Regularity of solutions to anisotropic nonlocal equations

https://doi.org/10.1007/s00209-020-02459-y

Mathematische Zeitschrift

Regularity of solutions to anisotropic nonlocal equations

Jamil Chaker¹

Received: 27 April 2018 / Accepted: 18 December 2019 / Published online: 28 January 2020

© The Author(s) 2020

Abstract

We study harmonic functions associated to systems of stochastic differential equations of the formd Xⁱ_t = Ai1(Xt−)d Z_t¹+· · ·+Ai d(Xt−)d Z_t^d,i∈ {1, . . . ,d}, whereZ_t^jare independent one-dimensional symmetric stable processes of orderαj ∈(0,2), j ∈ {1, . . . ,d}. In this article we prove Hölder regularity of bounded harmonic functions with respect to solutions to such systems.

Keywords Jump processes·Harmonic functions·Hölder continuity·Support theorem· Anisotropy·Nonlocal Operators

Mathematics Subject Classification Primary 60J75; Secondary 60H10·31B05·60G52

1 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 [14] for examples of financial models with jumps.

In the nineteen fifties, De Giorgi [15] and Nash [29] independently prove an a-priori Hölder estimate for weak solutionsuto second order equations of the form

div(A(x)∇u(x))=0

for uniformly elliptic and measurable coefficientsA. In [28], 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 [19]. For a comprehensive introduction into Harnack inequalities, we refer the reader e.g. to [20].

The corresponding case of operators in non-divergence form is treated in by Krylov and Safonov in [23]. The authors develop a technique for proving Hölder regularity and the

B

jchaker@math.uni-bielefeld.de 1

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 [6] to prove similar results for nonlocal operators of the form

L f(x)=

R^d\{0}[f(x+h)− f(x)−1{|h|≤1}h· ∇f(x)]a(x,h)dh (1.1) under suitable assumptions on the functiona. In [4] 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.

A one-dimensional Lévy process(Yt)_t≥0is calledsymmetric stable processes of order ∈ (0,2) if its characteristic function is given by

Ee^iξYt =e^−t|ξ|γ, ξ∈R.

The Lévy measure of such a process is given by ν(dh) = c_γ|h|^−1−γdh,where c_γ = 2^γ

1+γ 2

/

−^γ₂.

Letd∈Nandd≥2. We assume thatZ_tⁱ,i =1, . . . ,d, are independent one-dimensional symmetric stable processes of orderαi∈(0,2)and defineZ =(Z_t)t≥0=(Z_t¹, . . . ,Z^d_t)t≥0.

The Lévy-measure of this process is supported on the coordinate axes and is given by ν(dw)=

d k=1

⎛

⎝ c_αk

|wk|^1+αkdwk

⎛

⎝

j=k

δ_{0}(dwj)

⎞

⎠

⎞

⎠.

Thereforeν(A) = 0 for every set A ⊂ R^d, which has an empty intersection with the coordinate axes. The generatorLofZis given for f ∈C_b²(R^d)by the formula

L f(x)= d k=1

R\{0}(f(x+he_k)− f(x)−1{|h|≤1}∂kf(x)h) c_αk

|h|^1+αkdh. (1.2) For a deeper discussion on Lévy processes and their generators we refer the reader to [30].

Letx₀∈R^d andA:R^d →R^d×da matrix-valued function. We consider the system of stochastic differential equations

⎧⎪

⎪⎨

⎪⎪

⎩ d Xⁱ_t =

d j=1

Ai j(X_t−)d Zt^j,

Xⁱ₀=x₀ⁱ,

(1.3)

whereXt−=lim

s tXsis the left hand limit.

