On uniqueness of solutions to martingale problems --- counterexamples and sufficient criteria

The dynamics of a Markov process are often specified by its infinitesimal generator or, equivalently, its symbol. This paper contains examples of analytic symbols which do not determine the law of the corresponding Markov process uniquely. These examples also show that the law of a polynomial process is not necessarily determined by its generator. On the other hand, we show that a combination of smoothness of the symbol and ellipticity warrants uniqueness in law. The proof of this result is based on proving stability of univariate marginals relative to some properly chosen distance.


INTRODUCTION
Consider a system whose state at time t is represented by a vector X(t) in R d . In applications the dynamics of such a system are often described by specifying how X(t) changes as a function of the current state X(t). In a deterministic setup this is typically expressed in terms of an ordinary differential equation. If, on the other hand, X(t) is random, it may be viewed as a Markov process whose local dynamics can be specified in terms of a stochastic differential equation, its infinitesimal generator, its local semimartingale characteristics, or its symbol. As in the deterministic case, this immediately leads to the question of existence and uniqueness of a stochastic process exhibiting the given local dynamics.
This can be rephrased in terms of existence and uniqueness of the solution to a corresponding martingale problem. Existence is known to hold under relatively weak continuity conditions, cf. e.g. [8,Theorem 4.5.4], [13,Theorem 3.15], [3,Theorem 3.24], and Theorem 2.7 below. For continuous processes uniqueness holds for Lipschitz-resp. Hölder-continuous coefficients or under some ellipticity condition, cf. e.g. [22]. The situation is less obvious for processes with jumps. Lipschitz conditions only help for generators which have a natural representation as an SDE, which often is not the case. Ellipticity, on the other hand, requires a continuous martingale part to be present, which often is not the case either.
This piece of research is motivated by the desire to come up with a general uniqueness result for Markov processes that may not have a continuous martingale part or a natural representation as a SDE. In this context we share the point of view of [13,16,3] that it is natural to study Markov processes through their symbol. Indeed, e.g. weak convergence of a sequence of Levy processes corresponds to pointwise convergence of their symbols.
From the analogy to ODE's one may expect uniqueness to hold if the symbol of the process depends smoothly on the state X(t). Unfortunately, smoothness alone does not seem to suffice in order to warrant uniqueness. In Section 3 we present two examples of even analytic symbols where uniqueness in law of the corresponding Markov process does not hold. This is the first main result of this paper. These examples also show that the law of a polynomial process in the sense of [4,5] is not in general uniquely determined by its generator.
Section 4 contains a positive result, which is the second main contribution of this paper. It is shown that the combination of sufficient smoothness and ellipticity warrants uniqueness in law. In contrast to [25,Theorem 4.3] and related results, the continuous martingale part may vanish. The probably closest relative to our Theorem 4.4 below is [2, Theorem 2.8] which also relies on smoothness and ellipticity of the symbol. However, Böttcher requires a certain boundedness for derivatives of any order while we need this condition only for finitely many derivatives. Nevertheless, [2,Theorem 2.8] is not a special case of our Theorem 4.4 below. Another closely related result is [13,Theorem 5.24] which, however, requires the symbol to be real.
Uniqueness results have been obtained by a number of different approaches, cf. [16] for an overview. From a very rough perspective, the most commonly used techniques are • SDE methods where uniqueness is often obtained from fixed-point arguments, • construction of a solution to the backward equation, i.e. construction of solutions for the associated abstract Cauchy problem, and • so-called interlacing techniques which allow to add finitely many jumps.
By contrast, our approach is based on establishing stability of the univariate marginals relative to a properly chosen distance. This kind of reasoning seems to be new and it constitutes the third main contribution of this paper.
The paper is structured as follows. In Section 2 we recall various notions and properties concerning symbols and martingale problems. Moreover, we state an existence result which follows from [8,Theorem 4.5.4]. Subsequently, we present examples showing that smoothness of the symbol does not imply uniqueness of the solution to a martingale problem. In Section 4 a uniqueness result under smoothness and some mild ellipticity of the symbol is stated. Section 5 contains proofs. In the appendix, we recall some facts on complex measures.
1.1. Notation. d ∈ N generally denotes the dimension of the space under consideration. We denote the trace of a matrix C ∈ R d×d by Tr(C). For any two vectors x, y ∈ C d we define the standard bilinear form xy := d j=1 x j y j . Moreover, we set Cx := ( d k=1 C jk x k ) j=1,...,d and yCx := y(Cx) for any matrix C ∈ C d×d and vectors x, y ∈ C d . The set of positive semidefinite d × d-matrices is denoted by S d . We fix a truncation function χ : R d → R d , i.e. χ is measurable, bounded and it equals the identity in a neighbourhood of zero. W.l.o.g. we suppose that χ(x) = x for |x| ≤ 1, where |x| := ( d j=1 |x j | 2 ) 1/2 denotes the Euclidean norm on R d . We write B(x, r) := {y ∈ R d : |x − y| < r} for the open ball with radius r > 0 centered at x ∈ R d .
