asolution of the local martingale problem for L [ A,F,G ] , then under P the processes M f and M f are local martingales with respect to the canonical filtration B . Usingthat the coefficients F and G are bounded on bounded subsets, an approximation N A CLASS OF MARTINGALE PROBLEMS ON BANACH SPACES 9 argument shows that we can use τ n := inf { t > k x ( t ) k ≥ n } as localizing sequencefor both M f and M f . As in [17, Chapter 5, Problem 4.13] we see that the stoppedprocesses M f τ n and M f τ n are martingales with respect to filtration F := ( F t ), where F t is the augmentation of B t + by the P null sets. Hence M f and M f are localmartingales with respect to the filtration F , which satisfies the usual conditions. Itnow follows from [34, Lemma 34] that under P the process h x t , x ∗ i − h x , x ∗ i − Z t h x s , A ∗ x ∗ i + h F ( x s ) , x ∗ i ds is a continuous local martingale with quadratic variation R t k G ( x s ) ∗ x ∗ k H ds . By[35, Theorem 3.1], we find an extension (Ω , Σ , ˜ F , P ) of ( C ([0 , ∞ ); E ) , B , F , P ) onwhich a cylindrical Brownian motion W H is defined such that for all x ∗ ∈ D ( A ∗ )we have h x t , x ∗ i − h x , x ∗ i − Z t h x s , A ∗ x ∗ i + h F ( x s ) , x ∗ i ds = Z t G ( x s ) ∗ x ∗ dW H ( s ) P -almost everywhere for all t ≥ x , defined on this extension, isa weak solution of [ A, F, G ]. (cid:3) Corollary 3.7.
A measure P ∈ P ( C ([0 , ∞ ); E ) solves the local martingale problemfor L [ A,F,G ] if and only if it solves the local martingale problem for L min[ A,F,G ] . Motivated by Theorem 3.6 we will say that the local martingale problem for L [ A,F,G ] is the local martingale problem associated with equation [ A, F, G ]. Wewill say that equation [
A, F, G ] is (completely) well-posed if the associated localmartingale problem is (completely) well-posed.4.
Well-posed equations and the strong Markov property
In this section we prove that if equation [
A, F, G ] is well-posed, then it is com-pletely well-posed. The results of Section 2 then imply that solution of [
A, F, G ]is a strong Markov process with transition semigroup T := ( T ( t )) t ≥ , where T ( t ) f ( x ) = R E f ( x t ) d P x .The key step in the proof is is to show that it even suffices to consider the localmartingale problem for an operator L A,F,G ] , defined on a countable set, cf. [9,Theorem 4.4.6]. Lemma 4.1.
There exists a countable subset D of D such that a measure P solves the local martingale problem associated for L [ A,F,G ] if and only if it solvesthe martingale problem associated with L A,F,G ] := L [ A,F,G ] | D .Proof. Step 1: We construct the set D .First note that there exists a countable subset D of D ( A ∗ ) such that for every x ∗ ∈ D ( A ∗ ) there exists a sequence ( x ∗ n ) ⊂ D such that x ∗ n ⇀ ∗ x ∗ and A ∗ x ∗ n ⇀ ∗ A ∗ x ∗ . Here ⇀ ∗ refers to weak ∗ convergence in ˜ E ∗ . To see this, first note that thereis a countable set { z ∗ n : n ∈ N } ⊂ ˜ E ∗ which is sequentially weak ∗ -dense in ˜ E ∗ ,see § D := { R ( λ, A ∗ ) z ∗ n : n ∈ N } for some λ ∈ ρ ( A ∗ ). Usingthat R ( λ, A ∗ ) is σ ( ˜ E ∗ , ˜ E )-continuous as an adjoint operator, it is easy to see that D has the required properties. Replacing D with the set of all convex combinationsof elements of D with rational coefficients, we may (and shall) assume that suchconvex combinations belong to D again.Now choose a sequence ϕ n ∈ C ( R ) with the following properties:(1) ϕ n ( t ) = t for all − n ≤ t ≤ n and ϕ n ( t ) = 0 for t [ − n, n ].(2) sup n k ϕ ′ n k ∞ , sup n k ϕ ′′ n k ∞ < ∞ . We then define D := (cid:8) f = ϕ n ( h· , x ∗ i ) j for some n ∈ N , x ∗ ∈ D , j ∈ { , } (cid:9) . Clearly, D is countable. We define L A,F,G ] := L [ A,F,G ] | D . Step 2:
Now let P be a solution of the local martingale problem for L A,F,G ] .We prove that P solves the local martingale problem for L min[ A,F,G ] . This finishes theproof in view of Corollary 3.7.First note that M f is a local martingale for any f = h· , x ∗ i j , x ∗ ∈ D , j ∈ { , } .To see this, let σ n := inf { t > |h x t , x ∗ i| ∨ k x t k ≥ n } and put f n := ϕ n ( h· , x ∗ i ) j ∈ D . Clearly, M fσ n = M f n σ n . Since P solves the local martingale problem for L A,F,G ] ,the process M f n , hence by optional sampling also M f n σ n , is a local martingale under P . Since F and G are bounded on bounded sets, M f n σ n is uniformly bounded. Thus, M f n σ n is a true martingale by dominated convergence. This proves that M fσ n is atrue martingale under P and hence, since σ n ↑ ∞ pointwise, that M f is a localmartingale under P .It remains to extend this from x ∗ ∈ D to arbitrary x ∗ ∈ D ( A ∗ ). To that end, fix x ∗ ∈ D ( A ∗ ) and a sequence ( x ∗ n ) ⊂ D such that x ∗ n ⇀ ∗ x ∗ and A ∗ x ∗ n ⇀ ∗ A ∗ x ∗ . Bythe uniform boundedness principle, the sequences ( x ∗ n ) and ( A ∗ x ∗ n ) are bounded in˜ E ∗ , say by M . For m ∈ N put τ m := inf { t > k x ( t ) k ≥ m } .Let us first consider f := h· , x ∗ i . Arguing as above, we see that for f n := h· , x ∗ n i ,the stopped process M f n τ m is a martingale under P for all n, m ∈ N . Furthermore,since L [ A,F,G ] f n → L [ A,F,G ] f pointwise, it follows that M f n τ m ( t ) → M fτ m ( t ) pointwiseas n → ∞ , for all t ≥
0. Since F is bounded on ¯ B (0 , m ), say by C m , we find for t > s (cid:12)(cid:12) M f n τ m ( x )( t ) − M f n τ m ( x )( s ) (cid:12)(cid:12) ≤ ( t − s ) (cid:2) m · M + C m · M (cid:3) + 2 m · M for all n, m ∈ N . Applying the dominated convergence theorem to the sequence( M f n τ m ( t ) − M f n τ m ( s )) B , where B is an arbitrary set in B s , it follows that R B M fτ m ( t ) − M fτ m ( s )) d P = 0. Since 0 ≤ s < t and B ∈ B s were arbitrary, M fτ m is a B -martingale under P . As τ m ↑ ∞ almost surely, this proves that M f is a localmartingale under P .Next consider f := h· , x ∗ i . For f n := h· , x ∗ n i , the stopped process M f n τ m is amartingale under P for all n, m ∈ N . Similarly as above, one sees that for every m ∈ N the difference | M f n τ m ( t ) − M f n τ m ( s ) | may be majorized by a bounded functionindependent of n . However, due to the term k G ( · ) ∗ x ∗ n k H in L [ A,F,G ] f n , the weakconvergence x ∗ n ⇀ ∗ x ∗ does not suffice to conclude that L [ A,F,G ] f n → L [ A,F,G ] f pointwise. Hence we employ a different method here.We fix 0 ≤ s < t and m ∈ N . The dominated convergence theorem yields weakconvergence Z ts [0 ,τ m ] ( r ) G ( x r ) ∗ x ∗ n dr ⇀ Z ts [0 ,τ m ] ( r ) G ( x r ) ∗ x ∗ dr in L ( C ([0 , ∞ ); E ) , P ; H ) . Hence R ts [0 ,τ m ] ( r ) G ( x r ) ∗ x ∗ dr belongs to the weak closure of the tail sequence (cid:0) R ts [0 ,τ m ] ( r ) G ( x r ) ∗ x ∗ n dr (cid:1) n ≥ N , for any N ∈ N . By the Hahn-Banach theorem, itbelongs to the strong closure of that tail, whence we find vectors y ∗ N , belonging tothe convex hull the sequence ( x ∗ n ) n ≥ N , such that we have strong convergence Z ts [0 ,τ m ] ( r ) G ( x r ) ∗ y ∗ N dr → Z ts [0 ,τ m ] ( r ) G ( x r ) ∗ x ∗ dr in L ( C ([0 , ∞ ); E ) , P ; H ) . After passing to a subsequence, we may assume that this convergence holds point-wise P -a.e. Note that y ∗ N ⇀ ∗ x ∗ , as y ∗ N belongs to the tail ( x ∗ n ) n ≥ N . Hence it N A CLASS OF MARTINGALE PROBLEMS ON BANACH SPACES 11 follows that M g N τ m ( t ) − M g N τ m ( s ) → M fτ m ( t ) − M fτ m ( s )pointwise P -almost everywhere. Here, g N := h· , y ∗ N i .Note that we may assume without loss of generality that y ∗ N is a convex combi-nation of the ( x ∗ n ) n ≥ N with rational coefficients. Hence, y N ∈ D and thus g N ∈ D ,implying that M g N τ m is a martingale under P for all N ∈ N . Now, similarly as above,the dominated convergence theorem shows that M fτ m is a martingale under P forall m ∈ N . This finishes the proof. (cid:3) Now the announced result about the equivalence of well-posedness and completewell-posedness follows similar to the finite-dimensional case, cf. [16, Theorem 21.10].
