Regularity of solutions to anisotropic nonlocal equations

We study harmonic functions associated to systems of stochastic differential equations of the form $dX_t^i=A_{i1}(X_{t-})dZ_t^1+\cdots+A_{id}(X_{t-})dZ_t^d$, $i\in\{1,\dots,d\}$, where $Z_t^j$ are independent one-dimensional symmetric stable processes with indices $\alpha_j\in(0,2)$, $j\in\{1,\dots,d\}$. In this article we prove H\"older regularity of bounded harmonic functions with respect to solutions to such systems.


Introduction
The consideration of stochastic processes with jumps and anisotropic behavior is natural and reasonable since such objects arise in several natural and financial models. In certain circumstances Lévy processes with jumps are more suitable to capture empirical facts that diffusion models do. See for instance [CT04] for examples of financial models with jumps.
In the nineteen fifties, De Giorgi [DG57] and Nash [Nas57] independently prove an a-priori Hölder estimate for weak solutions u to second order equations of the form div(A(x)∇u(x)) = 0 for uniformly elliptic and measurable coefficients A. In [Mos61], Moser proves Hölder continuity of weak solutions and gives a proof of an elliptic Harnack inequality for weak solutions to this equation. This article provides a new technique of how to derive an a-priori Hölder estimate from the Harnack inequality. For a large class of local operators, the Hölder continuity can be derived from the Harnack inequality, see for instance [GT01]. For a comprehensive introduction into Harnack inequalities, we refer the reader e.g. to [Kas07].
The corresponding case of operators in non-divergence form is treated in by Krylov and Safonov in [KS79]. The authors develop a technique for proving Hölder regularity and the Harnack inequality for harmonic functions corresponding to non-divergence form elliptic operators. They take a probabilistic point of view and make use of the martingale problem to prove regularity estimates for harmonic functions. The main tool is a support theorem, which gives information about the topological support for solutions to the martingale problem associated to the corresponding operator. This technique is also used in [BL02] to prove similar results for nonlocal operators of the form under suitable assumptions on the function a. In [BC10] Bass and Chen follow the same ideas to prove Hölder regularity for harmonic functions associated to solutions of systems of stochastic differential equations driven by Lévy processes with highly singular Lévy measures. In this work we extend the results obtained by Bass and Chen to a larger class of driving Lévy processes.
Let d ∈ N and d ≥ 2. We assume that Z i t , i = 1, . . . , d, are independent one-dimensional symmetric stable processes with indices α i ∈ (0, 2) and define Z = (Z t ) t≥0 = (Z 1 t , . . . , Z d t ) t≥0 . The Lévy-measure of this process is supported on the coordinate axes and given by Therefore ν(A) = 0 for every set A ⊂ R d , which has an empty intersection with the coordinate axes. The generator L of Z is given for For a deeper discussion on Lévy processes and their generators we refer the reader to [Sat99].
Let x 0 ∈ R d and A : R d → R d×d a matrix-valued function. We consider the system of stochastic differential equations This system has been studied systematically in the case α 1 = α 2 = · · · = α d = α ∈ (0, 2) by Bass and Chen in the articles [BC06] and [BC10]. With the help of the martingale problem, Bass and Chen prove in [BC06] that for each x 0 ∈ R d there exists a unique weak solution (X = (X 1 t , . . . , X d t ) t≥0 , P x0 ) to (1.1). Furthermore the authors prove that the family {X, P x , x ∈ R d } forms a conservative strong Markov process on R d whose semigroup maps bounded continuous functions to bounded continuous functions (see Theorem 1.1, [BC06]). Consequently it follows that with the generator for any weak solution to (1.1), where a j (x) denotes the j th column of the matrix A(x). In [BC10] the authors prove Hölder regularity of harmonic functions with respect to L and give a counter example which shows that the Harnack inequality for harmonic functions is not satisfied. In [KR17] the authors study (1.1) in the case of diagonal matrices A and prove sharp two-sided estimates of the corresponding transition density p A (t, x, y) and prove Hölder and gradient estimates for the function x → p A (t, x, y).
In this paper we do not study unique solvability of (1.1) but prove an a-priori regularity estimate for harmonic functions if unique solutions to the system exist. The following assumptions will be needed throughout the paper.