We denote the gradient of f ∈ C 1 (R d , C) by ∇f (x) := (∂ 1 f, . . . , ∂ d f ), x ∈ R d , the Hessian of f ∈ C 2 (R d , C) by Hf (x) := (∂ 2 jk f (x)) d j,k=1 , x ∈ R d , and the Laplacian of is differentiable with respect to the first coordinate and ∆ 1 f (x, y) := (∆f (·, y))(x), x, y ∈ R d for sufficiently smooth f . If f is smooth enough in the second coordinate,∇ 2 f (x, y) and H 2 f (x, y) are defined accordingly. ByĈ(R d ) (resp.Ĉ(R d , C)) we denote the set of real-valued (resp. complex-valued) continuous functions on R d that vanish in ∞. The greatest integer less or equal x ∈ R is written as [x]. Further unexplained notation is used as in [8,18].

THE SYMBOL AND THE EXISTENCE THEOREM
We start by recalling the notion of the symbol and its associated martingale problem, cf. [16,3]. A systematic theory for symbols was first developed by Hoh [11,12,14]. Other important references include [15], which is more in view of strongly continuous semigroups, and [1], who developed a theory for symbols on nuclear separable spaces.
for any x, u ∈ R d . We call a symbol q : (2) If q denotes a symbol, an adapted càdlàg R d -valued stochastic process X is called solution to the q-martingale problem if the process is a local martingale for any u ∈ R d . Uniqueness for the q-martingale problem means that any two solutions X, Y to the q-martingale problem with the same initial law (i.e. X(0) has the same law as Y (0)) have the same distribution. Finally, we say that existence holds for the q-martingale problem if, for any probability measure µ, there is a solution X to the q-martingale problem with initial law P X(0) = µ.
for any Schwartz function f in the sense of [9, Definition 2.2.1] and any x ∈ R d .
In order to relate a symbol to a martingale problem in the sense of [8,Section 4.3], we define an operator corresponding to the symbol. Definition 2.4. Let q be a symbol. The operator A associated with q is defined as This operator can be expressed in terms of the Lévy-Khintchine triplet.
Lemma 2.5. Let q be a symbol, A the operator associated with q, and (b, c, F ) the triplet of q. Then for any real-valued Schwartz function f and any x ∈ R d .
Let q be a symbol with associated operator A. Moreover, denote by B the restriction of A to the set of real-valued Schwartz functions f such that Af is bounded. The following lemma shows that any solution X to the q-martingale problem is a solution to the martingale problem in the sense of [8,Section 4.3] for B. Under suitable conditions the converse is also true, cf. Theorem 2.7(2) below. Lemma 2.6. Let X be a solution to the q-martingale problem, f a real-valued Schwartz function such that Af is bounded, and Af (X(s))ds for any t ≥ 0. Then M f is a martingale.
Proof. Theorem 5.1 states that X is a semimartingale with local characteristics (b(X − ), c(X − ), F (X − , ·)). Thus Itō's formula for the local characteristics [19,Proposition 2.5] together with Lemma 2.5 yield that is a version of the local characteristics of f (X) relative to the truncation function h. By [18,Theorem II.2.42] Af (X(s))ds is a local martingale. However, M f is bounded on compact time intervals and hence it is a martingale.
Theorem 2.7 (Existence). Let q be a continuous symbol with associated operator A and triplet (b, c, F ). Assume that is bounded by some finite constant. Then the following statements hold.
(1) For any probability measure µ on R d there is a solution X to the q-martingale problem with P X(0) = µ. (2) A stochastic process X is a solution to the q-martingale problem if and only if Af (X(s))ds, t ∈ R + defines a martingale for any real-valued Schwartz function f (or, equivalently, any smooth function with compact support), i.e. if and only if X is a solution to the martingale problem A in the sense of [8,Section 4.3].
(3) The operator A has the following properties: (a) its range is contained inĈ(R d ), implies Af (x 0 ) ≤ 0 for any real-valued Schwartz function f and any x 0 ∈ R d , and (c) it is conservative, i.e. there is a bounded sequence of real-valued Schwartz functions (f n ) n∈N which converges pointwise to 1 such that (Af n ) n∈N is a bounded sequence which converges pointwise to 0. (4) It is possible to choose measures (P x ) x∈R d on the Skorokhod space such that the canonical process X is a solution to the q-martingale problem with X(0) = x a.s. under P x , x ∈ R d and such that x → P x (X(t) ∈ A) is measurable for any t ≥ 0 and any Borel set A ⊂ R d . Moreover, (X, (P x ) x∈R d ) is strong Markov.
