Existence of (Markovian) solutions to martingale problems associated with L\'evy-type operators

Let $A$ be a pseudo-differential operator with symbol $q(x,\xi)$. In this paper we derive sufficient conditions which ensure the existence of a solution to the $(A,C_c^{\infty}(\mathbb{R}^d))$-martingale problem. If the symbol $q$ depends continuously on the space variable $x$, then the existence of solutions is well understood, and therefore the focus lies on martingale problems for pseudo-differential operators with discontinuous coefficients. We prove an existence result which allows us, in particular, to obtain new insights on the existence of weak solutions to a class of L\'evy-driven SDEs with Borel measurable coefficients and on the the existence of stable-like processes with discontinuous coefficients. Moreover, we establish a Markovian selection theorem which shows that - under mild assumptions - the $(A,C_c^{\infty}(\mathbb{R}^d))$-martingale problem gives rise to a strong Markov process. The result applies, in particular, to L\'evy-driven SDEs. We illustrate the Markovian selection theorem with applications in the theory of non-local operators and equations; in particular, we establish under weak regularity assumptions a Harnack inequality for non-local operators of variable order.

In this paper, we are interested in the martingale problem associated with the Lévy-type operator, i. e. for a given initial distribution µ we study probability measures P µ on the Skorohod space D[0, ∞) such that the canonical process (X t ) t≥0 satisfies P µ (X 0 ∈ ⋅) = µ and Af (X s ) ds, t ≥ 0, is a P µ -martingale for any f ∈ C ∞ c (R d ); as usual we set P x ∶= P δx . If the martingale problem is wellposed (i. e. P µ is unique for any initial distribution µ), then this gives a lot of additional information on the stochastic process; for instance, well-posedness of the martingale problem implies the Markov property of (X t ) t≥0 , see e. g. [8,Theorem 4.4.2], and under some weak additional assumptions (X t ) t≥0 is a Feller process, cf. [21]. It is, however, in general difficult to prove the well-posedness of the martingale problem (see e. g. [5,23] for a survey on known results), and for many interesting examples it is known that well-posedness does not hold. It is therefore of great interest to study properties of solutions to martingale problems which are not necessarily well-posed. This paper has three parts.
Firstly, we are interested in finding sufficient conditions on the operator A (or its symbol q) which ensure the existence of a solution to the (A, C ∞ c (R d ))-martingale problem. If q has continuous coefficients, i. e. x ↦ q(x, ξ) is continuous for all ξ, then the existence of solutions is well understood, cf. [10,Theorem 3.2] and [21,Corollary 3.2]. The situation is more delicate if q is discontinuous, and we are not aware of a general existence result in the discontinuous setting (see Section 2 for a detailed discussion of known results). In this paper, we will show that a solution to the martingale problem for a Lévy-type operator A (with possibly discontinuous coefficients) exists if A can be approximated by a sequence of Lévy-type operators A n , n ≥ 1, satisfying a Krylov estimate, cf. Theorem 2.5 for the precise statement. Combining the result with heat kernel estimates obtained in [23], we obtain a new existence result for weak solutions to Lévy-driven SDEs with Borel measurable coefficients, cf. Corollary 2.10. Moreover, Theorem 2.5 allows us to prove the existence of stable-like processes with discontinuous coefficients, cf. Example 2.11 and Example 2.12.
Secondly, we will study under which assumptions a solution to the martingale problem gives rise to a (strong) Markov process. More precisely, we will investigate the following question: Assuming that for each initial distribution µ there exists a solution to the (A, C ∞ c (R d ))-martingale problem with initial distribution µ, i. e. Π µ ∶= {P; P is a solution to the martingale problem with initial distribution µ} ≠ ∅, then under which assumptions can we choose P x ∈ Π δx such that (X t , P x ; x ∈ R d , t ≥ 0) is a strong Markov process? Krylov [18] proved an abstract criterion for the existence of a Markovian selection for a large class of operators A (which need not be Lévy-type operators) and applied it to establish a Markovian selection theorem for diffusions (i. e. A is a local Lévy-type operator, ν = 0). Krylov's criterion has been refined by Ethier & Kurtz, cf. [8,Section 4.5]; roughly speaking, they show that, for "nice" operators A, a certain compact containment condition implies the existence of a Markovian selection. The result is the key tool to prove a Markovian selection theorem for Lévy-type operators; in particular, we obtain the following statement, which seems to be new.
Theorem Let A be a Lévy-type operator with symbol q. If x ↦ q(x, ξ) is continuous for all ξ ∈ R d , q is locally bounded, i. e.
q(x, ξ) < ∞, and q is locally uniformly continuous at ξ = 0 lim R→∞ sup y ≤R sup ξ ≤R −1 q(y, ξ) = 0, then there exists a conservative strong Markov process (X t , F t , P x ; x ∈ R d , t ≥ 0) such that P x is, for each x ∈ R d , a solution to the (A, C ∞ c (R d ))-martingale problem with initial distribution µ = δ x .
If the symbol q does not have continuous coefficients, we have to assume additionally the existence of a solution to the martingale problem for any initial distribution µ, cf. Theorem 3.1 for details. As a by-product, we obtain a sufficient condition for the existence of a Markovian (weak) solution to Lévy-driven SDEs, cf. Corollary 3.4.
In the third, and final, part of the paper, we will illustrate the well-established fact that there is a strong connection between probability theory and the analysis of PDEs and pseudo-differential operators. We will present two applications of Markovian selection theorems in the theory of non-local operators and equations. The first one is a Harnack inequality for a class of pseudodifferential operators, cf. Section 4.1, and the second one concerns viscosity solutions to a certain integro-differential equation, cf. Section 4.2.