Theorem 4.2.
Suppose that the local martingale problem for L [ A,F,G ] is well-posed. Then it is completely well-posed. Consequently, all weak solutions of equation [ A, F, G ] are strong Markov processes with a common transition semigroup T .Proof. We first prove the measurability of the map x P x . Consider the set V := { P x : x ∈ E } . We claim that V is a Borel subset of P ( C ([0 , ∞ ); E )). Indeed,by well-posedness, V = V ∩ V , where V is the set of all probability measures withdegenerate initial distributions and V is the set of all solutions to the martingaleproblem.Since the map P P ◦ x (0) − is measurable from P ( C ([0 , ∞ ); E )) to P ( E ),the measurability of V follows from [16, Lemma 1.39].By Lemma 4.1, P ∈ V if and only if M f is a local martingale under P for all f ∈ D . With τ n := inf { t > k x ( t ) k ≥ n } , this is equivalent with Z B M f ( t ∧ τ n ) d P = Z B M f ( s ∧ τ n ) d P ∀ s < t, B ∈ B s , n ∈ N . However, using continuity of t x ( t ) and the fact that the σ -algebra B s is count-ably generated for all s >
0, we see that M f is a local martingale under P wheneverthe above equality holds for n ∈ N , s, t ∈ Q with s < t and B in a countable subsetof B s . Hence the set V is determined by countably many ‘measurable relations’and hence measurable. It follows that V is measurable as claimed.Now define the map Φ : V → E by defining Φ( P ) as the unique x such that P ◦ x − = δ x . Clearly, Φ is injective. Furthermore, Φ is measurable as the compositionof the measurable map P ◦ x − and the inverse of the map x δ x , which establishesa homeomorphism between E and the range of that map. By the KuratowskiTheorem, see [36, Section 1.3], the inverse Φ − is measurable, i.e. x P x is ameasurable map from E to P ( C ([0 , ∞ ); E ))It remains to prove the uniqueness of solutions with arbitrary initial distributions µ for the martingale problem for L [ A,F,G ] . The existence of solutions with generalinitial distributions will then follow from Theorem 2.2.To that end, assume that P solves the local martingale problem for L [ A,F,G ] and that x (0) has distribution µ ∈ P ( E ). Let Q : E × B → [0 ,
1] be a regularconditional probability (under P ) for B given x . Then P ( A ) = Z E Q ( x, A ) dµ ( x ) ∀ A ∈ B . Now let t > s ≥ B ∈ B s be given. Then, for f ∈ D , we have Z B M f ( t ∧ τ n ) − M f ( s ∧ τ n ) d Q ( x, · ) = Z B ∩{ x (0)= x } M f ( t ∧ τ n ) − M f ( s ∧ τ n ) d P = 0for µ -almost every x . We note that the null-set outside of which this equation holdsdepends on t, s, n, B and the function f . However, arguing as above, we see thatfor fixed f , there exists a null-set N ( f ), such that the above equation holds outside N ( f ) for all t > s, n ∈ N and B ∈ B s . Putting N := S f ∈ D N ( f ), it follows thatoutside of N , the above holds for all t > s, n ∈ N , B ∈ B s and f ∈ D . Thisimplies that for µ -a.e. x the measure Q ( x, · ) solves the local martingale problem for L A,F,G ] and hence, by Lemma 4.1, the local martingale problem for L [ A,F,G ] . Bywell-posedness, Q ( x, · ) = P x ( · ) for µ -a.e. x . Hence we have(4.1) P ( A ) = Z E P x ( A ) dµ ( x ) ∀ A ∈ B , This shows that uniqueness of solutions of the local martingale problem for ( L , δ x )for all x ∈ E implies uniqueness of the solution of the local martingale problem for( L , µ ) for arbitrary initial distribution µ . (cid:3) We end this section by establishing a result which allows us to construct solutionsto equation [
A, F, G ] from solutions of approximate equations [
A, F n , G n ]. Lemma 4.3.
Suppose we are given sequences ( F n ) n ∈ N and ( G n ) n ∈ N which satisfythe assumptions of Hypothesis 3.1, are continuous and are uniformly bounded onbounded sets. Furthermore, assume that F n ( x ) converges to F ( x ) in ˜ E and G n ( x ) converges to G ( x ) in L ( H, ˜ E ) , both convergences being uniform on the compactsubsets of E .If P n solves the martingale problem associated with equation [ A, F n , G n ] and ifthe sequence ( P n ) n ∈ N is tight, then any accumulation point of the sequence solvesthe martingale problem associated with [ A, F, G ] .Proof. For a number M ∈ R we put τ M := inf { t > k x t k ≥ M } . Now fix0 ≤ s < · · · < s N ≤ s < t , N ∈ N , and for j = 1 , . . . , N functions h j ∈ C b ( E ) and f = ϕ ( h· , x ∗ i , . . . , h· , x ∗ m i ) ∈ D .We define Φ n : C ([0 , ∞ ); E ) → R byΦ n ( x ) := h f ( x t ∧ τ M ) − f ( x s ∧ τ M ) − Z ts [0 ,τ M ] ( r ) (cid:0) L n f (cid:1) ( x r ) dr i · N Y j =1 h j ( x s j ) , where L n := L [ A,F n ,G n ] . Similarly, we define the function Φ, replacing L n with L := L [ A,F,G ] .Using the assumption that F n and G n are uniformly bounded on bounded sub-sets, it is easy to see that the sequence Φ n is uniformly bounded.The assumptions on the convergence of F n and G n imply that L n f convergesto Lf , uniformly on the compact subsets of E . Now let a compact subset C of C ([0 , ∞ ); E ) be given. By the Arzel`a-Ascoli theorem, there exists a compact subset K of E such that x r ∈ K for all 0 ≤ r ≤ t , whenever x ∈ C . Let C := Q nj =1 k h k k ∞ .Given ε >
0, pick n such that | L n f ( x ) − Lf ( x ) | ≤ ε for all x ∈ K , whenever n ≥ n . Then, for x ∈ C and n ≥ n we have | Φ n ( x ) − Φ( x ) | ≤ Z ts [0 ,τ M ] ( r ) | L n f ( x r ) − Lf ( x r ) | dr · C ≤ | t − s | εC, proving that Φ n converges to Φ uniformly on compact subsets of C ([0 , ∞ ); E ).Now let P be an accumulation point of the sequence ( P n ). Passing to a subse-quence, we may assume that P n converges weakly to P . In particular, the sequence( P n ) is tight. Thus, given ε >
0, we find a compact set C of C ([0 , ∞ ); E ) such that2 c P n ( C c ) ≤ ε , where c is such that k Φ n k ∞ ≤ c . It follows that (cid:12)(cid:12)(cid:12) Z Φ d P − Z Φ n d P n (cid:12)(cid:12) ≤ (cid:12)(cid:12)(cid:12) Z Φ d P − Z Φ d P n (cid:12)(cid:12) + ε + sup x ∈ C | Φ( x ) − Φ n ( x ) | . To conclude that R Φ d P = lim n →∞ R Φ n d P n = 0, it remains to prove that R Φ d P n converges to R Φ d P . We know that P n converges weakly to P . Unfortunately, thefunction Φ is not continuous. However, it is continuous at all points y at which N A CLASS OF MARTINGALE PROBLEMS ON BANACH SPACES 13 the map x τ M ( x ) is continuous. Moreover, it can be proved that the set of all M such that P ( { y : τ M is discontinous at y } ) > M such that Φ is continuous except for a P -null set. As is well known, see [1, Cor.8.4.2], this together with the weak convergence of the P n suffices to conclude that R Φ d P n → R Φ d P , as desired and it follows that R Φ d P = 0.Since the sampling points ( s j ) and s, t as well as the functions h j were arbitrary,it follows from a monotone class argument that f ( x t ∧ τ M ) − f ( x ∧ τ M ) − Z t [0 ,τ M ] ( r ) Lf ( x r ) dr is a martingale under P . Since f was arbitrary, and we can pick a sequence M k ↑ ∞ such that the above is true, we have proved that P solves the local martingaleproblem associated with equation [ A, F, G ]. (cid:3) As a corollary, we obtain a sufficient condition for the Feller property of theassociated transition semigroup.
Corollary 4.4.