(5) If the q-martingale problem has several solutions for some initial law P X(0) = µ, then there are several families of measures (P x ) x∈R d as in (4).
where E x denotes expectation relative to P x .
The proof is to be found in Section 5.1.  (6) means that −q is a symbol in the sense of [16].
The assumption on the triplet in Theorem 2.7 can be replaced by a smoothness condition on the symbol: Corollary 2.9. Let q be a continuous symbol such that u → q(x, u) is twice differentiable with bounded gradient x → ∇ 2 q(x, 0) and bounded Hessian x → H 2 q(x, 0). Then statements (1)(2)(3)(4)(5)(6) in Theorem 2.7 hold.

Observe that
Dominated convergence and |y| 2 F x (dy) < ∞ yield thatF x is twice differentiable in 0. By [7, Lemma A.1], this implies that F x has finite second moments given by Again by dominated convergence we obtain Boundedness of H 2 q(·, 0) now yields that Tr(c(·)) and |y| 2 F (·, dy) are bounded as well.
Once more from dominated convergence we conclude Since (y − χ(y))F (·, dy) is a bounded function, b is bounded as well. Theorem 2.7 now yields the assertion.
Next we restate a result by van Casteren which shows that uniqueness implies the Feller property.
Proposition 2.10 (Feller property). Let q be a symbol satisfying the requirements of Theorem 2.7. If uniqueness holds for the q-martingale problem, there is a closed extension C of A which generates a strongly continuous positivity preserving contraction semigroup onĈ(R d ). In other words, any solution to the martingale problem A is a Feller process.
Proof. Boundedness of g in Theorem 2.7 implies that A maps Schwartz functions on a subset ofĈ(R d ). In view of [20,Theorem 3.1] or [13,Proposition 5.18], this yields the claim.

COUNTEREXAMPLES
In this section we provide an example of a real-valued analytic symbol which fails to have the uniqueness property in the sense of Definition 2.1. Moreover, we present a closely related example. Both correspond to polynomial processes in the sense of [4,5], i.e. the extended operator A in the sense of [5, Definition 2.3] maps polynomials on polynomials of at most the same degree.
Example 3.1. There is an analytic symbol, namely satisfying the requirements of Theorem 2.7 and having an entire extension to C×C where, however, uniqueness does not hold for the q-martingale problem. Moreover, there are solutions X, Y to the q-martingale problem with X(0) = 0 = Y (0) and P X(t) = P Y (t) for any t > 0. More generally, there are strong Markov processes X, Y on R d with the above symbol which do not have the same law. Moreover, X, Y are polynomial processes in the sense of [5, Definition 2.1]. Starting in X(0) = 0 = Y (0), their n-th moment at time t is given by Proof. Suppose that the truncation function χ is continuous and anti-symmetric. Define q as in (3.2). The function q has an obvious entire extension. Define Then (b, c, F ) is the corresponding triplet in the sense of Remark 2.2. For any n ∈ N, k ∈ R + we also define Then q k,n is a continuous symbol and the associated linear operator is given by The symbol q k,n satisfies the requirements of Theorem 2.7 whence there is a solution X k,n to the martingale problem (A k,n , δ 0 ) in the sense of [8,Section 4.3]. Since there is a subsequence of (X k,n ) n∈N which converges weakly to a solution X k of the martingale problem (A, δ 0 ). Note that X k takes values only in M k . Both X 1 , X √ 2 are solutions to the martingale problem related to A and initial law δ 0 , where X 1 takes values only in M 1 and X √ 2 only in M √ 2 . Their extended generator in the sense of [5, Definition 2.3] is defined for all polynomials. Polynomials are mapped to polynomials of at most the same degree, which means that X 1 , X √ 2 are polynomial processes by [5, Theorem 2.10]. [5, Theorem 2.7] and its proof yields the moments. In particular, We now turn to a related example with analytic symbol where uniqueness fails. It corresponds to an increasing process. It is once more polynomial in the sense of [5, Definition 2.3]. However, since its state space equals R + , it is not perfectly in line with the setup of this paper.
This function clearly has an entire extension to C × C. There are solutions X, Y to the qmartingale problem which start in 0 and are singular in the same sense as in the previous example. More generally, there are strong Markov processes X and Y with values in R + and symbol q, which do not have the same law. Once more X, Y are polynomial processes. Starting in 0, their n-th moment at time t is given by Exponent q(x, ·) has Lévy-Khintchine triplet (χ(x)/x, 0, δ x /x) for x > 0 and (1, 0, 0) for x = 0. Finally, observe that the continuous continuation given bỹ on state space R yields the same process. The picture shows simulated paths from three different Markov processes. Each of them uses the approximate generator A k,n appearing in the proof of Example 3.2 with k = 1 (black line), k = 3 √ 2 (red line) resp. k = 3 √ 4 (green line) and n = 10 for all of them.