Preliminaries
We consider R d endowed with the Borel σ-algebra B(R d ) and write B(x, r) for the open ball centered at x ∈ R d with radius r > 0; R d ∂ is the one-point compactification of R d . The transpose of a matrix A ∈ R d×d is denoted by A T . If a certain statement holds for x ∈ R d with x sufficiently large, we write "for x ≫ 1". We denote by is the space of continuous functions which vanish at infinity (resp. are bounded) , the space of two times continuously differentiable functions which are bounded (with its derivatives), we define a norm by here ∇f and ∇ 2 f are the gradient and Hessian of f , respectively. We write for the Hölder norm of a function f . The space of bounded Borel measurable functions is the σ-algebra associated with τ . For a probability measure P on (Ω, A) and a bounded Ameasurable random variable Y we denote by the expectation with respect to P; we write E P if we need to emphasize the underlying probability measure P.
If µ ∈ P(R d ) is a probability measure and (A, D) a linear operator with domain D ⊆ B b (R d ), then we say that a probability measure P µ on Ω = D[0, ∞) is a solution to the (A, D)-martingale problem with initial distribution µ if P µ (X 0 ∈ ⋅) = µ(⋅) and is a P µ -martingale with respect to the canonical filtration (F t ) t≥0 for all u ∈ D. Note that our definition entails, in particular, that (X t ) t≥0 does P µ -almost surely not explode in finite time, i. e. we consider only conservative solutions to the martingale problem. We write Π µ for the family of solutions to the (A, D)-martingale problem with initial distribution µ. If µ = δ x is a Dirac distribution, then we use the shorthand P x ∶= P δx and Π x ∶= Π δx . The (A, D)-martingale problem is well-posed if for any µ ∈ P(R d ) there exists a unique solution to the (A, D)-martingale problem with initial distribution µ. For a comprehensive study of martingale problems see [8,Chapter 4].
In this paper we are interested in martingale problems associated with pseudo-differential operators (also called Lévy-type operators), that is, operators of the form is the symbol of the pseudo-differential operator A. For each fixed x ∈ R d , q(x, ⋅) is a continuous negative definite function and (b(x), Q(x), ν(x, dy)) is a Lévy triplet, i. e. b(x) ∈ R d , Q(x) ∈ R d×d is a symmetric positive semidefinite matrix and ν(x, dy) is a σ-finite measure on satisfying ∫ y≠0 min{1, y 2 } ν(x, dy) < ∞. We call (b, Q, ν) the characteristics of q. Using properties of the Fourier transform, it is not difficult to see that (1) is equivalent to here tr(Q(x) ⋅ ∇ 2 f (x)) denotes the trace of the matrix Q(x) ⋅ ∇ 2 f (x). Throughout this paper, we will always assume that q(x, 0) = 0 for all x ∈ R d and that (3) holds for K = R d , then q has bounded coefficients. We say that q has continuous coefficients if x ↦ q(x, ξ) is continuous for all ξ ∈ R d , see [22,Theorem A.1] for a characterization in terms of the characteristics (b, Q, ν). Our standard references for martingale problems associated with pseudo-differential operators is the monograph [15], see also [11] and the references therein.
There is a close connection between Feller processes and martingale problems for pseudodifferential operators, cf. [5,21] for a detailed discussion. If the symbol q is of the form for the characteristic exponent ψ of some Lévy process (L t ) t≥0 , it is known that a solution to the (−q(x, D), C ∞ c (R d ))-martingale problem gives rise to a weak solution to the Lévy-driven SDE (4) and vice versa, cf. [28].