Assume that equation [ A, F, G ] is well-posed and that F and G arecontinuous. We denote by T the transition semigroup for the associated martingaleproblem for L [ A,F,G ] and by P µ the unique solution of the local martingale problemfor ( L [ A,F,G ] , µ ) . The following are equivalent (1) The map µ P µ is continuous from P ( E ) to P ( C ([0 , ∞ ); E )) where bothare endowed with their respective weak topology. (2) If x n → x in E , then the set { P x n : n ∈ N } is tight.In this case, the semigroup T has the Feller property, i.e. T ( t ) f ∈ C b ( E ) for all f ∈ C b ( E ) .Proof. (1) ⇒ (2): If x n → x then δ x n → δ x weakly. In particular, { δ x n : n ∈ N } is relatively weakly compact. By (1) the set { P x n : n ∈ N } is relatively weaklycompact hence tight.(2) ⇒ (1): Let x n → x . By (2), { P x n : n ∈ N } is tight. By Lemma 4.3 anyaccumulation point of the P x n must solve the local martingale problem for L A,F,G .Since every accumulation point also must have initial distribution δ x , well-posednessimplies that the only accumulation point is P x . Now a subsequence-subsequenceargument yields that P x n converges weakly to P x . This proves that the map x P x is continuous from E to P ( C ([0 , ∞ ); E )).It follows from the proof of uniqueness in Theorem 2.2, namely from equation(4.1), that Z Φ d P µ = Z E Z Φ d P x dµ ( x ) , for all bounded, continuous functions Φ on C ([0 , ∞ ); E ). With this representationthe continuity of µ P µ follows.If (1) or, equivalently, (2) is satisfied, then the Feller property of T follows fromthe identity T ( t ) f ( x ) = R f ◦ π t d P x and the fact that f ◦ π t is a bounded, continuousfunction on C ([0 , ∞ ); E ). (cid:3) Yamada-Watanabe theory
In view of Theorem 3.6, the uniqueness requirement for the local martingaleproblem associated with (1.1) is equivalent with the requirement that whenever X and X are weak solutions of (1.1), possibly defined on different probability spaces,such that X (0) and X (0) have the same distribution µ , then X and X have thesame distribution as C ([0 , ∞ ); E )-valued random variables. In this situation, onesays that uniqueness in law or uniqueness in distribution holds. In some cases, in particular in the case of Lipschitz continuous coefficients, it iseasier to verify a different notion of uniqueness.
Definition 5.1.
We say that pathwise uniqueness holds for solutions of equation(1.1) if whenever ((Ω , Σ , F , P ) , W H , X j ) are weak solution of (1.1) for j = 1 , X (0) = X (0) almost surely, then P ( X ( t ) = X ( t ) ∀ t ≥
0) = 1.A classical result of Yamada and Watanabe [39] asserts that in the case where E = R d and W H is a finite dimensional Brownian motion, i.e. H is finite-dimensional,pathwise uniqueness implies uniqueness in law. Pathwise uniqueness also has otherfar-reaching consequences, most notably, it implies the strong existence of solutions. Definition 5.2.
A weak solution ((Ω , Σ , F , P ) , W H , X ) is said to exist strongly if X is adapted to the filtration G := ( G t ) t ≥ , where G t is the augmentation of σ ( X (0) , W H h k ( s ) : s ≤ t, k ∈ I ). Here, ( h k ) k ∈ I is a finite or countably infiniteorthonormal basis of H .A priori, strong existence of solutions is a mere measurability requirement. Thisrequirement captures the idea that the information needed to construct a solutionto a stochastic differential equation is already contained in the initial datum andthe Wiener process. Of particular importance in applications is the fact that givenpathwise uniqueness solutions can be constructed on a given stochastic basis andwith respect to a given H -cylindrical Wiener process, see Corollary 5.4.Ondrej´at [33] has generalized the Yamada-Watanabe results to the situationwhere E is a 2-smoothable Banach space. One of the main difficulties he hadto overcome was to prove that distributional copies of solutions are again solutions.As he was working with the concept of mild solutions, this required a detailed studyof the distributions of Banach space valued stochastic integrals. In our situation,with the concept of weak solutions, the proof is easier and can in fact be reducedto the finite dimensional situation. Theorem 5.3.
Pathwise uniqueness for (1.1) implies uniqueness in law. Moreover,every solution of (1.1) exists strongly.
For the convenience of the reader, we include a full proof which follows closelythe proof in the finite dimensional situation. It is also possible to show that oursituation fits into the abstract framework considered in [22] and to obtain Theorem5.3 from the results proved there.
Proof.
Let two weak solutions ((Ω j , Σ j , F j , P j ) , W jH , X j ) of equation (1.1) be givensuch that X (0) and X (0) have the same distribution µ . We first define distribu-tional copies of these two solutions on a common stochastic basis.To that end, we fix an orthonormal Basis ( h n ) n ∈ N (the case where H is finitedimensional is similar) of H and define the measure P j on the Borel σ -algebra of˜Ω := C ([0 , ∞ ); E ) × E × C ([0 , ∞ ); R ∞ ) , viewed as the countable product of Polish spaces, as the image of P j under the map ω j (cid:0) X j ( · , ω j ) − X j (0 , ω j ) , X j (0 , ω j ) , ( H jH ( · , ω j ) h n ) n ∈ N (cid:1) A typical element of ˜Ω will be denoted by ( y , x , w ). Note that the projection of P j to C ([0 , ∞ ); R ∞ ) is the countable product of Wiener measure; we denote thismeasure by W . Thus, under P j , the random element ( x , w ) has distribution µ ⊗ W .We let Q j be a regular conditional distribution of y given ( x , w ) under P j ,i.e. Q j ( x , w , · ) is a probability measure on B ( C ([0 , ∞ ); E )) for all x ∈ E and w ∈ C ([0 , ∞ ); R ∞ ) and given sets A ∈ B ( C ([0 , ∞ ); E )), B ∈ B ( E ) and C ∈ B ( C ([0 , ∞ ); R ∞ )), we have P j ( A × B × C ) = Z B × C Q j ( x , w , A ) d ( µ ⊗ W )( x , w ) . N A CLASS OF MARTINGALE PROBLEMS ON BANACH SPACES 15
We now define distributional copies of the solutions on a common probability space.We put Ω := C ([0 , ∞ ); E ) × C ([0 , ∞ ); E ) × E × C ([0 , ∞ ); R ∞ ) , and denote a canonical element of Ω by ( y , y , x , w ). We define the measure P on the Borel σ -algebra Σ of Ω by P ( A × B × C × D ) := Z C × D Q ( x , w , A ) Q ( x , w , B ) d ( µ ⊗ W )( x , w ) . Finally, we define G t := σ ( x , y ( s ) , y ( s ) , w ( s ) : s ≤ t ), F t as the augmentation of G t + by the P -null sets and set F := ( F t ) t ≥ . As in the finite dimensional case, see[14, Lemma IV.1.2], we see that for every k ∈ N the k -th component w k of w is aBrownian motion with respect to F .As w k and w l are independent for k = l , we can define an H -cylindrical Wienerprocess with respect to F by setting, for f ∈ L (0 , ∞ ; H ) W H ( f ) := ∞ X k =1 Z ∞ [ f ( t ) , h k ] H d w k ( t ) . We claim that ((Ω , Σ , F , P ) , W H , x + y j ) is a weak solution of equation (1.1)for j = 1 ,
2. We will write x j := x + y j for j = 1 ,
2. To prove the claim, let x ∗ ∈ D ( A ∗ ) be fixed. Using the measurability of F and G , as well as the continuityof the functionals x ∗ resp. A ∗ x ∗ , it follows from the definitions above that the jointdistribution of (cid:0) h X j (0) , x ∗ i , h F ( X j ( · )) , x ∗ i , h X j ( · ) , A ∗ x ∗ i , ([ G ( X j ( · )) ∗ x ∗ , h k ]) k ∈ N , ( W jH ( · ) h k ) k ∈ N (cid:1) under P j is the same as that of (cid:0) h x , x ∗ i , h F ( x j ( · )) , x ∗ i , h x j ( · ) , A ∗ x ∗ i , ([ G ( x j ( · )) ∗ x ∗ , h k ]) k ∈ N , ( W H ( · ) h k ) k ∈ N (cid:1) under P . Thus, for fixed t ≥