This system has been studied systematically in the caseα1=α2= · · · =αd =α∈(0,2) by Bass and Chen in the articles [3] and [4]. With the help of the martingale problem, Bass and Chen prove in [3] that for eachx0 ∈ R^d there exists a unique weak solution(X = (X¹_t, . . . ,X_t^d)t≥0,P^x0)to (1.3). Furthermore the authors prove that the family{X,P^x,x ∈ R^d}forms a conservative strong Markov process onR^d whose semigroup maps bounded

continuous functions to bounded continuous functions (see Theorem 1.1, [3]). Consequently it follows that

Lf(x)= d

j=1

R\{0}(f(x+aj(x)h)− f(x)−h1{|h|≤1}∇f(x)·aj(x)) c_α

|h|^1+αdh coincides onC_b²(R^d)with the generator for any weak solution to (1.3), whereaj(x)denotes the j^{t h}column of the matrixA(x). In [4] the authors prove Hölder regularity of harmonic func- tions with respect toLand give a counter example which shows that the Harnack inequality for harmonic functions is not satisfied.

In this paper we do not study unique solvability of (1.3) but prove an a-priori regular- ity estimate for harmonic functions if unique solutions to the system exist. The following assumptions will be needed throughout the paper.

Assumption (i) For everyx∈R^dthe matrixA(x)is non-degenerate, that is det(A(x))=0.

(ii) The functionsx → A_{i j}(x)andx → A_{i j}⁻¹(x)are continuous and bounded for all 1≤ i,j≤dandx∈R^d.

(iii) For anyx0∈R^d, there exists a unique solution to the martingale problem for

Lf(x)= d

j=1

R\{0}(f(x+a_j(x)h)− f(x)−h1{|h|≤1}∇f(x)·a_j(x)) c_αj

|h|^1+αjdh (1.4) started atx₀. The operator_Lcoincides onC_b²(R^d)with the generator for the weak solution to (1.3).

For a comprehensive introduction into the martingale problem we refer the reader to [16].

Notation

LetAbe the matrix-valued function from (1.3). LetDbe a Borel set. Throughout the paper (D)denotes the modulus of continuity ofAand we write(D)for the upper bound ofA onD. We setαmin:=min{α1, . . . , αd}andαmax:=max{α1, . . . , αd}. Fori ∈Nwe write cifor positive constants and additionallyci =ci(·)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.

Definition 1.1 Letr∈(0,1]andα1, . . . , αd ∈(0,2). Fork>0, we define M_r^k(x):=

×

For brevity we writeM_r(x)instead ofM_r¹(x).

Note thatM_r^kis increasing inkandr. Forz∈R^dandr∈(0,1], the setMr(z)is a ball with radiusrand centerzin the metric space(R^d,d), where

d(x,y)= sup

k∈{1,...,d}{|xk−y_k|^αk/αmax1_{{|xk−yk|≤1}}(x,y)+1_{{|xk−yk|>1}}(x,y)}.

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 processZand compensate them in an appropriate way, see for instance

The purpose of this paper is to prove the following result.

Theorem 1.2 Let r∈(0,1], s>0and x₀∈R^d.Suppose h is bounded inR^dand harmonic in M_r^1+s(x0)with respect to X . Then there exist c1 =c1((M_r^1+s(x0)), (M_r^1+s(x0))) >0 andβ=β((M_r^1+s(x0)), (M_r^1+s(x0))) >0, independent of h and r , such that

|h(x)−h(y)| ≤c₁

|x−y| r^αmax/αmin

_β sup

R^d |h(z)| for x,y∈M_r(x₀).

We want to emphasize, that in the caseα1 = · · · =αd the setMr(x0)reduces to a cube with radiusrand hence this result coincides with [4, Theorem 2.9], when one chooses cubes instead of balls.

Let us briefly discuss selected related results in the literature.

As previously mentioned, in [6] the authors study operators of the form (1.1) for coeffi- cientsa :R^d ×R^d →Rwhich are assumed to be symmetric in the second variable and satisfya(x,h) |h|^−d−αfor allx,h∈R^d, whereα∈(0,2). Using probabilistic techniques they prove a Harnack inequality and derive Hölder regularity estimates for bounded harmonic functions. The results of this work have been extended to more general kernels by several authors. For instance, in [5] the authors establish a Hölder estimate for harmonic functions to operators of the form (1.1), where they replace the jump measurea(x,h)dhby a family of measuresn(x,dh), which is not required to have a density with respect to the Lebesgue meaure. Furthermore, [32] extends the method of [6] to prove the Harnack inequality for more general classes of Markov processes. In [7] the authors construct and study the heat kernel a class of highly anisotropic integro-differential operators, where the Lévy measure does not have to be absolutely continuous with respect to the Lebesgue measure.