Proof. For any n ∈ N, k ∈ R + and x, u ∈ R we define Then q k,n is a continuous symbol with corresponding triplet ( The associated linear operator is given by By Theorem 2.7 there is a solution X k,n for the martingale problem related to A k,n and initial law δ 0 . As in the previous example, we conclude that X k,n (t) = s≤t ∆X k,n (s) and that X k,n takes values only in the closed set M k := { k2 n : ∈ N} ∪ {k2 z : z ∈ Z} ∪ {0}. Also as in Example 3.1, for any real-valued Schwartz function f on R we have The statements on preservation of polynomials and the moments of X follow as in the previous example.

THE SYMBOL AND THE UNIQUENESS PROBLEM
The obvious question to ask is what conditions are needed to ensure uniqueness of the q-martingale problem for a given symbol q. For continuous processes the situation is well understood, The fact that SDE's have unique solutions under Lipschitz conditions directly yields uniqueness for C 2 -symbols without jump part.
Then uniqueness holds for the q-martingale problem.
Proof. Observe that c takes actually values in the set of positive semidefinite matrices. Let σ(x) be the positive square root of c(x) for all x ∈ R d . Then [23,Theorem V.12.12] implies that σ is Lipschitz. [23, Theorem V.12.1 and Section V.22] applied to (b, σ) yields the claim.
As Example 3.1 indicates, the previous theorem does not hold for processes with jumps. One could express q-martingale problems for general symbols q in terms of an SDE, but it is not clear what conditions on q warrant Lipschitz continuity of the corresponding coefficients. Stroock-Varadhan type results, however, require the diffusion part of the symbol to be non-singular. A systematic study of existence and uniqueness property has been undertaken by Hoh in a number of papers, see e.g. [ One of the main contributions of the present paper is the following uniqueness result.
be a Hölder-continuous symbol satisfying the requirements of Theorem 2.7. Moreover, suppose that there are K ∈ R + and complex measures (P t,u ) t∈[0,1],u∈R d such that Then existence and uniqueness holds for the q-martingale problem. In particular, there is a unique Feller process starting in any given distribution and having symbol q.
Proof. The proof is to be found in Section 5.2.
where ψ is a real-valued Lévy exponent on R which satisfies the requirements of Theorem 2.7, e.g. ψ(u) = −u 2 2 . In particular, q(0, u) = 0 for any u ∈ R. Then existence and uniqueness for the q-martingale problem hold.
Proof. Instead of verifying the conditions in the previous theorem directly, we refer to Lemma 5.14 below, which follows from Theorem 4.2.
If the measure P t,u in Theorem 4.2 happens to be nonnegative as in Example 3.1, we have In this case condition (2) in Theorem 4.2 can be interpreted as first and second moment condition on P t,u , which can be vaguely viewed as a "smoothness" condition on q. However, in particular for complex P t,u it is less obvious how restrictive condition (2) is and how one can verify it. We therefore provide a second uniqueness result which follows from Theorem 4.2, but which is stated directly in terms of q.
Theorem 4.4. Let q be a continuous symbol with q(·, u) ∈ C [d/2]+3 (R d , C) for all u ∈ R d and such that q satisfies the requirements of Theorem 2.7, cf. also Corollary 2.9. Let ϕ : R d → C be a characteristic exponent satisfying the following conditions for some bounded functions g 1 , g 2 : Then existence and uniqueness holds for the q-martingale problem.
Proof. The proof is to be found in Section 5.2.
Condition (4.6) is a uniform smoothness requirement. Condition (4.5), however, means that the symbol is bounded from below in an appropriate sense. Such an ellipticity condition occurs in the Stroock-Varadhan existence and uniqueness result, cf. [25,Theorem 4.3]. The advantage of the result in [25] is that continuity suffices and no extra smoothness is needed. Moreover, the drift only needs to be measurable. However, Stroock requires an ellipticity condition with respect to the explicit symbol ϕ(u) = − 1 2 u 2 , which means that a continuous diffusion part is present everywhere. His proof also uses some extra regularity for the jump measure which, however, could be relaxed.
Since Stroock and Varadhan have published their result, some variants of the Stroock-Varadhan theorem with a more general ellipticity condition have been established by several authors, i.e. with a more general function ϕ than for the original result, cf. [16] for an overview. A recent result is due to Böttcher, cf. [2, Theorem 2.8], who requires equations (4.5,4.6) for arbitrary α and, moreover, a certain boundedness for the derivatives with respect to u. Theorem 4.4 may be easier to apply in practice because it involves only finitely many derivatives and no smoothness in u.
is a version of the local characteristics of X relative to truncation function χ. We continue with a lemma which yields a sufficient condition for the existence of second moments.
Lemma 5.2. Let q be a symbol which satisfies the requirements of Theorem 2.7. Moreover, let X be a solution to the q-martingale problem. Then E(sup s∈[0,t] |X(s) − X(0)| 2 ) < ∞ for any t ∈ R + .