0, we infer as in the finite dimensional situation thatfor j = 1 , n ∈ N the distribution of Z j,n ( t ) := X j ( t ) −h X j (0) , x ∗ i − Z t h X j ( s ) , A ∗ x ∗ i ds − Z t h F ( X j ( s )) , x ∗ i ds − n X k =1 Z t [ G ( X j ( s )) ∗ x ∗ , h k ] dW jH ( s ) h k under P j is the same as that of z j,n ( t ); = x j ( t ) −h x j (0) , x ∗ i − Z t h x j ( s ) , A ∗ x ∗ i ds − Z t h F ( x j ( s )) , x ∗ i ds − n X k =1 Z t [ G ( x j ( s )) ∗ x ∗ , h k ] dW jH ( s ) h k under P . Since X j is a solution of equation (1.1), Z j,n ( t ) → P j -almost surely as n → ∞ , hence z j,n converges to 0 in distribution and thus P -almost surely. Since t ≥ x ∗ ∈ D ( A ∗ ) were arbitrary, this proves that x j is indeed a weak solution.As x + y and x + y are weak solutions defined on the same stochastic basisand with respect to the same H -cylindrical Wiener process, pathwise uniquenessimplies that x + y = x + y P -almost surely. This, in turn, implies that therandom elements X j have the same distribution.As for the strong existence of solutions, define for x ∈ E and w ∈ C ([0 , ∞ ); R ∞ )the measure R ( x , w , · ) on the Borel σ -algebra S of C ([0 , ∞ ); E ) × C ([0 , ∞ ); E ) as the product of Q ( x , w , · ) and Q ( x , w , · ). Then, for G ∈ S , C ∈ B ( E ) and D ∈ B ( C ([0 , , ∞ ); R ∞ ) we have P ( G × C × D ) = Z C × D R ( x , w , G ) d ( µ ⊗ W )( x , w ) . Now consider Λ := { ( y , y ) : y = y } . It follows from the first part of theproof that R ( x , w , Λ) = 1 for ( µ ⊗ W )-almost every ( x , w ), say outside the set N ∈ B ( E ) ⊗ B ( C ([0 , ∞ ); E )) with ( µ ⊗ W )( N ) = 0. Using Fubini’s theorem, wefind for ( x , w ) ∈ N c R ( x , w , Λ) = Z C ([0 , ∞ ); E ) Q ( x , w , { y } ) Q ( x , w , d y ) . As all measures involved in this equation are probability measures, this can onlyhappen if Q ( x , w , { y } ) = Q ( x , w , { y } ) = 1 for a certain y = Φ( x , w ) ∈ C ([0 , ∞ ); E ).A straightforward generalization of the proof in the finite-dimensional case, see[17, Section 5.3.D], shows that the map Φ : E × C ([0 , ∞ ); R ∞ ) → C ([0 , ∞ ); E ) is B ( E ) ⊗ B ( C ([0 , ∞ ); R ∞ )) / B ([0 , ∞ ); E )-measurable. Moreover, if we define H t asthe augmentation of B ( E ) ⊗ σ ( w ( s ) : s ≤ t ) by the µ ⊗ W -null sets and I t := σ ( y ( s ) : s ≤ t ), then Φ is H t / I t -measurable for every t > x + y j = x + Φ( x , w ) P -almost surly. Thus, for j = 1 ,
2, we have X j = X j (0) + Φ( X j (0) , ( W jH ( · ) h n ) n ∈ N ) P j -almost surely. Themeasurability properties of Φ now imply that the solution ((Ω j , Σ j , F , P j ) , W jH , X j )exists strongly for j = 1 , (cid:3) As a consequence of pathwise uniqueness, we find solutions of equation (1.1) ona given probability space and with respect to a given H -cylindrical Wiener process. Corollary 5.4.
Assume that pathwise uniqueness holds for equation [ A, F, G ] andthat for some µ ∈ P ( E ) , there exists a weak solution of [ A, F, G ] with initial distri-bution µ . Then, given any stochastic basis (Ω , Σ , F , P ) on which an H -cylindricalWiener process W H with respect to F is defined and on which an F -measurablerandom variable ξ with distribution µ is defined, there exists a process X such that ((Ω , Σ , P ) , F , W H , X ) is a weak solution of equation [ A, F, G ] with X (0) = ξ .Proof. Let ((Ω ′ , Σ ′ , P ′ ) , F ′ , W ′ H , X ′ ) be a weak solution of [ A, F, G ] with X ′ (0) ∼ µ .The proof of Theorem 5.3 yields that X ′ = X (0) + Φ( X ′ (0) , ( W ′ H ( · ) h n ) n ∈ N ). Weput X := ξ + Φ( ξ, ( W H ( · ) h n ) n ∈ N ).Then the distribution of ( X ′ (0) , X ′ , ( W ′ H ( · ) h n ) n ∈ N ) under P ′ is the same as thedistribution of ( ξ, X , ( W H ( · ) h n ) n ∈ N ) under P . Arguing as in the first part of theproof of Theorem 5.3, it follows that ((Ω , Σ , P ) , F , W H , X ) is a weak solution ofequation [ A, F, G ] with X (0) = ξ . (cid:3) Stochastic integration and mild solutions
We now address the question whether weak solutions of (1.1) are also mild solu-tions, i.e. for all t ≥ L ( H, E )-valued process s S t − s G ( X s ) is stochasticallyintegrable (in a sense to be made precise below) and we have, almost surely,(6.1) X t = X + Z t S t − s F ( X s ) ds + Z t S t − s G ( X s ) dW H ( s ) . Having mild solutions, rather than weak solutions, has many advantages. Inparticular, one can make use of the factorization method [7]. The factorizationmethod is useful to prove continuity of the paths of solutions which we have as-sumed throughout and also to establish the tightness assumption in Lemma 4.3,thus enabling us to construct solutions to stochastic differential equations.
N A CLASS OF MARTINGALE PROBLEMS ON BANACH SPACES 17
In this section, we prove the equivalence of weak and mild solutions under addi-tional assumptions on either equation [
A, F, G ] or the state space E . As an inter-mediate step, we first consider weakly mild solutions in which we only require (6.1)to hold when tested against functionals x ∗ ∈ ˜ E ∗ .6.1. Weakly mild solutions.Definition 6.1.
A tuple (cid:0) (Ω , Σ , F , P ) , W H , X (cid:1) , where (Ω , Σ , F , P ) is stochastic basissatisfying the usual conditions, W H is an H -cylindrical Wiener process with respectto F and X is a continuous, F -progressive, E -valued process is called a weakly mildsolution of (1.1) if for all x ∗ ∈ ˜ E ∗ and t ≥ h X t , x ∗ i = h S t X , x ∗ i + Z t h S t − s F ( X s ) , x ∗ i ds + Z t G ( X s ) ∗ S ∗ t − s x ∗ dW H ( s ) . P -a.e. Remark . By our assumptions on the coefficients
A, F and G , the Lebesgue-integral and the stochastic integral in (6.2) are well-defined for all t ≥ x ∗ ∈ E ∗ .Indeed, the map ( s, ω ) F ( X ( s, ω )) is measurable as a composition of twomeasurable maps. Hence, it is the limit of a sequence of simple functions f n almosteverywhere with respect to ds ⊗ P . Thus h S ( t − · ) F ( X ) , x ∗ i = lim h f n , S ( t − · ) ∗ x ∗ i ds ⊗ P − a.e. We have h f n , S ( t − · ) ∗ x ∗ i = P N n j =1 A jn h x jn , S ( t − · ) ∗ x ∗ i for certain measurablesets A jn and vectors x jn ∈ ˜ E and this is measurable since s
7→ h x, S ( t − s ) ∗ x ∗ i is continuous for all x ∈ ˜ E and x ∗ ∈ ˜ E ∗ . Hence h S ( t − · ) F ( X ) , x ∗ i is the limitof measurable functions ds ⊗ P almost everywhere and thus measurable. In viewof the continuity of the paths of X , the boundedness of F on bounded sets andthe boundedness of S on finite time intervals, it follows that for almost all ω thefunction s
7→ h S ( t − s ) F ( X ( s, ω )) , x ∗ i is bounded, hence integrable.The stochastic integral can be dealt with similarly, using the series expansion G ( X ( s, ω )) ∗ S ( t − s ) ∗ x ∗ = X k h G ( X ( s, ω )) h k , S ( t − s ) ∗ x ∗ i H h k where ( h k ) is a finite or countably infinite orthonormal basis of H .We now prove that the notions ‘weak solution’ and ‘weakly mild solution’ areequivalent. Under additional assumptions which ensure that the stochastic convo-lution is well-defined, variations of this result (for mild solutions) have been provedin various settings, see [8, Theorem 5.4], [32, Theorem 7.1] or [37, Proposition 3.3].Assuming that G is constant or that E is a UMD Banach space, in the followingsubsection we prove that weakly mild solutions are mild solutions. In particular, it follows that the stochastic convolution is well-defined.We note that the adjoint semigroup S ∗ may not be strongly continuous, whichcauses technical difficulties. To overcome these, we will use results about the ⊙ -dualsemigroup S ⊙ . We recall some basic definitions and properties and refer the readerto [27] for more information.Define ˜ E ⊙ := D ( A ∗ ). Then ˜ E ⊙ is a closed, weak ∗ -dense subspace of ˜ E ∗ whichis invariant under the adjoint semigroup. The restriction of the adjoint semigroupto ˜ E ⊙ , denoted by S ⊙ , is strongly continuous. In fact, ˜ E ⊙ = { x ∗ ∈ ˜ E ∗ : t S ( t ) ∗ x ∗ is strongly continuous } . We denote by A ⊙ the generator of S ⊙ . Note that A ⊙ is exactly the part of A ∗ in ˜ E ⊙ . Proposition 6.3.