Existence of solutions to martingale problems with discontinuous coefficients
Let (q(x, ⋅)) x∈R d be a family of continuous negative definite functions represented by (2) such that q(x, 0) = 0 for all x ∈ R d . If x ↦ q(x, ξ) is continuous, then there are general existence results for solutions to the (−q(x, D), C ∞ c (R d ))-martingale problem. The key tool is the following statement, cf. [8,Theorem 4.5.4].

Theorem Let
be a linear operator such that D(A) ⊆ C ∞ (R d ), and let µ ∈ P(R d ) be a probability measure. If A satisfies the positive maximum principle and D(A) is dense in C ∞ (R d ), then there exists an R d ∂ -valued solution to the (A, D(A))-martingale problem with initial distribution µ.
If A is a pseudo-differential operator with symbol q and domain D( in Theorem 2.1 means, in particular, that x ↦ q(x, ξ) has to be continuous, i. e. Theorem 2.1 allows us only to derive existence results for martingale problems with continuous coefficients. Hoh [11,Theorem 3.15] used Theorem 2.1 to establish the existence of solutions to the (A, C ∞ c (R d ))-martingale problem under the assumption that x ↦ q(x, ξ) is continuous and q has bounded coefficients, i. e. q(x, ξ) ≤ c(1 + ξ 2 ), x, ξ ∈ R d , for some absolute constant c > 0. The following refinement has recently been obtained in [21, Corollary 3.2].

2.2.
Theorem Let A be a pseudo-differential operator with symbol q, q(x, 0) = 0. If q has continuous coefficients, is locally bounded and satisfies the linear growth condition then there exists for any µ ∈ P(R d ) a (non-explosive) solution to the (A, C ∞ c (R d ))-martingale problem with initial distribution µ.
Let us mention that the growth condition (5) can be formulated in terms of the characteristics (b, Q, ν) of q, cf. [21,Lemma 3.1].
For martingale problems with discontinuous coefficients we are not aware of general statements on the existence of solutions. The publication [13] is concerned with such an existence result but, unfortunately, there seems to be a doubt about its proof. For the particular case that the symbol q of the pseudo-differential operator A is of the form x, ξ ∈ R d for the characteristic exponent ψ of a Lévy process (L t ) t≥0 , it is known that solving the (A, C ∞ c (R d )) is equivalent to studying weak solutions to the SDE There are, however, only few results on the existence of weak solutions to SDEs with discontinuous coefficients b, σ, and they are mostly restricted to SDEs driven by isotropic α-stable Lévy processes. Kurenok [27] used a timechange method to study SDEs of the form (6) driven by a one-dimensional isotropic α-stable Lévy process, α ∈ [1,2], and for Borel measurable coefficients b, σ. For the particular case that there is no drift part (i. e. b ∶= 0) and (L t ) t≥0 is a one-dimensional isotropic Lévy process, Zanzotto [32] obtained an Engelbert-Schmidt-type result which gives a necessary and sufficient condition for the existence of the weak solution. Moreover, a result by Kurenok [25] states that the SDE In this section we will be derive a new existence result for martingale problems with discontinuous coefficients, cf. Theorem 2.5. This will allow us to establish a new existence result for Lévy-driven SDEs with discontinuous coefficients, see Corollary 2.10. As usual we denote by (X t ) t≥0 the canonical process on Ω ∶= D[0, ∞). We start with the following, rather simple observation.

Proposition Let
)martingale problem and (X t ) t≥0 admits a transition density p with respect to Lebesgue measure, Roughly speaking, the process (X t ) t≥0 does not "see" Lebesgue null sets (since it has a transition density with respect to Lebesgue measure), and therefore we can modify Af on a Lebesgue null set.

Proof of Proposition 2.3. For any
Using that (X t ) t≥0 is a solution to the (A, C ∞ c (R d ))-martingale problem which has the Markov property we find 3 is a useful tool to derive existence results for the particular case that the symbol q is "nice" up to a null set.

2.4.
Example (Isotropic stable-like process) Let α ∶ R d → (0, 2] be a Hölder continuous mapping which is bounded away from from 0. If β ∶ R d → (0, 2] satisfies α = β Lebesgue-almost everywhere, then there exists a Feller process which solves the martingale problem for the pseudo-differential Proof. It is known that there exists a Feller process with symbol q(x, ξ) = ξ α(x) and that the process admits a transition density, cf. [20] or [23]. As for Lebesgue almost all x ∈ R d , we have Af = Lf almost everywhere; applying Proposition 2.3 finishes the proof.
Let us remark that Example 2.4 works in a similar fashion for other stable-like processes, for instance relativistic stable-like processes or Lamperti stable-like processes, cf. [23].
The main result in this section is the following existence result. Recall that (X t ) t≥0 denotes the canonical process.