This article studies regularity for operators in non-divergence form given by (1.4). Hölder regularity results have intensively been studied for linear and nonlinear nonlocal equations governed by operators in non-divergence form. [31] provides a purely analytic proof of Hölder continuity for harmonic functions with respect to a class of integro differential equations given by (1.1), where no symmetry on the kernelais assumed. In [9], the authors study viscosity solutions to fully nonlinear integro-differential equations and prove a nonlocal version of the Aleksandrov-Bakelman-Pucci estimate, a Harnack inequality and a Hölder estimate.

There are many more important results concerning Hölder estimates and Harnack inequalities for integro-differential equations in non-divergence form including [1,8,10,22,27] and [33].

Hölder regularity estimates have also been intensely studied for operators in divergence form. We would like to mention two works, where the corresponding jump intensities are similar to the ones we study in this article. In [12] and [13] the authors study nonlocal elliptic resp. parabolic equations for families of operators which can be of the form (1.2). They prove a weak Harnack inequality and Hölder regularity estimates for weak solutions to the corresponding equations.

Let us give a short survey to known results related to systems of stochastic differential equations given by (1.3). We first discuss some results in the caseα1= · · · =αd. In [3] the authors prove unique weak solvability for (1.3). [4] shows Hölder regularity estimates for bounded harmonic functions. Furthermore, in [26] the authors prove the strong Feller property for the corresponding semigroup for (1.3). Sharp lower bounds for the transition densities for the processZ_t=(Z¹_t, . . . ,Z_t^d)are studied in [17] and sharp upper bounds in [21].

The existence of a unique solution to the martingale problem for (1.3) in the case of different orders of differentiability, i.e. αi = αj fori = j, is shown in [11] under the additional assumption that the matrixAis diagonal. [24] also studies the system (1.3) in the case of diagonal matricesA. The authors prove sharp two-sided estimates of the corresponding

transition densityp^A(t,x,y)and prove Hölder and gradient estimates for the functionx → p^A(t,x,y). In [18] the authors study the existence of densities for solutions (1.3) with Hölder continuous coefficients. They allow for a wide class of Lévy processes including the anisotropic processes Zt with different orders of differentiability. In [25] the authors study systems of the form (1.3) whereZ_t¹, . . . ,Z^d_t are independent one-dimensional Lévy processes with characteristic exponentsψ1, . . . , ψd. Under scaling conditions and regularity properties on the characteristic function they prove semigroup properties for solutions.

Structure of the article

This article is organized as follows. In Sect.2we provide definitions and auxiliary results.

We constitute sufficient preparation and study the behavior of the solution to the system. In Sect.3we 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 support theorem. Sect.4contains the proof of Theorem1.2.

2 Definitions and auxiliary results

In this section we provide important definitions and prove auxiliary results associated to the solution of the system (1.3).

LetA^τ(x)denote the transpose of the matrixA(x)and(a^τ_j(x))⁻¹the j^throw of(A^τ(x))⁻¹. For a Borel setD, we denote the first entrance time of the processXinDbyTD:=inf{t≥ 0:X_t ∈D}and the first exit time ofXofDbyτD:=inf{t≥0:X_t ∈/ D}.

Let us first recall the definition of harmonicity with respect to a Markov process.

Definition 2.1 A bounded functionh :R^d →Ris called harmonic with respect toXin a domainD⊂R^dif for every bounded open setUwithU D

h X_t∧τU

is aP^x-martingale for everyx ∈U.

ForR=Ms(y)we use the notationR=M_s³(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 [2, Proposition V.7.2], which can be easily adjusted to our case.

Proposition 2.2 Let r ∈(0,1],q ∈(0,1)and x₀ ∈R^d. If A⊂ M_r(x₀)and|A|<q, then there exists a set D⊂Mr(x0)such that

(1) D is the union of rectanglesRisuch that the interiors of the Riare pairwise disjoint, (2) |A| ≤ |D∩Mr(x0)|and

(3) for each i ,|A∩R_i|>q|Ri|.