The weak and the weakly mild solutions of (1.1) coincide.
Proof.
First assume that X is a weak solution. For n ∈ N , define τ n := inf { t > k X ( t ) k ≥ n } . Since X is a weak solution, we have for x ∗ ∈ D ( A ∗ ) and t ≥ h X t ∧ τ n , x ∗ i = h X ∧ τ n , x ∗ i + Z t [0 ,τ n ] ( s ) h X s , A ∗ x ∗ i ds + Z t [0 ,τ n ] ( s ) h F ( X s ) , x ∗ i ds + Z t [0 ,τ n ] ( s ) G ( X s ) ∗ x ∗ dW H ( s )almost surely. In view of Remark 3.5, we may (and shall) assume that the excep-tional set does not depend on t . Below, we will suppress the statement P -almostsurely.Fix t > f ∈ C ([0 , t ]) and x ∗ ∈ D ( A ∗ ). Putting ϕ := f ⊗ x ∗ , Itˆo’sformula yields(6.3) h X t ∧ τ n , ϕ ( t ) i = h X ∧ τ n , ϕ (0) i + Z t h X s ∧ τ n , ϕ ′ ( s ) i ds + Z t ∧ τ n h X s , A ∗ ϕ ( s ) i ds + Z t ∧ τ n h F ( X s ) , ϕ ( s ) i ds + Z t [0 ,τ n ] ( s ) G ( X s ) ∗ ϕ ( s ) dW H ( s ) . By linearity, the above equation holds for ϕ = P Nk =1 f k ⊗ x ∗ k where f k ∈ C ([0 , t ])and x ∗ k ∈ D ( A ∗ ). Since D ( A ⊙ ) is a Banach space with respect to the graph norm, sois C ([0 , t ]; D ( A ⊙ )). Functions of the form ϕ := P nk =1 f k ⊗ x ∗ k with f k ∈ C ([0 , t ])and x ∗ k ∈ D ( A ⊙ ) for 1 ≤ k ≤ n are dense in C ([0 , t ]; D ( A ⊙ )) and hence anapproximation argument shows that (6.3) holds for all ϕ ∈ C ([0 , t ]; D ( A ⊙ )).Now let x ∗ ∈ D (( A ⊙ ) ) and ϕ ( s ) = S ∗ t − s x ∗ . Then ϕ ∈ C ([0 , t ]; D ( A ⊙ )) with ϕ ′ ( s ) = − S ∗ t − s A ∗ x ∗ . Let us note that R t h X s ∧ τ n , ϕ ′ ( s ) i ds = R t ∧ τ n h X s , ϕ ′ ( s ) i ds + R tt ∧ τ n h X τ n , ϕ ′ ( s ) i ds , where the last term is zero if τ n ≥ t . Thus equation (6.3)yields for this ϕ (6.4) h X t ∧ τ n , x ∗ i = h S t X ∧ τ n , x ∗ i + Z t ∧ τ n h S t − s F ( X s ) , x ∗ i ds − Z tt ∧ τ n h S t − s X τ n , A ∗ x ∗ i ds + Z t [0 ,τ n ] G ( X s ) ∗ S ∗ t − s x ∗ dW H ( s ) . We next want to extend (6.4) to arbitrary x ∗ ∈ ˜ E ∗ . Obviously, the term R tt ∧ τ n h S t − s X τ n , A ∗ x ∗ i is not well-defined for arbitrary x ∗ ∈ ˜ E ∗ . However, usingthe well-known fact that for 0 ≤ a < b and x ∈ ˜ E the integral R ba S ( s ) x ds belongsto the domain of the generator A and A R ba S ( s ) x ds = S ( b ) x − S ( a ) x , it follows that Z tt ∧ τ n h S t − s X τ n , A ∗ x ∗ i ds = h S t − t ∧ τ n X τ n − X τ n , x ∗ i . Since D (( A ⊙ ) ) is sequentially weak ∗ -dense in ˜ E ∗ , given z ∗ ∈ ˜ E ∗ , we find a sequence x ∗ k ∈ D (( A ⊙ ) ) such that x ∗ k ⇀ ∗ z ∗ . Arguing similar as in the proof of Lemma 4.1,we find a sequence y ∗ m in the convex hull of the ( x ∗ k ) such that y ∗ m ⇀ ∗ z ∗ and [0 ,τ n ] ( · ) G ( X ( · )) ∗ S ( t − · ) ∗ y ∗ m → [0 ,τ n ] ( · ) G ( X ( · )) ∗ S ( t − · ) ∗ z ∗ in L (Ω). Thus, since E (cid:12)(cid:12) R t Φ( s ) dW H ( s ) (cid:12)(cid:12) = k Φ k L (Ω; L ([0 ,t ]; H )) we see that Z t [0 ,τ n ] G ( X ( s )) ∗ S ( t − s ) ∗ y ∗ m dW H ( s ) → Z t [0 ,τ n ] G ( X ( s )) ∗ S ( t − s ) ∗ z ∗ dW H ( s ) N A CLASS OF MARTINGALE PROBLEMS ON BANACH SPACES 19 in L (Ω; L (0 , t ; H )). Passing to a subsequence, we may assume that we have con-vergence almost everywhere. Moreover, since (6.4) also holds for x ∗ = y ∗ m , for all m ∈ N , noting that [0 ,τ n ] ( s ) (cid:12)(cid:12) h S ( t − s ) F ( X ( s )) , y ∗ m i (cid:12)(cid:12) ≤ [0 ,τ n ] ( s ) M e ω ( t − s ) B n · sup m ∈ N k y ∗ m k , where M and ω are such that k S ( t ) k ≤ M e ωt for t ≥ B n := sup {k F ( x ) k : k x k ≤ n } , is follows from dominated convergence that R t ∧ τ n h S t − s F ( X s ) , y ∗ m i ds converges to R t ∧ τ n h S t − s F ( X s ) , z ∗ i ds almost surely. It altogether we see that(6.5) h X t ∧ τ n , z ∗ i = h S t X ∧ τ n , z ∗ i + Z t ∧ τ n h S t − s F ( X s ) , z ∗ i ds + h X τ n − S t − t ∧ τ n X τ n , z ∗ i + Z t [0 ,τ n ] G ( X s ) ∗ S ∗ t − s z ∗ dW H ( s ) . Upon letting n → ∞ , (6.2) is proved for arbitrary x ∗ = z ∗ .We now prove the converse and assume that X is a weakly mild solution of (3.1).Fix x ∗ ∈ D ( A ∗ ) and t >
0. Then for 0 < s < t we have(6.6) h X s , A ∗ x ∗ i = h S s X , A ∗ x ∗ i + Z s h S s − r F ( X r ) , A ∗ x ∗ i dr + Z s G ( X r ) ∗ S ∗ s − r A ∗ x ∗ dW H ( r )almost surely. We note that the exceptional set may depend s . However, all termsin this equation are jointly measurable in s and ω . Hence, the left-hand side and theright-hand side of (6.6) are equal as elements of L ((0 , t ); L (Ω)). By the canonicalisomorphism L ((0 , t ); L (Ω)) ≃ L (Ω; L (0 , t )), there exists a set N ⊂ Ω with P ( N ) = 0 such that outside N equation (6.6) holds as an equation in L (0 , t ), i.e.for almost every s ∈ (0 , t ), where the exceptional set may depend on ω . Next notethat by the continuity of the paths, the local boundedness of S and the boundednessof F on bounded sets, the first three terms are, as functions of s , P -almost surelybounded on (0 , t ) and hence belong to L (0 , t ). Possibly enlarging N , we mayassume that outside N equation (6.6) holds as an equation in L (0 , t ). Integratingfrom 0 to t , we find that, P -almost surely, we have(6.7) Z t h X s , A ∗ x ∗ i ds = Z t h S s X , A ∗ x ∗ i ds + Z t Z s h S s − r F ( X r ) , A ∗ x ∗ i dr ds + Z t Z s G ( X r ) ∗ x ∗ S ∗ s − r A ∗ x ∗ dW H ( r ) ds . Recall that for x ∗ ∈ D ( A ∗ ) we have R t S ( s ) ∗ A ∗ x ∗ ds = S ( t ) ∗ x ∗ − x ∗ for all t ≥ ∗ -integral. Using this, we obtain,pathwise, Z t h S s X , A ∗ x ∗ i ds = D X , Z t S ∗ s A ∗ x ∗ ds E = h X , S ∗ t x ∗ − x ∗ i = h S t X − X , x ∗ i . Using Fubini’s theorem, we have Z t Z s h S s − r F ( X r ) , A ∗ x ∗ i dr ds = Z t D F ( X r ) , Z tr S ∗ s − r A ∗ x ∗ E ds dr = Z t h S t − r F ( X r ) , x ∗ i dr − Z t h F ( X r ) , x ∗ i dr pathwise. Using the stochastic Fubini theorem [29, Theorem 3.5], it follows that Z t Z s G ( X r ) ∗ S ∗ s − r A ∗ x ∗ dW H ( r ) ds = Z t Z tr G ( X r ) ∗ S ∗ s − r A ∗ x ∗ ds dW H ( r )= Z t G ( X r ) ∗ S ∗ t − r x ∗ dW H ( r ) − Z t G ( X r ) ∗ x ∗ dW H ( r ) P -almost surely.Plugging these three identities into (6.7) and using that X is a mild solution,(3.1) follows. (cid:3) Since all terms appearing in (3.1) are almost surely continuous, there is no prob-lem in writing an equation for the stopped process h X t ∧ τ , x ∗ i and we did this inthe proof of Proposition 6.3. On the other hand, for weakly mild solutions, theintegrand in the stochastic integral changes with t , causing problems to obtain anequation for the stopped process. In [3, Appendix], this problem was solved underthe assumption that the stochastic convolution is almost surely continuous. In theproof of Proposition 6.3, we have shown that for a weak solution, (6.5) holds for all x ∗ ∈ ˜ E ∗ . Given a stopping time τ , we can repeat the arguments with τ n replacedwith τ n ∧ τ to obtain Corollary 6.4. If X is a weak (equivalently, weakly mild) solution of (1.1) and τ is a stopping time, then for all t ≥ and x ∗ ∈ ˜ E ∗ we have (6.8) h X t ∧ τ , x ∗ i = h S t X ∧ τ , x ∗ i + Z t ∧ τ h S t − s F ( X s ) , x ∗ i ds + h X τ − S t − t ∧ τ X τ , x ∗ i { τ< ∞} + Z t [0 ,τ ] ( s ) G ( X s ) ∗ S ∗ t − s x ∗ dW H ( s ) . almost surely. The question arises whether (6.2) can be extended to hold for all x ∗ ∈ E ∗ . Thisis indeed the case under the following additional assumption. Hypothesis . Assume Hypothesis 3.1, that S ( t ) ⊂ L ( ˜ E, E ) for all t > x ∈ ˜ E the E -valued map t S ( t ) x is continuous on (0 , ∞ ). Furthermore,assume that for all t > , t ) ∋ s
7→ k S ( s ) k L ( ˜ E,E ) is square integrable.Assuming Hypothesis 6.5, a slight variation of the arguments in Remark 6.2shows that in this case the integrals in (6.2) are well-defined for x ∗ ∈ E ∗ . Corollary 6.6.