Assumption.
(i) For every x ∈ R d the matrix A(x) is non-degenerate, that is det(A(x)) = 0.
(ii) The functions x → A ij (x) and x → A −1 ij (x) are continuous and bounded for all 1 ≤ i, j ≤ d and x ∈ R d .
(iii) For any x 0 ∈ R d , there exists a unique solution to the martingale problem for with the generator for the weak solution to (1.1).
For a comprehensive introduction into the martingale problem we refer the reader to [EK86].
Notation. Let A be the matrix-valued function from (1.1). Let D be a Borel set. Throughout the paper ̟(D) denotes the modulus of continuity of A and we write Λ(D) for the upper bound of A on D. We set α min := min{α 1 , . . . , α d } and α max := max{α 1 , . . . , α d }. For i ∈ N we write c i for positive constants and additionally c i = c i (·) if we want to highlight all the quantities the constant depends on.
In order to deal with the anisotropy of the process we consider a corresponding scale of cubes.
Note that M k r is increasing in k and r. For z ∈ R d and r ∈ (0, 1], the set M r (z) is a ball with radius r and center z in the metric space This metric is useful for local considerations only, that is studies of balls with radii less or equal than one. The advantage of using these sets is the fact that they reflect the different jump intensities of the process Z and compensate them in an appropriate way, see for instance Proposition 2.4. The purpose of this paper is to prove the following result. r (x 0 ))) > 0, independent of h and r, such that We want to emphasize, that in the case α 1 = · · · = α d the set M r (x 0 ) reduces to a cube with radius r and hence this result coincides with [BC10, Theorem 2.9], when one chooses cubes instead of balls.
Structure of the article. This article is organized as follows. In Section 2 we provide definitions and auxiliary results. We constitute sufficient preparation and study the behavior of the solution to the system. In Section 3 we study the topological support of the solution to the martingale problem associated to the system of stochastic differential equations. The aim of this section is to prove a Krylov-Safonov type support theorem. Section 4 contains the proof of Theorem 1.2.
Acknowledgement. This work is part of the author's Master thesis, written under the supervision of Moritz Kassmann at Bielefeld University. Financial support by the German Science Foundation DFG (SFB 701) is gratefully acknowledged.

Definitions and auxiliary results
In this section we provide important definitions and prove auxiliary results associated to the solution of the system (1.1).
Let A τ (x) denote the transpose of the matrix A(x) and (a τ j (x)) −1 the j th row of (A τ (x)) −1 . For a Borel set D, we denote the first entrance time of the process X in D by T D := inf{t ≥ 0 : X t ∈ D} and the first exit time of Let us first recall the definition of harmonicity with respect to a Markov process.
For R = M s (y) we use the notation R = M 3 s (y). The next Proposition is a pure geometrical statement and not related to the system of stochastic differential equations. We skip the proof and refer the reader to [Bas98, Proposition V.7.2], which can be easily adjusted to our case.
Following the ideas of the proof of [BL02, Proposition 2.3], we next prove a Lévy system type formula.
Proposition 2.3. Suppose D and E are two Borel sets with dist(D, E) > 0. Then By Assumption (iii) for each x ∈ R d the probability measure P x is a solution to the martingale problem for L.
Since the stochastic integral with respect to a martingale is itself a martingale, is a P x -martingale. Since X s = X s− for only countably many values of s, Note, that c α k /|h| 1+α k is integrable over h in the complement of any neighborhood of the origin for any k ∈ {1, . . . , d}. Since D and E have a positive distance from each other, the sum in (2.1) is finite. Hence is a P x -martingale, which is equivalent to our assertion.
The next Proposition gives the behavior of the expected first exit time of the solution to (1.1) out of the set M r (·). This Proposition highlights the advantage of M r (·) and shows that the scaling of the cube in the different directions with respect to the jump intensity compensates the different jump intensities in the different directions.
Let j ∈ {1, . . . , d} be fixed but arbitrary. The aim is to show that there exists c 2 > 0 such that Since we reduced the problem to a one-dimensional one, we may suppose by scaling r = 1. Let By Assumption (i), we have κ > 0. There exists a c 3 ∈ (0, 1) with The independence of the one-dimensional processes implies that with probability zero at least two of the Z i 's make a jump at the same time. This leads to Our aim is to show that the probability of the process X for leaving M 1 (x) in the j th coordinate after time m is bounded in the following way

Assertion (2.3) follows by
where we used the fact that the sum on the right hand side is a geometric sum. Thus the assertion follows by (2.2) and (2.3).
We close this section by giving an estimate for leaving a rectangle with a comparatively big jump.
j (x)) −1 |c αj . By Proposition 2.3 and optional stopping we get for Using the monotone convergence on the right and dominated convergence on the left, we have for t → ∞ where c 3 is the constant showing up in the estimate E z (τ Mr (x) ) ≤ c 3 r α of Proposition 2.4.