Following the ideas of the proof of [6, Proposition 2.3], we next prove a Lévy system type formula.

Proposition 2.3 Suppose D and E are two Borel sets withdist(D,E) >0. Then

s≤t

1{Xs−∈D,Xs∈E}− _t

0 1D(X_s)

E

d k=1

⎛

⎝|(a^τ_k(Xs))⁻¹|^1+αkc_αk

|h_k−X^k_s|^1+αk dh_k

⎛

⎝

j=k

δ_{X^j

s}(dh_j)

⎞

⎠

⎞

⎠ds

is aP^x-martingale for each x.

Proof Let f ∈C_b²(R^d)with f =0 onDand f =1 onE. Moreover set M_t^f := f(X_t)− f(X₀)−

_t

0

Lf(X_s)ds.

By Assumption (iii) for eachx ∈R^dthe probability measureP^xis a solution to the martingale problem forL. Since the stochastic integral with respect to a martingale is itself a martingale,

_t

0

1D(Xs−)d Ms^f

is aP^x-martingale. Rewriting f(Xt)− f(X0)=

s≤t(f(Xs)− f(Xs−))leads to

s≤t

(1D(Xs−)(f(Xs)− f(Xs−)))− _t

0

1D(Xs−)Lf(Xs)ds

is aP^x-martingale. SinceXs =X_s−for only countably many values ofs,

s≤t

(1D(X_s−)(f(X_s)− f(X_s−)))− _t

0 1D(X_s)Lf(X_s)ds (2.1) is also aP^x-martingale. Letw=(w1, . . . , wd)andu =(u1, . . . ,u_d). By definition of f, forx ∈Dwe have f(x)=0 and∇f(x)=0. Hence

Lf(x)= d k=1

R\{0} f(x+a_k(x)h) c_αk

|h|^1+αkdh

= d k=1

R^d\{0}

⎛

⎝f(x+A^τ(x)w) c_αk

|w|^1+αk

⎛

⎝

j=k

δ{0}(dwj)

⎞

⎠

⎞

⎠dwk

= d k=1

R^d\{0} f(u)|(a^τ_k(x))⁻¹|^1+αkc_αk

|u−x|^1+αk

⎛

⎝

j=k

δ{xj}(duj)

⎞

⎠duk.

Note, thatc_αk/|h|^1+αk is integrable overh in the complement of any neighborhood of the origin for anyk ∈ {1, . . . ,d}. SinceDandEhave a positive distance from each other, the sum in (2.1) is finite. Hence

s≤t

(1D(Xs−)(1E(Xs)−1E(Xs)))

− _t

0 1D(X_s)

E

d k=1

⎛

⎝|(a^τ_j(Xs))⁻¹|^1+αjc_αk

|h_k−X_s^k|^1+αk dh_k

⎛

⎝

j=k

δ_{X^j

s}(dh_j)

⎞

⎠

⎞

⎠ds

is aP^x-martingale, which is equivalent to our assertion.

The next Proposition gives the behavior of the expected first exit time of the solution to (1.3) out of the setMr(·). This Proposition highlights the advantage of Mr(·)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.

Proposition 2.4 Let x ∈ R^d and r ∈ (0,1]. Then there exists a constant c₁ = c₁((M_r(x)),d) >0such that for all z∈M_r(x)

E^z τMr(x)

≤c₁r^αmax. Proof First note

E^z τM_r(y)

=E^z

1≤i≤dmin inf{t≥0:X_tⁱ∈/(yi−r^(αmax/αi),yi−r^(αmax/αi))}

≤ 1 d

d i=1

E^z

inf{t≥0:X_tⁱ∈/(yi−r^(αmax/αi),yi−r^(αmax/αi))}

=: 1 d

d i=1

E^z[ϒi].