2.5.
Theorem Let A n , n ≥ 1, be a pseudo-differential operator with symbol q n such that q n (x, 0) = 0. For fixed µ ∈ P(R d ) let P n , n ≥ 1, be a solution to the (A n , C ∞ c (R d ))-martingale problem with initial distribution µ. Assume that the following assumptions are satisfied.
We will construct the solution P as the weak limit of (a subsequence of) (P n ) n∈N ; (C1) and (C2) give tightness of (P n ) n∈N whereas (C3) and (8) are used to show that the weak limit P is indeed a solution to the (L, This condition is automatically satisfied if m is a finite measure; indeed, if m is finite, [30,Lemma 6.2], the boundedness condition (C1) is equivalent to (iii) Condition (C3) implies, by the Radon-Nikodým theorem, that the distribution P n (X t ∈ ⋅) is absolutely continuous with respect to m for Lebesgue almost every t > 0.
(iv) We will see in the proof of Theorem 2.5 that the solution P satisfies the Krylov estimate In particular, P(X t ∈ ⋅) is absolutely continuous with respect to m for Lebesgue almost all t > 0. (v) Inequality (7) is automatically satisfied for functions u ≥ 0 such that u L p (m) = ∞.
The local finiteness of m ensures that L p (m) is sufficiently rich (in particular, For the proof of Theorem 2.5 we need some auxiliary statements.

2.7.
Lemma Let A be a pseudo-differential operator with symbol q, q(x, 0) = 0, and characteristics Moreover, there exist absolute constants C 1 , C 2 > 0 (not depending on R and f ) such that and On the other hand, we have for and combining the estimates gives (9). Since  (10).
The following maximal inequality is a crucial tool for the proof of Theorem 2.5 but also for the proof of the Markovian selection theorem in Section 3.

Proposition Let
be a pseudo-differential operator with symbol q, q(x, 0) = 0, and let P µ be a solution to the (A, C ∞ c (R d ))-martingale problem with initial distribution µ. Then there exists an absolute constant c > 0 (not depending on µ or P µ ) such that for any t > 0 and R ≥ 2r > 0.
For Feller processes (X t ) t≥0 the maximal inequality goes back to Schilling [29] (see also [5,Theorem 5.1]), and has been refined in [23]. A localized maximal inequality was derived in [24], and [22] gives a maximal inequality for solutions to martingale problems.
Proof of Proposition 2.8. The reasoning is similar to the proof of [5, Theorem 5.1] (see also [22, Lemma 3.1]) but for the readers' convenience we sketch the idea of the proof.
then it follows from the optional stopping theorem that Using (12) and exactly the same reasoning as in [5, Proof of Theorem 5.1], we get From Proposition 2.8 we can deduce the following statement on the tightness of a sequence of solutions to martingale problems.
2.9. Corollary For k ≥ 1 let (q k (x, ⋅)) x∈R d be a family of continuous negative definite mappings with for some absolute constant c > 0. By (14) this implies that the compact containment condition lim R→∞ sup k≥1 P k sup t≤T X t ≥ R = 0 holds for any T > 0. Moreover, it is not difficult to see that (14) gives and therefore we find from Lemma 2.7 that sup k≥1 − q k (x, D)f ∞ < ∞. Now the assertion follows from Aldous tightness condition, cf. [15,Theorem 4.1.16].
We are now ready to prove Theorem 2.5.
Proof of Theorem 2.5. It follows from (C1),(C2) and Corollary 2.9 that the sequence (P n ) n≥1 is tight, and therefore the weak limit P = lim k→∞ P n k exists for a suitable subsequence. It remains to prove that P is a solution to the (L, C ∞ c (R d ))-martingale problem. We claim that P satisfies the Krylov estimate Indeed: If u = 1 A for some open set A ⊆ R d , then this is a direct consequence of the Portmanteau theorem, Fatous lemma and (C3); for general u ≥ 0 the Krylov estimate then follows from a straight-forward application of the monotone class theorem.
In order to show that P is a solution to the (L, C ∞ c (R d ))-martingale problem, it suffices to prove that We estimate the terms separately. It follows from (15) and (16) that Since g is continuous, the weak convergence of P n k to P gives Using that P n k solves the (A n k , C ∞ c (R d ))-martingale problem we obtain Thus, by (C3) and (16), Let us illustrate Theorem 2.5 with some examples. We obtain the following existence result for solutions to SDEs with not necessarily continuous coefficients.