Assume that Hypothesis 6.5 holds. If X is a weak (equivalently,weakly mild) solution of (1.1) , then (6.2) and (6.8) hold for all x ∗ ∈ E ∗ .Proof. Define V := { x ∗ ∈ E ∗ : (6.2) holds a.e. } . By Proposition 6.3, ˜ E ∗ ⊂ V and hence V is weak ∗ -dense in E ∗ . The claim is provedonce we show that V is weak ∗ -closed in E ∗ . By the Krein-Smulyan theorem (see,e.g., § V is weak ∗ -closed in E ∗ if and only if B V := { x ∗ ∈ V : k x ∗ k E ∗ ≤ } is weak ∗ -closed in E ∗ . However, since the weak ∗ -topology is metrizableon bounded sets, it suffices to prove that B V is sequentially weak ∗ -closed.Using Hypothesis 6.5, this can be proved similarly as when extending equation(6.4) from x ∗ ∈ D (( A ⊙ ) ) to arbitrary x ∗ ∈ ˜ E ∗ in the proof of Proposition 6.3. Theproof for (6.8) is similar. (cid:3) N A CLASS OF MARTINGALE PROBLEMS ON BANACH SPACES 21
Mild solutions.
We begin by recalling some facts about stochastic integrationof operator-valued processes. For time being, B denotes a general separable Banachspace and H a separable Hilbert space. We also fix a stochastic basis (Ω , Σ , F , P )satisfying the usual condition on which an H -cylindrical Wiener process with respectto F is defined.An elementary process is a process Φ : [0 , T ] × Ω → L ( H, B ) of the formΦ( t, ω ) = N X n =1 M X m =1 ( t n − ,t n ] × A mn ( t, ω ) K X k =1 h k ⊗ x kmn , where 0 ≤ t < · · · < t N ≤ T , A n , · · · , A Mn ∈ F t n − are disjoint for all n and thevectors h , · · · , h K are orthonormal in H . If Φ does not depend on ω we also saythat Φ is an elementary function . For an elementary process, the stochastic integral R T Φ( t ) dW H ( t ) is defined by Z T Φ( t ) dW H ( t ) := N X n =1 M X m =1 A mn K X k =1 (cid:2) W H ( t n ) h k − W H ( t n − ) h k (cid:3) x kmn Now let Φ : [0 , T ] × Ω → L ( H, B ) be an H -strongly measurable and adaptedprocess which belongs to L (0 , T ; H ) scalarly, i.e. Φ ∗ x ∗ ∈ L (Ω; L (0 , T ; H )) forall x ∗ ∈ B ∗ . Then Φ is called stochastically integrable (on (0 , T )) if there exists asequence Φ n of elementary processes and an C ([0 , T ]; E )-valued random variable η such that(1) h Φ n h, x ∗ i → h Φ h, x ∗ i in L (Ω; L (0 , T )) for all h ∈ H and x ∗ ∈ B ∗ and(2) We have η ( · ) = lim n →∞ Z · Φ n ( t ) dW H ( t ) in L (Ω; C ([0 , T ]; B )) . In this case, η is called the stochastic integral of Φ and we write R t Φ( t ) dW H ( t ) := η ( t ). In the case where Φ does not depend on ω , we also require that the approxi-mating sequence Φ n does not depend on ω .Having defined stochastic integrability, we can now define what we mean by a mild solution . Definition 6.7.
A tuple ((Ω , Σ , F , P ) , W H , X ) where (Ω , Σ , F , P ) is stochastic basissatisfying the usual conditions, W H is an H -cylindrical Wiener process with respectto F and X is a continuous, F -progressive, E -valued process is called a mild solution of (1.1) if for all t ≥ s S ( t − s ) G ( X ( s )) is stochastically integrableand (6.1) holds almost surely.It is clear from the definition of stochastic integrability, that every mild solution ofequation [ A, F, G ] is also a weakly mild solution of [
A, F, G ] and thus, by Proposition6.3, also a weak solution of [
A, F, G ]. Moreover, if X is a mild solution, then (6.2)even holds for all x ∗ ∈ E ∗ (rather than for x ∗ ∈ ˜ E ∗ ) and the exceptional setoutside of which (6.2) holds can be chosen independently of x ∗ . We also notethat if X is a weak (hence a weakly mild) solution and it is known a priori that s S ( t − s ) G ( X ( s )) is stochastically integrable, then X is a mild solution.The obvious question is whether for a weak solution X the process s S t − s G ( X s )is automatically stochastically integrable. As we shall see, this is indeed the case intwo important cases. The proof relies on a characterization of stochastic integrabil-ity of a process Φ. Let us first discuss the case of L ( H, B )-valued functions , whichwas considered in [32]. It was proved there that a function Φ : [0 , T ] → L ( H, B ) isstochastically integrable if and only if there exists an B -valued random variable ξ such that(6.9) h ξ, x ∗ i = Z T Φ( s ) ∗ x ∗ dW H ( s ) . This, in turn, is equivalent with Φ representing a γ -Radonifying operator R ∈ γ ( L (0 , T ; H ) , B ). For the definition of γ -Radonifying operators and more in-formation, we refer to the survey article [28]. That Φ represents an operator R ∈ γ ( L (0 , T ; H ) , B ) means that for all x ∗ ∈ B ∗ the function t Φ ∗ ( t ) x ∗ be-longs to L (0 , T ; H ) and we have(6.10) h Rf, x ∗ i = Z T [ f ( t ) , Φ ∗ ( t ) x ∗ ] H dt ∀ f ∈ L (0 , t ; H ) , x ∗ ∈ B ∗ . Note that if Φ is H -strongly measurable, then the operator R is uniquely determinedby Φ.Using the results of [32], we obtain for (1.1) with additive noise: Proposition 6.8.