(2.2)

Let j ∈ {1, . . . ,d}be fixed but arbitrary. The aim is to show that there existsc2 >0 such that

E^z(ϒj)≤c2r^αmax. (2.3)

Since we reduced the problem to a one-dimensional one, we may suppose by scalingr=1.

Let

κ:=inf

|A(x)ej| :x∈M1(x) . By Assumption (i), we haveκ >0. There exists ac3∈(0,1)with

P^z(∃s∈ [0,1] :Z_s^j ∈R\ [−3/κ,3/κ])≥c₃.

The independence of the one-dimensional processes implies that with probability zero at least two of theZⁱ’s make a jump at the same time. This leads to

P^z(∃s∈ [0,1] :Z_s^j > 3

κ andZ_sⁱ =0 fori ∈ {1, . . . ,d} \ {j})≥c3. (2.4) Our aim is to show that the probability of the processXfor leavingM1(x)in the j^thcoordinate after timemis bounded in the following way

P^z(ϒj >m)≤(1−kj)^m for all m∈N.

Suppose there existss∈ [0,1]such thatZs^j > ³_κ,Z_sⁱ =0 fori ∈ {1, . . . ,d} \ {j}, and Xs− ∈M1(x).Then

|Xs^j| = |Zs^j| |A(Xs−)(ej)|>3.

Note, that we leaveM₁(x)by this jump. By (2.4)

P^z(ϒj ≤1)≥c₃⇔P^z(ϒj >1)≤(1−c₃).

Let{θt :t≥0}denote the shift operators forX. Now assumeP^z(ϒj >m)≤(1−c3)^m. By the Markov property

P^z(ϒj >m+1)≤P^z(ϒj >m;ϒj◦θm>1)

=E^z[P^Xm(ϒj >1);ϒj >m]

≤(1−c3)P^z(ϒj >m)

≤(1−c3)^m+1.

Assertion (2.3) follows by E^x[ϒj] =

_∞

0

P^z(ϒj >t)dt ≤^∞

m=0

P^z(ϒj >m)≤^∞

m=0

(1−c₃)^m=c₂,

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.

Proposition 2.5 Let x ∈ R^d, r ∈ (0,1] and R ≥ 2r . There exists a constant c1 = c₁((R^d),d) >0, such that for all z∈M_r(x)

P^z(X_τMr(x) ∈/ M_R(x))≤c₁ r

R _αmax

. Proof Let

Cj :=R\ [xj−R^αmax/αj,xj+R^αmax/αj]

and for 1≤ j≤dletkj =sup_x∈R|(a^τ_j(x))⁻¹|cαj. By Proposition2.3and optional stopping we get forc2=_d

j=1((2kj2^α_max)/cαj)≤8dsup_x∈R|(a^τ_j(x))⁻¹| P^z

X_t∧τMr(x)∈/M_R(x)

=E^z

⎡

⎣_t∧τMr(x)

0

MR(x)^c

d j=1

|(a^τ_j(Xs))⁻¹|cαj

|hj−Xs^j|^1+αj

⎛

⎝

i=j

δ_{Xⁱ_s}(dh_i)

⎞

⎠dh_jds

⎤

⎦

≤E^z

⎡

⎣ _t∧τMr(x)

0

d j=1

Cj

kj

|hj−Xs^j|^1+αjdhjds

⎤

⎦

≤E^z

⎡

⎣ t∧τMr(x) 0

d j=1

Cj

kj

|hj−(xj+r^αmax/αj)|^1+αjdhjds

⎤

⎦

=E^z[t∧τMr(x))] d

j=1

2kj

αj(R^αmax/αj−r^αmax/αj)^αj

≤E^z[t∧τMr(x))] d

j=1

2kj

αj((R/2)^αmax/αj)^αj = c₂

R^αmaxE^z[t∧τMr(x))].

Using the monotone convergence on the right and dominated convergence on the left, we have fort→ ∞

P^z(X_t∧τMr(x)∈/M_R(x))≤ c₂

R^αmaxE^z(τMr(x))≤c₂c₃ r

R _αmax

,

wherec3is the constant showing up in the estimateE^z(τMr(x))≤c3r^αof Proposition2.4.