Proof. Proposition 5.1 implies that X is a semimartingale with local characteristics of the form (5.7). Boundedness of g implies that it is a special semimartingale, cf. e.g. [18, Proposition II.2.29(a)]. The finite variation part A in its canonical decomposition X = X(0) + M + A is of the form and hence bounded on any compact interval. For the local martingale part M we have which is bounded on any compact interval as well. Doob's inequality yields that for any t ∈ R + , which yields the claim.
Lemma 5.3. Assume that the requirements of [8,Theorem 4.5.19] are met where A 0 denotes the corresponding operator on the state space R d and T is the set of all bounded stopping times relative to the natural filtration on the Skorokhod space. Then it is possible to choose measures (P x ) x∈R d on the Skorokhod space such that the canonical process X is a solution to the A 0 -martingale problem with X(0) = x a.s. under P x , x ∈ R d and such that x → P x (X(t) ∈ A) is measurable for any t ≥ 0 and any Borel set A ⊂ R d . Moreover, (X, (P x ) x∈R d ) is strong Markov. Finally, if the q-martingale problem has several solutions for some initial law P X(0) = ν, then there are several such families of measures (P x ) x∈R d . Proof.
Step 1: Let A be an extension of A 0 as in [8,Theorem 4.5.19]. Moreover, let P x ∈ Γ δx for x ∈ R d such that P x solves the A-martingale problem, which exists by [8,Theorem 4.5.19(c)]. We define the law P (x, t, dy) := P x (X(t) ∈ dy) for x ∈ R d , t ≥ 0. Then P (x, 0, ·) = δ x . Let f : R d → R be continuous and bounded, which implies that t → f (X(t)) is right-continuous and bounded. By dominated convergence = E x (P X(t) (X(s) ∈ A)) = P y (X(s) ∈ A)P X(t) x (dy) = P (y, s, A)P (x, t, dy) and hence P is a transition function in the sense of [8, Page 156].
Step 2: From [8, Theorem 4.5.19(d)] we also get that X is a strong Markov process in the sense of [8, Page 158] with transition function P . Indeed, let P be a solution to the A-martingale problem, τ a finite stopping time and C ∈ B(D R d [0, ∞)). Then we have E(1 {X((τ ∧n)+·)∈C} |F τ ∧n ) = P X(τ ∧n) (C) for any n ∈ N by [8, Theorem 4.5.19(d)] because τ ∧ n is bounded. For the strong Markov property we have to show that equality holds for τ instead of τ ∧ n. Clearly, P X(τ ∧n) (C) → P X(τ ) (C) pointwise for n → ∞. Moreover, Y n := 1 {X((τ ∧n)+·)∈C} → Y ∞ := 1 {X(τ +·)∈C} pointwise and hence in L 2 (P ) for n → ∞. We obtain Step 3: Now we turn to the last part of the claim. Here, we assume that Γ ν contains more than one element for some Borel measure ν on R d . The proof of [8,Lemma 4.5.19] actually constructs extensions A = B given in [8,Lemma 4.5.19(b)] which are maximal in the sense that any further extension of A or B does not meet [8,Lemma 4.5.19(c)] any more.
Then, B ⊆ A due to maximality. Thus there is (f, g) ∈ B with (f, g) / ∈ A. Assume by contradiction that M (t) := f (X(t)) − t 0 g(X(s))ds is a P x -martingale for any x ∈ R d . Then is a strict extension of A which is a linear operator and such that the canonical process X solves the A + -martingale problem under P x for any x ∈ R d . [8,Proposition 4.3.5] yields that A + is dissipative. This contradicts the maximality of A. We conclude that there is Proof of Theorem 2.7.
Step 1: Let f be a real-valued Schwartz function and > 0. Boundedness of g implies sup x∈R d F (x, {y ∈ R d : |y| ≥ a}) < for some sufficiently large a > 0 such that the support of χ is contained in B(0, a). Lemma 2.5 yields for any x ∈ R d . Hence Af (x) → 0 for x → ∞.
Step 2: Let f be any real-valued Schwartz function with maximum x 0 ∈ R d , i.e. sup x∈R d f (x) = f (x 0 ). Then ∇f (x 0 ) = 0 and Hf (x 0 ) is negative semidefinite. Therefore i.e., A satisfies the positive maximum principle in the sense of [8, p. 165], whence statement (3b) holds. Moreover, it is defined on a dense subset ofĈ(R d ) because its domain are the real-valued Schwartz functions.