The support theorem
In this section we prove the main ingredient for the proof of the Hölder regularity estimate for harmonic functions. The so-called support theorem states that sets of positive Lebesgue measure are hit with positive probability.
This theorem was first proved in [KS79] for the diffusion case. In the article [BC10], Bass and Chen prove the support theorem in the context of pure jump processes with singular and anisotropic kernels. They consider the system (1.1) in the case α i = α for all i ∈ {1, . . . , d} and use the Krylov-Safonov technique to prove Hölder regularity with the help of the support theorem. The idea we use to prove the support theorem is similar in spirit to the one in [BC10].
The following Lemma is a statement about the topological support of the law of the stopped process. It gives the existence of a bounded stopping time T such that with positive probability the stopped process stays in a small ball around its starting point up to time T , makes a jump along the k th coordinate axis and stays afterwards in a small ball.

Proof. Let
We assume ξ ∈ [0, r αmax/αmin ]. The case ξ ∈ [−r αmax/αmin , 0] can be proven similar. Let us first suppose ξ ≥ γ/(3 A ∞ ) and let β ∈ (0, ξ), which will be chosen later. We decompose the process Z i t in the following way: Let (X t ) t≥0 be the solution to The continuity of A allows us to find a δ < γ/(6 A ∞ ), such that ∆ Z j s = 0 for all s ≤ t 0 and all j = k , E = Z i s = 0 for all s ≤ t 0 and i = 1, . . . , d . Since A is bounded, we can find c 2 > 0, such that Note, that β ∈ (0, ξ) ⊂ (0, r αmax/αmin ) ⊂ (0, 1). Therefore, we get By Tschebyscheff's inequality and Doob's inequality, we get Choose β ∈ (0, ξ) such that For Z k to have a single jump before time t 0 , and for that jump's size to be in the interval [ξ, ξ + δ], then up to time t 0 Z k t must have (i) no negative jumps, (ii) no jumps whose size lies in [β, ξ), (iii) no jumps whose size lies in (ξ + δ, ∞), (iv) precisely one jump whose size lies in the interval [ξ, ξ + δ].
For all j = k, the probability that Z j does not have a jump before time t 0 , is the probability that a Poisson random variable with parameter 2c 6 t 0 β −αj is equal to 0. Using the indepence of Z j for j = 1, . . . , d, we can find a c 8 > 0 such that P x0 (∆ Z j s = 0 for all s ≤ t 0 and all j = k) ≥ c 8 .
Similary we obtain P x0 (E) ≥ c 10 and P x0 (C ∩ E) ≥ c 11 . (3.5) Let T be the time, when Z k jumps the first time, i.e. Z k makes a jump greater then β. Then Z s− = Z s− for all s ≤ T and hence X s− = X s− for all s ≤ T. So up to time T , X s does not move away more than δ away from its starting point. Note ∆X T = A(X T − )∆Z T . By (3.2), we obtain on C ∩ D Appling the strong Markov property at time T , we get by (3.5) All in all we get by the strong Markov property which proves the assertion. Now suppose ξ < γ/(3 A ∞ ). Then |x 0 − (x 0 + ξv k )| < γ/3. We can choose T ≡ 0 and by (3.5) we get: which finishes the proof.
We need two simple geometrical facts from the field of linear algebra, whose proofs can be found in [BC10] (Lemma 2.4 and Lemma 2.5).
Lemma 3.3. Let v be a vector in R d , u k = Ae k , and p k the projection of v onto u k for k = 1, . . . , d. Then there exists ρ = ρ(Λ(R d )) ∈ (0, 1), such that for some k, For a given time t 1 > 0 the following lemma shows that solutions stay with positive probability in an ε-tube around a given line segment on [0, t 1 ]. The case of α 1 = · · · = α d was considered in [BC10]. We follow their technique.
Theorem 3.5. Let r ∈ (0, 1], x 0 ∈ R d , ε ∈ (0, r αmax/αmin ), t 0 > 0 and x 0 ∈ R d . Let ϕ : [0, t 0 ] → R d be continuous with ϕ(0) = x 0 and the image of ϕ contained in M r (x 0 ). Then there exists c 1 = c 1 (Λ(M 2 r (x 0 )), ̟(M 3 r (x 0 )), ϕ, ε, t 0 ) > 0 such that Proof. Let ε > 0. We define and approximate ϕ within U by a polygonal path. Hence we can assume that ϕ is polygonal by changing ε to ε/2 in the assertion. We subdivide [0, t 0 ] into n subintervals of the same length for n ≥ 2 such that where L denotes the length of the line segment. Let Using Lemma 3.4, there exists a constant c 2 > 0 such that By the strong Markov property at time t 0 /n we get Using the Iteration as in the proof of Lemma 3.4, we get for all k ∈ {1, . . . , d} Hence the assertion follows by We state two corollaries, which follow immediately from Theorem 3.5.
Proof. Follows immediately by Corollary 3.6.
We now prove the main ingredient for the proof of the Hölder regularity. It states, that sets of positive Lebesgue measure are hit with positive probability.
By Proposition 2.2 there exists a set D ⊂ M r (x 0 ) such that we get |E| > q. By the definition of ϕ, we have P x (T E < τ Mr (x0) ) ≥ ϕ(q). We will first show (3.8) P y (T A < τ Mr (x0) ) ≥ ρϕ(q) for all y ∈ E.
Let y ∈ ∂E, then y ∈ R i for some R i ∈ R and dist(y, ∂M r (x 0 )) ≥ (1 − β)r αmaxαmin . Define R * i as the cube with the same center as R i but sidelength half as long. By Corollary 3.7 P y (T R * i < τ Mr (x0) ) ≥ ρ. By Proposition 2.2 (3) for all R i ∈ R |A ∩ R i | ≥ q|R i | and therefore P x0 (T A∩Ri < τ Mr (x0) ) ≥ ϕ(q) for x 0 ∈ R * i .
Using the strong Markov property, we have for all y ∈ E P y (T A < τ Mr (x0) ) ≥ E y P