3 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 [23] for the diffusion case. In the article [4], Bass and Chen prove the support theorem in the context of pure jump processes with singular and anisotropic kernels. They consider the system (1.3) in the caseαi =αfor alli∈ {1, . . . ,d} and use the technique by Krylov and Safonov 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 [4].

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 timeTsuch that with positive probability the stopped process stays in a small ball around its starting point up to timeT, makes a jump along the k^thcoordinate axis and stays afterwards in a small ball.

Lemma 3.1 Let r ∈(0,1], x0 ∈R^d,k∈ {1, . . . ,d}, vk= A(x0)ek, γ ∈(0,r^αmax/αmin),t0>

0 and ξ ∈ [−r^αmax/αmin,r^αmax/αmin]. There exists a constant c1 > 0 = c1(γ,t0, ξ,r, (M_r²(x0))), (M_r²(x0))) >0and a stopping time T ≤t₀, such that

P^x0

"

sup

s<T|X_s−x₀|< γ and sup

T≤s≤t₀|X_s−(x₀+ξvk)|< γ

#

≥c₁. (3.1) Proof Let

A_∞:=1∨

⎛

⎝^d

i,j=1

sup

x∈Mr²(x0)|Ai j(x)|

⎞

⎠.

We assumeξ ∈ [0,r^αmax/αmin]. The caseξ ∈ [−r^αmax/αmin,0]can be proven similar. Let us first supposeξ ≥γ /(3A_∞)and letβ∈(0, ξ), which will be chosen later. We decompose the processZⁱ_t in the following way:

$Zⁱ_t =

s≤t

Zⁱ_s1_{|Zⁱ_s|>β}, Zⁱ_t =Zⁱ_t−$Z_tⁱ.

Let(Xt)t≥0be the solution to d Xⁱ_t =

d j=1

Ai j(X_t−)d Z_t^j,Xⁱ₀=x₀ⁱ.

The continuity ofAallows us to find a δ < γ /(6A∞), such that sup

i,j

sup

|x−x0|<δ|A_{i j}(x)−A_{i j}(x0)|< γ

12d. (3.2)

Consider C =

%

s≤tsup0

|Xs−X0| ≤δ

&

,

D=

$Z^khas precisely one jump before timet0with jump size in[ξ, ξ+δ], $Z_s^j =0 for alls≤t₀and all j =k

,

E=

$Z_sⁱ =0 for alls≤t0andi=1, . . . ,d .

SinceAis bounded, we can findc₂>0, such that

Xⁱ,Xⁱ

t≤c2

d j=1

Z^j,Z^j

t. Note, thatβ∈(0, ξ)⊂(0,r^αmax/αmin)⊂(0,1). Therefore, we get

E^x0 Xⁱ,Xⁱ

t ≤c₂ d

j=1

E^x0 Z^j,Z^j

t =c₂ d

j=1

_t

0

" _β

−β

c_αjh²

|h|^1+αjdh

#

dt ≤c₃tdβ^2−αmax.

By Tschebyscheff’s inequality and Doob’s inequality, we get P^x0

sup

s≤t₀|Xⁱ_s−Xⁱ₀|> δ

≤ 1 δ²E^x0

sup

s≤t₀

Xⁱ_s−Xⁱ₀

₂

≤ 1 δ²4E^x0

Xⁱ_t

0−Xⁱ_{0 2}

≤ c4t0dβ^2−αmax

δ² .

Chooseβ∈(0, ξ)such that

c₅t₀β^2−αmax≤ δ²

2d (3.3)

holds. Then by (3.3), we get

P^x0(C)=1−P^x0

sup

s≤t₀|Xⁱ_s−Xⁱ₀|> δ

≥ 1

2. (3.4)

For$Z^k to have a single jump before timet0, and for that jump’s size to be in the interval [ξ, ξ+δ], then up to timet0 $Z_t^kmust 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].