Step 3: Let ϕ : R d → [0, 1] be an infinitely differentiable function which is constant 1 on the unit ball in R d and whose support is contained in the centered ball with radius 2. For any n ∈ N define the real-valued Schwartz function Then f n → 1 pointwise and the second derivatives of f n are bounded by k/n 2 , where k is a common bound for the first two partial derivatives of ϕ. Lemma 2.5 yields Af n (x) = (f n (x + y) − 1)F (x, dy) for x ∈ R d and n > |x|. Due to a remainder estimate for the Taylor series we have |f n (x + y) − 1| ≤ k|y| 2 2n 2 . Thus the dominated convergence theorem yields whence Af n (x) → 0 pointwise. Similar arguments yield |Af n (x)| ≤ Kg(x) for any x ∈ R d , n ∈ N and some constant K > 0 which does not depend on x and n. Thus we have f n → 1 and Af n → 0 for n → ∞, where the convergence holds relative to the bp-topology, cf. Similarly as in Step 4 one shows that there is a bound B < ∞ such that |f n (x)| ≤ 1, |Af n (x)| ≤ B for any x ∈ R d , n ∈ N, and a.s. for any t ∈ R + . By dominated convergence, M u is a martingale which shows that X is a solution to the q-martingale problem. Altogether, we obtain both the direct implication of statement (2) and statement (1).
Step 6: Let x, u ∈ R d and (P x ) x∈R d measures on the canonical space such that the canonical process X is a solution to the q-martingale problem with X(0) = x P x -a.s. for any x ∈ R d . Then Hence right-continuity of X in 0 yields for t ↓ 0, which is statement (6).

5.2.
Proof of the uniqueness theorems. The remainder of the paper is devoted to the proof of Theorems 4.2 and 4.4. The idea is as follows. We aim at proving uniqueness of univariate marginals in the Fourier domain, i.e. we show that for two solutions X, Y the characteristic functions ϕ X(t) , ϕ Y (t) coincide. This will be done by a Grönwall argument. We proceed in two steps. First we show that any solution can be approximated locally by a conditional Lévy process, cf. Lemmas 5.7, 5.8. Secondly we try to find bounds for the deviation rate of two piecewise Lévy processes, which leads to Theorem 4.2. In order to derive Theorem 4.4, the conditions are first verified for simple symbols. Moreover, the set of symbols meeting the requirements has a certain closedness property, cf. Lemma 5.11. Then we can deduce Lemma 5.14 which states that uniqueness holds if the symbol can be locally approximated with a Fourier series satisfying some positivity condition. Finally, we construct such a Fourier series for elliptic symbols, cf. Lemma 5.15. The localisation procedure for martingale problems reveals that uniqueness is a local property, cf. [8,Section 4.6] and [3,Theorem 3.28]. We restate a localisation theorem suitable for our applications with slightly different assumptions than the very related [3,Theorem 3.28]; the proof however is basically the same.
Proposition 5.4. Let q be a symbol such that existence holds for the q-martingale problem. Let U be an open covering for R d and for all U ∈ U let q U be a symbol such that existence and uniqueness holds for the q U -martingale problem, (3) q(·, u) is bounded for any u ∈ R d and (4) q U (·, u) is bounded for any u ∈ R d , U ∈ U. Then existence and uniqueness hold for the q-martingale problem.
Proof. W.l.o.g. we may assume that U is countable. Let µ be a probability measure on R d and U ∈ U. Define The next result is a Grönwall-type theorem with perturbation which will be useful later.
In the sequel we will work with the norm Remark 5.6. A sequence of characteristic functions which converges with respect to · , converges uniformly on compact sets. Lévy's continuity theorem [17,Theorem 19.1] yields weak convergence of the corresponding sequence of random variables.
As mentioned above, the proof of Theorem 4.2 is based on local comparison to conditional Lévy processes.
Lemma 5.7 (Comparison to conditional Lévy process I). Let q be a continuous symbol such that q(·, u) is bounded for all u ∈ R d . Moreover, let X be a solution to the qmartingale problem and for s ≥ 0 let Q s be a regular version of the conditional law of X given F s , i.e.
The fundamental theorem of calculus yields Thus we obtain = E e (t−s)q(X(s),u)+iuX(s) + E (Γ(t, s, u)). for all s, t ∈ I, u ∈ R d with s < t.

Finally, we have
Proof. Let f be a bounded and continuous function such that

Proposition 5.1 states that X is quasi-left continuous. Hence
is continuous and H(s, s) = 0 for all s ∈ R + . The mean value theorem theorem yields the claim for β(t) := t sup{H(r, s) : r, s ∈ I and |r − s| ≤ t}.
We can now show that the univariate marginals of solutions to the martingale problem are uniquely determined under certain conditions. Lemma 5.9. Let q ∈ C(R d × R d , C) be a continuous and Hölder-continuous symbol. Moreover, let K ∈ R + and I = [0, t 0 ] for some t 0 > 0. Assume that for any t ∈ I, u ∈ R d there is a complex measure P t,u such that (1) P t,u (x) = e tq(x,u) for all x ∈ R d and (2) 1+|u+v| 2 1+|u| 2 |P t,u |(dv) ≤ 1 + Kt. If X, Y are solutions to the q-martingale problem with the same initial distribution, X(t) and Y (t) have the same distribution for all t ∈ I.