2.10.
Corollary Let (L t ) t≥0 be a one-dimensional Lévy process with characteristic exponent ψ satisfying the following assumptions.
Corollary 2.10 applies, for instance, if (L t ) t≥0 is isotropic stable, relativistic stable, Lamperti stable or a truncated Lévy process; see [23,  Proof of Corollary 2.10. Let (L t ) t≥0 be a Lévy process satisfying (L1)-(L3). We split the proof in two parts; in the first part we will derive a Krylov estimate for SDEs with Hölder continuous coefficients, and in the second part we will approximate the coefficients b, σ by Hölder continuous functions in order to apply Theorem 2.5.
Step 1: Let f, g be bounded Hölder continuous functions such that inf x g(x) > 0. In [23] (see also [20]) it was shown that there exists a Feller process (X t , F t , P x ; x ∈ R d , t ≥ 0) which is the unique weak solution to the SDE (19) dX The Feller process (X t ) t≥0 has a continuous transition probability p t (x, y). Using the heat kernel estimates from [23] we find that there exists a continuous function C such that where (f ) and (g) denote the Hölder exponent of f and g, respectively, and see the appendix for details. Since the transition probability p is continuous, it is not difficult to see that x ↦ P x (A) is measurable for any A ∈ F ∞ , and therefore P µ ∶= ∫ P x µ(dx) defines a probability measure; it is a weak solution to (19) with initial distribution µ ∈ P(R d ), and for any B ∈ B(R d ), s > 0. Consequently, we obtain from Fubini's theorem and (20) for any function u ≥ 0, u ∈ B b (R d ), i. e. a Krylov estimate holds for p = 1 and the measure Note that m is a finite measure since, by Tonelli's theorem and the invariance of Lebesgue measure under translations, Step 2: Let b and σ be as in Corollary 2.10. By Lemma A.1, we can choose sequences (f n ) n∈N , (g n ) n∈N ⊆ C(R) and (α n ) n∈N , (β n ) n∈N ⊆ (0, 1] such that sup and n→∞ → σ(x) Lebesgue almost everywhere. If we denote by A n the pseudo-differential operator with symbol q n (x, ξ) ∶= −if n (x)ξ + ψ(g n (x)ξ), then Step 1 shows that there exists for each n ∈ N a solution P n to the (A n , C ∞ c (R d )) martingale problem with initial distribution µ which satisfies the Krylov estimate t 0 E Pn u(X s ) ds ≤ C(T, f n αn , g n βn , 1 inf for C and m defined in Step 1. Because of (22) and the continuity of C we can choose a constant This shows that (C3) in Theorem 2.5 holds for p = 1 and the finite measure m. Moreover, it can be easily verified that (22) gives (C1), (C2). If we denote by L the pseudo-differential operator with symbol q(x, ξ) ∶= −ib(x)ξ + ψ(σ(x)ξ), then q n (x, ξ) → q(x, ξ) for Lebesgue-almost all x ∈ R, and so lim sup Since m is a finite measure, we know that Applying Theorem 2.5 we find that there exists a solution to the (L, C ∞ c (R))-martingale problem with initial distribution µ. It is known that the solution is a weak solution to (18), see [28]. The absolute continuity of the distribution follows from Remark 2.6(iii).
Using the heat kernel estimates in [23, Section 5.3] and an approximation procedure as in the proof of Corollary 2.10 we can use Theorem 2.5 to derive results on mixed processes and stable-like processes.

Example
For a Borel measurable function ϕ ∶ R d → J denote by A the pseudo-differential operator with symbol (24) q(x, ξ) ∶= Then there exists for any µ ∈ P(R d ) a solution to the (A, C ∞ c (R d ))-martingale problem with initial distribution µ.

Remark It follows from the well-known identity
that we can write the symbol (24) in the form

Markovian solutions to martingale problems for Lévy-type operators
Throughout this section, we denote by (X t ) t≥0 the canonical process on Ω = D[0, ∞), and (q(x, ⋅)) x∈R d is a family of continuous negative definite functions such that q(x, 0) = 0.
The aim of this section is to establish a condition which ensures the existence of a Markovian solution to the (−q(x, D), C ∞ c (R d ))-martingale problem. It is well-known, see e. g. [8], that the Markov property holds if the martingale problem is well-posed. It is, however, in general hard to verify the well-posedness of a martingale-problem. Our main result in this section, Theorem 3.1, states that a Markovian selection exists if the symbol q satisfies a certain continuity condition at ξ = 0.