Assume Hypotheses 3.1 and 6.5 and that G ∈ L ( H, ˜ E ) is con-stant. Then the weak, the weakly mild and the mild solutions of (1.1) coincide.Furthermore, if there exist solutions, the function s S t − s G represents an elementof γ ( L (0 , t ; H ) , E ) for all t > .Proof. Let X be a weak (equivalently, a weakly mild) solution of (1.1). If no suchsolution exists, there is nothing to prove since every mild solution is also a weaklymild solution.Arguing as Remark 6.2, using that as a consequence of Hypothesis 6.5 themap s
7→ h x, S ∗ t − s x ∗ i is continuous even for x ∗ ∈ E ∗ and x ∈ ˜ E , we see that( s, ω )
7→ h S ( t − s ) F ( X ( s, ω )) , x ∗ i is measurable for all x ∗ ∈ E ∗ . By Hypothesis 6.5, k S s k L ( ˜ E,E ) is majorized on (0 , t ) by a square integrable function, say g . Hence, bythe boundedness of F on bounded sets we have k S t − s F ( X ( s, ω )) k ≤ g ( t − s ) sup r ∈ (0 ,t ) k F ( X ( r, ω ) k ∈ L (0 , t ) . This implies that R t S t − s F ( X s ) ds can be defined pathwise as an E -valued Bochnerintegral. Furthermore, this integral is a weakly measurable function of ω . Since E is separable, R t S t − s F ( X s ) ds is a strongly measurable function of ω by the Pettismeasurability theorem. Consequently, ξ := X t − S t X − R t S t − s F ( X s ) ds is an E -valued random variable. Since X is a weakly mild solution, (6.9) holds for T := t, Φ : s S t − s G and all x ∗ ∈ E ∗ by Corollary 6.6. The claim follows from theresults of [32]. (cid:3) Let us now return to our discussion of stochastic integrability in a general sepa-rable Banach space B . In order to have a powerful integration theory for L ( H, B )-valued processes , we need an additional geometric assumption on B . Of particularimportance are the so-called UMD Banach spaces . For the definition of UMD spacesand more information, we refer to the survey article [4]. We here confine ourselvesto note that every Hilbert space is a UMD space as are the reflexive L p and Sobolevspaces.The importance of the UMD property for stochastic integration is that it allowsfor so-called decoupling, see [10, 23]. Roughly speaking, this enables us to replacethe cylindrical Wiener process W H by an independent copy ˜ W H and thus use theresults of [32] pathwise. This program was carried out in [30] and yields a similarcharacterization of stochastic integrability as in [32] in the case of processes whichbelong scalarly to L p (Ω; L (0 , T ; H )). We recall that Φ : [0 , T ] × Ω → L ( H, E )) issaid to belong to L p (Ω; L (0 , T ; H )) scalarly , if for every x ∗ ∈ E ∗ the function t Φ ∗ ( t, ω ) x ∗ belongs to L (0 , T ; H ) for almost every ω and the map ω Φ ∗ ( · , ω ) x ∗ belongs to L p (Ω; L (0 , T ; H )). N A CLASS OF MARTINGALE PROBLEMS ON BANACH SPACES 23
It is proved in [30] that an H -strongly measurable and adapted process Φ :[0 , T ] × Ω → L ( H, E ) which belongs to L p (Ω; L (0 , T ; H )) scalarly is stochasticallyintegrable if and only if there is a random variable ξ ∈ L p (Ω; E ) such that (6.9)holds for all x ∗ ∈ E ∗ . This in turn is the case if and only if Φ represents a randomvariable R ∈ L p (Ω; γ ( L (0 , T ; H ) , E )). Here ‘represents’ means that (6.10) holdsfor almost every ω .A characterization of stochastic integrability for processes Φ which belong scalarlyto L (Ω; L (0 , T ; H )) is also contained in [30], however, in this characterization oneneeds information about the whole integral process R · Φ( s ) dW H ( s ); when dealingwith weakly mild solutions, such information is not available, whence this charac-terization cannot be used for our purposes. Therefore, in the proposition below, weuse a stopping time argument to reduce to the L p (Ω)-case. Proposition 6.9.
Assume Hypotheses 3.1 and 6.5 and that E is a UMD Banachspace. Then the weak, the weakly mild and the mild solutions of (1.1) coincide.Furthermore, if X is a weak solution, then for all t ≥ the function s S t − s G ( X s ) represents an element of the space L (Ω , γ ( L (0 , t ; H ) , E )) .Proof. Let X be a weak (equivalently, a weakly mild) solution of (1.1). If no weaksolution exists, there is nothing to prove.For n ∈ N and define τ n := inf { s > k X s k ≥ n } . Fix t >
0. Arguing similaras in the proof of Proposition 6.8, we see that ξ n := X t ∧ τ n − ( X τ n − S t − t ∧ τ n X τ n ) { τ n < ∞} − S t X ∧ τ n − Z t [0 ,τ n ] S t − s F ( X s ) ds is a well-defined, bounded, E -valued random variable. It follows from Corollary 6.6,that for x ∗ ∈ E ∗ , h ξ n , x ∗ i = Z t [0 ,τ n ] G ( X s ) ∗ S ∗ t − s x ∗ dW H ( s ) . almost surely. Since X has continuous paths and G is bounded on bounded subsets,Φ n : s [0 ,τ n ] S t − s G ( X s ) belongs to L ∞ (Ω; L (0 , t ; H )) scalarly. Hence, by [30,Theorem 5.9], Φ n is stochastically integrable and(6.11) X t ∧ τ n = S t X ∧ τ n + X τ n − S t − t ∧ τ n X τ n + Z t ∧ τ n S t − s F ( X s ) ds + Z t [0 ,τ n ] S t − s G ( X s ) dW H ( s ) . Furthermore, Φ n represents an element of L p (Ω; γ ( L (0 , t ; H ) , E )) for all p ≥ N be a set with P ( N ) = 0 such that for ω N the map s Φ n ( s, ω )represents an element R n ( ω ) of γ ( L (0 , t ; H ) , E ). Such a set exists by [30, Lemma2.7].Note that by the continuity of the paths, Φ n ( s, ω ) = Φ( s, ω ) := S t − s G ( X ( s, ω ))for all s ∈ (0 , t ) and n ≥ n = n ( ω ). Thus, Φ( s, ω ) represents an element R ( ω ) of γ ( L (0 , t ; H ) , E ) for all ω N . Since R n ( ω ) → R ( ω ) for all ω N , it follows that R is a strongly measurable γ ( L (0 , t ; H ) , E )-valued random variable. Furthermore, R is represented by Φ. By [30, Theorem 5.9], Φ is stochastically integrable and [30,Theorem 5.5] shows that Z t Φ n ( s ) dW H ( s ) → Z t Φ( s ) dW H ( s ) in L (Ω; E ) . On the other hand, ξ n → X ( t ) − S ( t ) X (0) − Z t S ( t − s ) F ( X ( s )) ds pointwise a.e. and hence in L (Ω; E ). Thus, letting n → ∞ in (6.11) finishes theproof. (cid:3) Applications
We end this article by discussing some examples of stochastic partial differentialequations where the results of this article can be applied.7.1.
Equations with measurable semilinear term and additive noise.
In[20], we are concerned with the following equation(7.1) dX ( t ) = (cid:2) AX ( t ) + F ( X ( t ) (cid:3) + GdW H ( t )where E, ˜ E, H and A are as in Hypothesis 3.1, the semilinear term F : E → E isbounded and measurable, W H is an H -cylindrical Wiener process and G ∈ L ( H, ˜ E ).In the case where F ≡
0, this is an Ornstein-Uhlenbeck equation, which is wellunderstood. If the Ornstein-Uhlenbeck equation associated with (7.1), i.e. equation[ A, , G ] is well-posed, the associated transition semigroup T ou is known explicitly.Namely, T ou ( t ) f ( x ) = Z E f ( S ( t ) x + y ) d N Q t ( y )where N Q denotes the centered Gaussian measure with covariance operator Q and Q t : E ∗ → E is given as Q t x ∗ := Z t S ( s ) GG ∗ S ( s ) ∗ x ∗ ds. By H Q t , we denote the reproducing kernel Hilbert space associated with Q t . In[20], the following theorem is proved. Theorem 7.1.
Let E, ˜ E, H and A as in Hypothesis 3.1, G ∈ L ( H, ˜ E ) and assumethat also Hypothesis 6.5 is satisfied. Moreover, assume that the Ornstein-Uhlenbeckequation [ A, , G ] is well-posed and that S ( t ) E ⊂ H Q t for all t > with (7.2) Z T k S ( t ) k L ( E,H Qt ) dt < ∞ for all T > . Then for every bounded, measurable F : E → E equation (7.1) iswell-posed. The solutions are strong Markov processes with a strong Feller transitionsemigroup. This extends earlier results from [6, 11, 12] where the corresponding equationwas studied for bounded and continuous (resp. bounded and weakly continuous) F under similar assumptions in the case where E = ˜ E is a Hilbert space. Theassertion that (7.1) is well-posed even for bounded measurable F appears to be neweven in the case of Hilbert spaces since existence of solutions cannot be inferredfrom the Girsanov theorem, as G is, in general, not invertible.The assumption that (7.2) holds implies that the transition semigroup T ou isstrongly Feller and is satisfied in many important examples, for example for theone-dimensional stochastic heat equation driven by space-time white noise, i.e. A is the L p -realization of the Dirichlet Laplacian on the interval (0 ,
1) and for p ≤ G is the injection from L (0 ,
1) to L p (0 , p > E = L (0 ,
1) and G the identity. It is also possible to consider thestochastic heat equation on C (0 , A, F, G ]. N A CLASS OF MARTINGALE PROBLEMS ON BANACH SPACES 25
The first step to prove uniqueness for solutions of (7.1) is to prove a Miyadera-Voigt type perturbation result for strongly Feller semigroups. For the generator A ou of the Ornstein-Uhlenbeck semigroup T ou , this result can be used to show that A pert , defined by A pert u ( x ) := A ou u ( x ) + h F ( x ) , ∇ u ( x ) i , generates a strongly Fellersemigroup T pert . A detailed analysis of the operator A pert shows that a probabilitymeasure P on C ([0 , ∞ ); E ) solves the local martingale problem associated withequation [ A, F, G ] if and only if it solves the true martingale problem (in the senseof [9]) for the operator A pert . Thus a well-known result [9, Theorem 4.4.1] yieldsthat the one-dimensional distributions of a solution P of the martingale problem for A pert are determined by the distribution of x (0) under P and the semigroup T pert .By Theorem 2.2, this implies uniqueness in law for the solutions of equation (7.1).Moreover, if solutions exist, then the associated transition semigroup is T pert , whichis strongly Feller.It thus remains to prove existence of solutions. If F is additionally Lipschitzcontinuous, then solutions can be constructed using Banach’s fixed point theoremin a standard way. Thus, for bounded, Lipschitz continuous F , equation (7.1) iswell-posed. To extend the existence result to general bounded, measurable F , arefinement of Lemma 4.3 is used. Indeed, making use of the strong Feller property,it can be proved that if F n is a sequence of bounded measurable functions suchthat equation [ A, F n , G ] is well-posed for every n and the sequence F n is uniformlybounded and converges pointwise to the bounded function F , then also equation[ A, F, G ] is well-posed. The tightness of the solutions to the local martingale problemfor [
A, F n , G ] can be proved using that these measures are distributions of mildsolutions of the equation. Using the approximation result, well-posedness of (7.1)can be extended from bounded, Lipschitz continuous F to bounded, measurable F via a monotone class argument.7.2. Stochastic reaction-diffusion systems with H¨older continuous multi-plicative noise.