Proof. Observe that condition (2) implies that the total variation measure |P t,u | is finite. Define d(t) := ϕ X (t, ·) − ϕ Y (t, ·) for all t ∈ R + where ϕ X (t, ·) and ϕ Y (t, ·) denote the characteristic functions of X(t) resp. Y (t). Let g t,u (x) := e tq(x,u)+iux . Lemmas 5.7, 5.8 yield for all s, t ∈ I with s < t, where β is a function as in Lemma 5.8 with lim t→0 β(t)/t = 0. Moreover, condition (1) and Fubini's theorem imply E (g t−s,u (X(s))) = e ivX(s) P t−s,u (dv)e iuX(s) dP = ϕ X (s, u + v)P t−s,u (dv) and likewise for Y . We obtain By Lemma 5.5 we have d(t) = 0 for all t ∈ I. Thus the characteristic functions of X(t) and Y (t) coincide, whence they have the same law.
Corollary 5.10. Assume that the requirements of Lemma 5.9 are fulfilled and that existence holds for the q-martingale problem. Then uniqueness holds for the q-martingale problem.
Proof. Let X, Y be solutions with the same initial law and let T ∈ [0, ∞] be maximal such that X(t), Y (t) have the same law for all t ∈ [0, T ). By Proposition 5.1 X and Y are quasi-left continuous, which implies that X(T ) and Y (T ) have the same law. Assume by contradiction that T = ∞. Then X(t) := X(T + t), Y (t) := Y (T + t) are solutions to the q-martingale problem with the same initial law. Lemma 5.9 yields that X, Y have the same one-dimensional distribution up to t 0 and hence X, Y have the same univariate marginals up to T + t 0 . This contradicts the maximality of T . Hence [8,Theorem 4.4.2] yields the claim.
Proof of Theorem 4.2. Theorem 2.7 implies existence and Lemma 5.10 yields uniqueness. The second statement follows from Proposition 2.10.
We now turn to the proof of Theorem 4.4.
By (4), (6) and Proposition B.6 the infinite convolutions P t,u of (P t,u,n ) n∈N and Q t,u of (|P t,u,n |) n∈N exist and we have |P t,u | ≤ Q t,u in the sense that the density is bounded by one. Moreover, e ivx P t,u (dv) = lim l→∞ e ivx (P t,u,1 * · · · * P t,u,l ) (dv) for any x ∈ R d . Hence, P t,u satisfies (1).
By Proposition B.4 the infinite product measure P t,u of (P t,u,n ) n∈N exists. Let π n : and π n π m d|P t,u | = c |P t,u,n |(R d )|P t,u,m |(R d ) v|P t,u,n |(dv) w|P t,u,m |(dw) where we used (3) and (4). We conclude that Similar arguments show that Q t,u (R d ) ≤ 1 and vQ t,u (dv) = 0. Thus we have Recall from Theorem 4.2 that uniqueness holds for symbol q in Lemma 5.11 if it satisfies the requirements of Theorem 2.7. The functions q n will later be chosen from the following lemma.
Lemma 5.12. Let n ∈ R d , a, b ∈ C, k ∈ R such that Re(a) ≥ |b|. Then there is a complex measure P t for any t ∈ [0, 1] such that Moreover, there is a complex measure Q t such that Proof. Let µ be a complex measure on R d with total variation less or equal 1. Moreover, let P t := exp(−ta) exp(tbµ), cf. Appendix A. Then we have for all x ∈ R d P t (x) = exp(−ta) exp(tbμ(x)), |P t | ≤ exp(−tRe(a)) exp(t|b||µ|), for any x ∈ R d , where (5.8) means that the measure on the left is absolutely continuous with density at most one relative to the measure on the right. The last inequality follows from Remark A.4. For the specific choice µ = 1 2 (δ nk + δ −nk ), Lemma A.2 yields v|P t |(dv) = 0, which shows the first assertion. For the specific choice µ = 1 2i (δ nk − δ −nk ), Lemma A.3 implies v|P t |(dv) = 0 and hence the second claim.
1+|u| 2 are bounded. Then q is (f -)Hölder continuous for In particular, if q satisfies the requirements of Theorem 4.4 for some ϕ, then q is Höldercontinuous.
Lemma 5.14 (Fourier conditions). Let q ∈ C (1,0) (R d × R d , C) be a symbol with the following properties.
(1) q satisfies the requirements of Theorem 2.7.