Theorem (Markovian selection theorem) Let
A be a pseudo-differential operator with symbol q, q(x, 0) = 0, such that for any µ ∈ P(R d ) there exists a solution to the (A, C ∞ c (R d ))-martingale problem with initial distribution µ. If q is locally bounded and satisfies (25) lim then there exists a strongly Markovian solution (X t , F t , P x ; x ∈ R d , t ≥ 0) to the (A, C ∞ c (R d ))martingale problem, i. e. there exists a family of probability measures is a solution to the (A, C ∞ c (R d ))-martingale problem with initial distribution µ. Moreover, the following statement holds true: (iv) For any fixed f ∈ C ∞ (R d ), f ≥ 0, and λ > 0, the family (P x ) x∈R d can be chosen in such a way that where Π x is the family of all probability measures P solving the (A, C ∞ c (R d ))-martingale problem with initial distribution δ x .
If q has continuous coefficients, then the assumption on the existence of a solution is automatically satisfied (cf. Corollary 3.2). For symbols q with discontinuous coefficients we refer to Section 2 for sufficient conditions ensuring the existence.
We will see in Section 4.2 that the representation (26) is useful in order to study properties of the function which can be understood as the resolvent with respect to the sublinear expectation E Q x ∶= sup P∈Πx E P .

3.2.
Corollary (Markovian selection for symbols with continuous coefficients) Let A be a pseudodifferential operator with symbol q, q(x, 0) = 0. If q is locally bounded, has continuous coefficients and lim We will first prove Theorem 3.1 and Corollary 3.2, and then we will present some examples illustrating both results. The following result is compiled from Ethier & Kurtz [8]; it is the key tool for the proof of Theorem 3.1.

Theorem Let
be a linear operator, and denote by Π µ the family of solutions to the (A, C ∞ c (R d ))-martingale problem with initial distribution µ. If Π µ ≠ ∅ for any initial distribution µ and if the compact containment condition is a strong Markov process and x ↦ P x is measurable. For any fixed f ∈ C b (R d ), f ≥ 0, and λ > 0 the Markovian selection (P x ) x∈R d can be chosen in such a way that Proof. The first statement follows from Theorem 4.5.19 and (the proof of) Lemma 4.5.11(b) in [8]. Let us remark that Ethier & Kurtz assume in Lemma 4.5.11(b) that Af is continuous; a close look at the proof shows, however, that this condition is not needed. The existence of a Markovian selection satisfying (28)  Proof of Theorem 3.1. Since q is locally bounded and satisfies (25), Lemma 2.7 shows that Af is bounded for any f ∈ C ∞ c (R d ). Moreover, it follows from (25) and Proposition 2.8 that the compact containment condition (27) is satisfied. Applying Theorem 3.3 finishes the proof.
Proof of Corollary 3.2. The assertion is a direct consequence of Theorem 2.2 and Theorem 3.1.
Let us illustrate the Markovian selection theorems with some examples. Since there is a close connection between weak solutions to Lévy-driven SDEs and martingale problems, Theorem 3.1 allows us to deduce the following statement.
3.4. Corollary (Markovian selection for Lévy-driven SDEs) Let (L t ) t≥0 be a d-dimensional Lévy process with characteristic exponent ψ. If b, σ ∶ R k → R k×d are functions of sublinear growth such that the Lévy-driven SDE has a weak solution for any µ ∈ P(R d ), then there exists a conservative strong Markov process (X t , F t , P x ; x ∈ R d , t ≥ 0) such that (X t ) t≥0 is a weak solution to (29) with respect to P µ ∶= ∫ P x µ(dx) for any µ ∈ P(R d ).
The assumption on the existence of a weak solution to (29) is, in particular, satisfied if b and σ are continuous. For Lévy-driven SDEs with discontinuous coefficients we refer to Theorem 2.6 for a sufficient condition for the existence. Note that (29) covers SDEs of the form where (J t ) t≥0 is a pure jump Lévy process and (B t ) t≥0 an independent Brownian motion; simply choose L t = (B t , J t ) in (29). Let us mention that Anulova & Pragarauskas [1] proved a Markovian selection theorem for SDEs (30) for the particular case that f is uniformly elliptic.
Proof of Corollary 3.4. Set q(x, ξ) ∶= −ib(x) ⋅ ξ + ψ(σ(x) T ⋅ ξ), x, ξ ∈ R k , and denote by A the pseudo-differential operator with symbol q. Since b and σ are of sublinear growth, it follows easily that q satisfies the assumptions of Theorem 3.1, and therefore there exists a conservative strongly Markovian solution to the (A, C ∞ c (R d ))-martingale problem. It is known, see e. g. [28], that any solution to the (A, C ∞ c (R d ))-martingale problem with initial distribution µ is a weak solution to (29); this finishes the proof.
If there exist finite constants c 1 , c 2 > 0 and α, β ∈ (0, 2) such that then there exists a conservative strongly Markovian solution to the martingale problem for the pseudo-differential operator with symbol We will see in Section 4.1 that Corollary 3.5 can be used to establish a Harnack inequality. Corollary 3.5 applies, in particular, to stable-like processes. If we choose, for instance, κ(x, y) = y −d−α(x) for a continuous mapping α ∶ R d → (0, 2) satisfying inf x α(x) > 0, we find that there exists a strongly Markovian solution to the the martingale problem for the pseudo-differential operator with symbol q(x, ξ) = ξ α(x) .
Proof of Corollary 3.5. Using the elementary estimates it is not difficult to see that q has bounded coefficients and satisfies the continuity condition (25). Applying Corollary 3.2 proves the assertion.
The next example shows that the Markovian selection from Theorem 3.1 fails, in general, to be unique. Moreover, it shows that we can, in general, not choose the Markovian selection in such a way that the associated semigroup has nice mapping properties (e. g. the Feller property or the C b -Feller property).
3.6. Example Consider the martingale problem for the pseudo-differential operator A with symbol q(x, ξ) = −2iξ sgn(x) x . Clearly, (the distribution of) a process (X t ) t≥0 is a solution to the (A, C ∞ c (R))-martingale problem with initial distribution µ = δ x if (X t ) t≥0 satisfies the ordinary differential equation (31) dX t = 2 sgn(X t ) X t dt, It is not difficult to check that both are Markovian solutions, and hence uniqueness of a Markovian selection fails. Moreover, we note that (31) has a unique solution for any x ≠ 0, and therefore it follows easily that lim x↓0 E x f (X t ) = f (t 2 ) and lim x↑0 E x f (X t ) = f (−t 2 ) for any selection (X t , P x ) t≥0,x∈R d of solutions to the (A, C ∞ c (R))martingale problem; in particular, the C b -Feller property and the Feller property fail to hold for any Markovian selection.