Proof of Theorem 1.2
In this section we prove our main result.
Let R be a collection of N equal sized rectangles as in Definition 1.1 with disjoint interiors and R ⊂ S. Moreover let R to be a covering of S ′ . For at least one rectangle Q ∈ R Let Q ′ be the rectangle with the same center as Q but each sidelength half as long. By Corollary 3.6 there exists a c 2 > 0 such that (4.1) P x (T Q ′ < τ S ) ≥ c 2 , x ∈ M 1/2 s (y). Using Theorem 3.8 and the strong Markov property there exists a constant c 3 > 0 with (4.2) P x (T A < τ S ) ≥ c 3 , x ∈ M 1/2 s (y). Let R ≥ 2r. By Proposition 2.5 there exists a c 4 > 0 such that for all z ∈ M r (x 0 ). By linearity it suffices to suppose 0 ≤ h ≤ M on R d . We first consider the case r = 1. Let M i = M ρ i (x 0 ) and τ i = τ Mi . We will show that for all k ∈ N 0 To shorten notation, we set a i = inf Without loss of generality, assume 2|A ′ | ≥ |M k+1 |. If this assumption does not hold, we consider M − h instead of h. Let A ⊂ A ′ be compact such that 3|A| ≥ |M k+1 |. By (4.2) there exists a c 3 > 0 such that P y (T A < τ k ) ≥ c 3 for all y ∈ M k+1 .
Since h is harmonic, h(X t ) is a martingale. We get by optimal stopping h(y) − h(z) =E y (h(X TA ) − h(z); τ k > T A ) We will now study these three components on the right hand side seperately. Note h(z) ≥ a k ≥ a k−i−1 for all i ∈ N (1) In the first component, X enters A ⊂ A ′ before leaving M k . Hence (2) In the component X leaves M k before entering A. While leaving M k , X does not make a big jump in the following sense: X is at time τ k in M k−1 . Hence in this case h(X τ k ) ≤ b k−1 . This yields to (3) In the third component X τ k ∈ M k−i−1 for i ∈ N. Therefore h(τ k ) ≤ b k−i−1 .