Reaction-diffusion systems and stochastic perturbations of themplay an important role in applications in chemistry, biology and physics [25]. Inan abstract form, a stochastic reaction-diffusion system takes the form (1.1), wherethe state space E is a Banach space of R r -valued functions, defined on a domain O ⊂ R d . Typically, the reaction term F is a vector of composition operators withpolynomial entries.Such systems with locally Lipschitz continuous multiplicative noise where studiedin [5]. In the case where the noise term G is merely H¨older continuous, only partialresults are available and, to the best of our knowledge, only for r = 1, i.e. a singlereaction-diffusion equation rather than a system. In [2], existence of solutions forsuch an equation was proved under an additional boundedness assumption on G .However, a uniqueness result is missing, except for the case of locally Lipschitzcontinuous G .In [19], we prove pathwise uniqueness and strong existence of solutions for aclass of stochastic reaction-diffusion equations with H¨older continuous multiplicativenoise. Let us here present an example which fits into the framework of [19] andexplain how results of this article are used in the proof of existence and uniqueness.Let O ⊂ R d be an open domain with Lipschitz boundary. Moreover, we let a = ( a (1) ij ) , a = ( a (2) ij ) ∈ L ∞ ( O ; R d × d ) be symmetric and uniformly elliptic, i.e.there exists η > ξ ∈ R d and almost all x ∈ O we have d X i,j =1 a ( l ) ij ( x ) ξ i ξ j ≥ η | ξ | for l = 1 ,
2. Let R , R be Hilbert-Schmidt operators on L ( O ) such that R j isdiagonalized by an orthonormal basis ( e ( j ) n ) n ∈ N of L ( O ) which consists of functions in C ( O ) and satisfies P ∞ n =1 k R j e ( j ) n k ∞ < ∞ for j = 1 ,
2. Finally, we let g , g : R → R be of linear growth and locally -H¨older continuous. We consider the followingstochastic reaction-diffusion system(7.3) (cid:26) du ( t ) = (cid:2) div ( a ∇ u ( t )) + u ( t ) − u ( t ) + u ( t ) (cid:3) dt + g ( u ( t )) R dW ( t ) du ( t ) = (cid:2) div ( a ∇ u ( t )) + u ( t ) − u ( t ) (cid:3) dt + g ( u ( t )) R dW ( t )complemented with conormal boundary conditions.To reformulate the above system in our abstract framework, we set ˜ E = E = C ( O ) × C ( O ) and A = diag( A , A ), where A j is the C ( O )-realization of the dif-ferential operator div ( a j ∇· ) under conormal boundary conditions. We set H = L ( O ) × L ( O ). By the assumption on R j , for h ∈ L ( O ) we find that R j h ∈ C ( O ).We may thus define G : E → L ( H, E ) by[ G ( u, v ) h ]( x ) := ( g ( u ( x )) R h ( x ) , g ( u ( x )) R h ( x ))for h , h ∈ L ( O ) and x ∈ O . The reaction term F is given by [ F ( u, v )]( x ) :=( u ( x ) − u ( x ) + v ( x ) , u ( x ) − v ( x )). This reaction Term is of Fitzhugh-Nagumo typeand equations with this reaction term are generic excitable systems [25].In [19] we prove Theorem 7.2.
Under the assumptions above, equation (7.3) is well-posed on thestate space E = C ( O ) × C ( O ) . The solutions exist strongly, they are pathwise uniqueand strong Markov processes. The proof of Theorem 7.2 is in spirit rather different from the proof of well-posedness of (7.1), insofar as we work directly with solutions of the equation, ratherthan with solutions of the associated local martingale problem. In the proof, weuse the equivalence of weak and mild solutions. Indeed, in the proof of pathwiseuniqueness, we use weak solutions, whereas in the proof of existence of solutions,we use mild solutions. We also employ the Yamada-Watanabe theory from Section5. The proof of pathwise uniqueness is an adaption of the proof of [39, Theorem1]. The main difficulty in extending the proof from the finite-dimensional settingto an infinite dimensional setting is to handle the differential operators involvedin (7.3). In [19], we use the concept of a weak solution and test solutions againstfunctionals x ∗ = ( λR ( λ, A ) ∗ δ x , x ∗ = (0 , λR ( λ, A ) ∗ δ x ), where A j are therealizations of of the differential operator div ( a j ∇· ) on C ( O ). This approach shouldbe compared with [26], where pathwise uniqueness was proved for stochastic heatequations on O = R d , namely du ( t ) = ∆ u ( t ) + σ ( u ( t )) dW ( t ) , where ∆ is the Laplacian on R d , W is a colored noise and σ : R → R is γ -H¨oldercontinuous, where the allowed value of γ depends on the noise W . To prove pathwiseuniqueness in [26], the authors convolute solutions of the stochastic heat equationwith a mollifier ϕ n . In their variational framework, this yields the term u ∗ ∆ ϕ n in the equation for the resulting process. It is then used that, as a consequenceof its translation invariance, the Laplacian commutes with convolutions, i.e. wehave u ∗ (∆ ϕ n ) = ∆( u ∗ ϕ n ). This is no longer true for differential operators withnonconstant coefficients as in (7.3).Let us also note that a recent result [24] for the stochastic heat equation that inthe case of d = 1 shows that we cannot hope for pathwise uniqueness in the case ofspace-time white noise.Note that by Theorem 5.3, pathwise uniqueness implies uniqueness in law, hencethe strong Markov property of solutions follows from Theorem 2.2 once we haveestablished existence of solutions. To that end, we approximate the function f in N A CLASS OF MARTINGALE PROBLEMS ON BANACH SPACES 27 the reaction term and the functions g , g with bounded functions by cutting offthe functions. Existence of solutions for the approximate problems with boundedcoefficients and deterministic initial values follows from the results of [2]. We couldthen use Lemma 4.3 to infer existence of solutions for the limit problem (7.3).However, in [19] we choose a different approach and use that, as a consequence ofpathwise uniqueness and Corollary 5.4, the approximate solutions can be realizedon a common stochastic basis and with respect to a common H -cylindrical Wienerprocess. This allows us to adopt the strategy from [5, 21] to prove existence ofsolutions. Indeed, as the approximate solutions exist on a common stochastic basisand are pathwise unique, they can be ‘glued together’ to a ‘maximal solution’ ofequation (7.3). To prove existence of solutions in the sense used here, we haveto prove that the ‘maximal solution’ exists globally. By the results of [21], tothat end, we have to prove uniform boundedness of the approximate solutions in L p (Ω; C ([0 , T ]; E )) for a suitable p >
1, all
T > p -integrable initial data. Asthe approximate solutions are also mild solutions, the uniform boundedness can beproved using estimates for deterministic and stochastic convolutions, see [31].We note that, in comparison with [2], in Theorem 7.2 we do not need that theterm G is bounded. Moreover, with the above arguments, we initially prove exis-tence of solutions only for initial data with a certain integrability, thus in particularfor deterministic initial data. However, by Theorem 2.2, we automatically obtainexistence of solutions for all initial distributions. Acknowledgment.
I would like to thank Jan van Neerven for several helpful dis-cussions and also for reading an earlier version of this article. I am also grateful tothe anonymous referees for the critical comments, which helped improve this article.
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Institute of Applied Analysis, University of Ulm, 89069 Ulm, Germany
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