Applications
In this section we present two applications of Markovian selection theorems in the theory of nonlocal operators and equations. The first one is a Harnack inequality for pseudo-differential operators of variable order, cf. Section 4.1, and the second one concerns viscosity solutions to a certain integro-differential equation, cf. Section 4.2.

Harnack inequality for non-local operators of variable order
Harnack inequalities are an important tool in the study of partial differential equations. In the last years there has been an increasing interest in Harnack inequalities for functions that are harmonic with respect to a Lévy-type operator. Due to the non-local nature of these operators, it is not possible to use the same techniques as for differential operators. It has turned out that the probabilistic approach via martingale problems is very powerful. In order to use this method, it is, however, necessary to know that there exists a strongly Markovian solution to the martingale problem, and many important contributions, e. g. [3,4,16,31], have to assume the existence of a strongly Markovian solution. It is, in general, difficult to prove that the martingale problem is well-posed, and this made it so far difficult to check this assumption. Our Markovian selection theorem, Theorem 3.1, allows us to prove the existence without the much harder well-posedness of the martingale problem.
In this section we combine the Markovian selection theorem with the results from [3] to prove a Harnack inequality for operators of variable order. In [3] Bass and Kaßmann established a Harnack inequality for pseudo-differential operators of the form their result requires only weak assumptions on the kernel κ, but for the proof they have to assume that there exists a strongly Markovian solution to the (A, C ∞ c (R d ))-martingale problem. Thanks to the Markovian selection theorem, we can give mild assumptions which ensure the existence of such a strongly Markovian solution.
We recall the following definition. As usual, (X t ) t≥0 denotes the canonical process with canonical filtration F t ∶= σ(X s ; s ≤ t).

Definition For a linear operator
The following theorem is the main result in this section.
Then the following statements hold for the pseudo-differential operator A with symbol 1 − e iy⋅ξ + iy ⋅ ξ1 (0,1) ( y ) κ(x, y) dy, x, ξ ∈ R d , (i) There exists a conservative strong Markov process (X t , F t , P x ; x ∈ R d , t ≥ 0) such that P x solves for each x ∈ R d the (A, C ∞ c (R d ))-martingale problem with initial distribution δ x . (ii) If β − α < 1 then a Harnack inequality holds; more precisely, if u ∈ B b (R d ), u ≥ 0, is harmonic on an open ball B(x 0 , 2r), then there exists C > 0 such that the constant C depends on r and c 1 , . . . , c 4 , but not x 0 and u.
Theorem 4.2 is a direct consequence of Corollary 3.5 and [3]. Let us point out that the assumptions (H2), (H3) and β − α < 1 are taken from [3] and are thus needed to prove the Harnack inequality. The condition on the large jumps (H1) is slightly stronger than in [3]; this is the price which we have to pay for the existence of a strongly Markovian solution to the martingale problem (more precisely, (H1) is needed to ensure that the continuity condition (25) holds).
Note that the Harnack inequality (32) can be used to study the regularity of harmonic functions, see [16,Section 4].