(2) There is a constant c > 0 such that |q(x, u)| + |∇ 1 q(x, u)| ≤ c(1 + |u| 2 ) for all x, u ∈ R d . (3) It has Fourier series representation, i.e. there are a n (u), b n (u) ∈ C for all n ∈ Z d and a constant k > 0 such that (a n (u) cos(knx) + b n (u) sin(knx)) (5.9) and the family (a n , b n ) n∈Z d satisfies: Then existence and uniqueness holds for the q-martingale problem.
Proof. Lemma 5.13 states that q is Hölder continuous.
The Fourier conditions in Lemma 5.14 might seem hard to verify. However, ellipticity and Fourier ellipticity are almost equivalent as can be seen from the proof of the next lemma.
Lemma 5.15. Let q be a continuous symbol such that q(·, u) ∈ C [d/2]+3 (R d , C) for all u ∈ R d and such that (1) q satisfies the requirements of the existence theorem 2.7 and (2) for every x 0 there is a neighbourhood V of x 0 and L < ∞ such that ψ := q(x 0 , ·) satisfies for all x ∈ V, u ∈ R d , β, α ∈ N d with |β| ≤ [d/2] + 1 and |α| ≤ [d/2] + 3. Then existence and uniqueness hold for the q-martingale problem. Proof.
Moreover, q (·, u) satisfies the requirements of Theorem 2.7 and, together with Steps 2, 3, those of Lemma 5.14. The localisation theorem 5.4 yields that existence and uniqueness holds for the q-martingale problem. The total variation of µ is defined by µ := |µ|(R d ). The product measure of complex measures µ, ν on R d resp. R n is the complex measure µ ⊗ ν on R d × R n given by The convolution of complex measures µ, ν on R d is the complex measure µ * ν on R d defined by Complex measures µ, ν on R d are called orthogonal if there is A ∈ B(R d ) such that µ(B) = 0 for any Borel set B ⊂ A and ν(C) = 0 for any Borel set C ⊂ R d \A. The Dirac measure concentrated in a ∈ R d is denoted by δ a . The Fourier transform of a complex measure µ on R d is the functionμ : R d → C given bŷ µ(u) := e i u,x µ(dx).
Let us recall several properties of complex measures, which can be found or easily derived from results in [6].
Lemma A.1. Let µ, ν be complex measures on R d and η a complex measure on R n . Then the following statements hold.
(4) · is a complete norm on C(R d ).
For an introduction to analytic functional calculus see e.g. [21, Definition 3.3.1]. For a complex measure µ and a function f : C → C which is holomorphic on a neighbourhood of the spectrum of µ we write f (µ) for the complex measure obtained by the analytic functional calculus applied to µ and f . Lemma A.2. Let µ be a complex measure on R d . If µ is symmetric or anti-symmetric, then |µ| is symmetric. If f : C → C is holomorphic on a neighbourhood of the spectrum of µ and µ is symmetric, then f (µ) is symmetric. If f is an odd entire function and µ is anti-symmetric, then f (µ) is anti-symmetric. If f is an even entire function and µ is anti-symmetric, then f (µ) is symmetric.
By Lemma A.1(15) α z := R(z, µ) − R(−z, µ) is symmetric. Similar arguments yield that β z := R(z, µ) + R(−z, µ) is anti-symmetric and we obviously have R(z, µ) = 1/2(α z + β z ). Let Γ be a symmetric path around the spectrum of µ. Then (A.14) Observe that the first summand is symmetric and the second summand is anti-symmetric. If f is even, then the first summand vanishes and hence f (µ) is symmetric. If f is odd, then the second summand vanishes and hence f (µ) is anti-symmetric.
Lemma A.2 yields that | exp(µ)| is the sum of symmetric measures and hence symmetric.

APPENDIX B. INFINITE PRODUCT MEASURES AND CONVOLUTIONS
In this section we recall the definition and properties for infinite product measures and infinite convolution for complex Borel measures on R d . Denote by B(R d ) N the σ-algebra on (R d ) N which is generated by the mappings It is also generated by the algebra R := n∈N σ(π 1 , . . . , π n ).
If it exists, the infinite product measure is unique because it is unique on R. It is denoted by ⊗ ∞ n=1 µ n . The following existence statement is classical. Proposition B.2. Let (P n ) n∈N be a sequence of probability measures on R d . Then there is a unique probability measure P on ((R d ) N , B(R d ) N ) such that (π n ) n∈N is a sequence of independent random variables with P πn = P n .
This can be easily lifted to finite measures as long as the product of their total mass converges to a finite non-zero number.
Lemma B.3. Let (µ n ) n∈N be a sequence of finite Borel measures on R d and define a n := µ n (R d ) for any n ∈ N. Assume that a := Π ∞ n=1 a n ∈ (0, ∞). Then the infinite product measure of (µ n ) n∈N exists.
Proof. Define P n := µ n /a n . Then (P n ) n∈N satisfies the requirements of Proposition B.2 and, hence, there is a probability measure P as in Proposition B.2. The measure µ := aP has the required property.