Viscosity solutions
Viscosity solutions were originally introduced by Lions & Crandall to study non-linear PDEs of the form F (x, u(x), ∇u(x), ∇ 2 u(x)) = 0.
The concept has been successfully to extended to nonlinear non-local equations (33) G(x, u(x), ∇u(x), ∇ 2 u(x), u(⋅)) = 0, and over the last two decades viscosity solutions have turned out to be one of the most important notions for generalized solutions. Non-linear non-local equations (33) appear naturally in the theory of stochastic processes, for instance in the study of Feller processes (cf. [5]) and sublinear Markov processes (see e. g. [12,6] and the references therein). Costantini & Kurtz [7] showed that there is a close connection between martingale problems and viscosity solutions to the integro-differential equation roughly speaking, they showed that a comparison principle for (34) implies the well-posedness of the martingale problem for the operator A. Recently, this result has been used by Zhao [33] to derive the existence of a unique weak solution to the SDE driven by an α-stable Lévy process, α ∈ (0, 1), under weak regularity assumptions on the drift b.
We will use the Markovian selection theorem, Theorem 3.1, to give a sufficient condition which ensures that the function is a viscosity solution to (34); here (X t ) t≥0 is the canonical process and Π x is the family of probability measures on Ω = D[0, ∞) which solve the (A, C ∞ c (R d ))-martingale problem with initial distribution δ x .

Definition Let
be a linear operator, λ > 0 and f ∈ C ∞ (R d ).
(i) An upper semicontinuous bounded function u is a viscosity subsolution to λu − Au = f if the implication subsolution and a viscosity supersolution.
For a pseudo-differential operator A with negative definite symbol q, the assumption A(C 2 b (R d )) ⊆ C b (R d ) in Definition 4.4 means that the symbol q is continuous (with respect to x and ξ) and has bounded coefficients.

4.5.
Theorem Let A be a pseudo-differential operator with symbol q of the form (2), q(x, 0) = 0, and let f ∈ C ∞ (R d ), f ≥ 0. Assume that x ↦ q(x, ξ) is continuous, q has bounded coefficients and here Π x denotes the set of probability measures on D[0, ∞) which are a solution to the (A, C ∞ c (R d ))-martingale problem with initial distribution δ x . (ii) If u is lower semicontinuous, then u is a viscosity solution to λu − Au = f .
If u is lower semicontinuous (hence, by (i), continuous), then a result by Barles et al. [2] shows that -under rather general assumptions -u is Hölder continuous. This, in turn, would allow us to derive new results on the well-posedness of martingale problems using a similar approach as in [33]. It seems, however, that the lower semicontinuity of u is, in general, difficult to check; in fact, Example 3.6 shows that u fails, in general, to be (lower semi)continuous.
Proof of Theorem 4.5. The first statement is a direct consequence of [7, Lemma 3.5] and Proposition 2.8 but we prefer to give a direct proof which gives both (i) and (ii). By Theorem 2.2 we have Π x ≠ ∅ and therefore u(x) ∈ R is well-defined for each x ∈ R d . If (x n ) n∈N ⊆ R d is such that x n → x ∈ R d , then Proposition 2.8 shows that for any P n ∈ Π xn , n ≥ 1, the sequence (P n ) n∈N is tight. It is not difficult to see that this implies that u is upper semicontinuous, see [7,Lemma 3.4] for more details.
is a strong Markov process with respect to the canonical filtration F t ∶= σ(X s ; s ≤ t) and here (and throughout the remaining part of the proof) we use the shorthand E x ∶= E P x . Using a standard approximation procedure and the fact that Af (2) ≤ c f (2) , f ∈ C 2 b (R d ), for some absolute constant c > 0, it follows easily that P x is a solution to the (A, C 2 b (R d ))-martingale problem with initial distribution δ x . Consequently, Aφ(X s ) ds is a P x -martingale, and this implies that cf. [7, Lemma 2.9]. Thus, by (36) and (37), If we define τ x r ∶= τ r ∶= min{t > 0; X t − x > r} ∧ r then we find from the strong Markov property of (X t ) t≥0 that Invoking (36) and (37) we find Now let φ ∈ C 2 b (R d ) and x 0 ∈ R d be such that sup Without loss of generality, we may assume u(x 0 ) = φ(x 0 ). Then u ≤ φ yields u(X τr ) − ϕ(X τr ) ≤ 0, and so By the right-continuity of the sample paths of (X t ) t≥0 we have E x0 (τ r ) > 0 for r > 0 sufficiently small; moreover, trivially E x0 (τ r ) ≤ r < ∞. Dividing both sides of the previous equation by E x0 τ r and letting r → 0 we obtain 0 ≤ f (x 0 ) − λφ(x 0 ) + Aφ(x 0 ). Appendix A.