We can use the fact, that$Z^kis a compound Poisson process and use the knowledge about Poisson random measures. The events descriped in (i)-(iv) are the probabilities that Poisson random variablesP₁,P₂.P₃andP₄of parametersλ1=c₆t₀β^−αk,λ2=c₆t₀(β^−αk−ξ^−αk), λ3=c6t0(ξ+δ)^−αk, andλ4 =c6t0(ξ^−αk−(ξ+δ)^−αk), respectively, take the values 0,0,0, and 1, respectively.

So there exists a constantc7=c7(αk,t0, δ, ξ, β) >0 such that

P^x0$Z^khas a single jump before timet0, and its size is in[ξ, ξ+δ]

≥c7. For all j=k, the probability that$Z^j does not have a jump before timet₀, is the probability that a Poisson random variable with parameter 2c₆t₀β^−αj is equal to 0. Using the indepence of$Z^j forj=1, . . . ,d, we can find ac₈>0 such that

P^x0($Zs^j =0 for alls≤t0and all j=k)≥c8. Thus we obtain

P^x0(D)≥c₉

for ac₉=c₉(α1, . . . , αd,t₀, δ, ξ, β) >0. Furthermore theZⁱ’s are independent of the$Z^j’s for alli,j ∈ {1, . . . ,d}, soC andDare independent and we obtain

P^x0(C∩D)≥c9/2.

Similary we obtain

P^x0(E)≥c₁₀ and P^x0(C∩E)≥c₁₁. (3.5) LetTbe the time, when$Z^kjumps the first time, i.e.Z^kmakes a jump greater thenβ. Then Zs− =Zs−for alls≤T and henceXs−=Xs−for alls≤T.So up to timeT,Xsdoes not move away more thanδaway from its starting point. NoteXT =A(XT−)ZT.By (3.2), we obtain onC∩D

|XT−(x0+ξvk)| ≤ |XT−−x0| + |XT−ξA(x0)ek))|

= |X_T−−x₀| + |A(X_T−)Z_T−ξA(x₀)e_k)|

≤ |XT−−x0| +ξ|(A(XT−)−A(x0))ek| + |A(XT−)(ZT −ξek)|

≤δ+ξdγ

12d +δA∞≤ γ 6

1 A∞ +γ

2 +1

≤γ 2. Appling the strong Markov property at timeT, we get by (3.5)

P^x0

"

sup

T≤s≤T+t₀

|Xs−X_T|< δ

#

≥P^XT(C∩E)≥c₁₁.

Note, that|X_T −(x₀+ξvk)| < γ /2 and |X_s − X_T| < δfor all T ≤ s ≤ t₀ imply

|X_s−(x₀+ξvk)|< γ.

All in all we get by the strong Markov property P^x0

"

sup

s<T|X_s−x₀|< γand sup

T≤s≤t₀|X_s−(x₀+ξvk)|< γ

#

≥ c₉c₁₁ 2 , which proves the assertion.

Now supposeξ < γ /(3A∞). Then|x₀−(x₀+ξvk)|< γ /3. We can chooseT ≡0 and by (3.5) we get:

P^x0

s≥tsup0

|Xs−x0|< δ

≥c11,

which finishes the proof.

We need two simple geometrical facts from the field of linear algebra, whose proofs can be found in [4] (Lemmas 2.4 and 2.5).

Lemma 3.2 Suppose u, vare two vectors inR^d,η∈(0,1), and p is the projection ofvonto u. If|p| ≥η|v|, then

|v−p| ≤'

1−η²|v|.

Lemma 3.3 Letv be a vector inR^d, uk = Aek, and pk the projection of vonto uk for k=1, . . . ,d.Then there existsρ=ρ((R^d))∈(0,1), such that for some k,

|v−p | ≤ρ|v|.

For a given timet₁ >0 the following lemma shows that solutions stay with positive prob- ability in anε-tube around a given line segment on[0,t₁]. The case ofα1 = · · · =αd was considered in [4]. We follow their technique.

Lemma 3.4 Let r ∈(0,1], x₀ ∈R^d, t₁ >0, ε ∈(0,r^αmax/αmin), ξ ∈(0, ε/4)andγ >0. Moreover letψ: [0,t₁] →R^dbe a line segment of lengthξstarting at x₀. Then there exists c1=c1((M_r²(x0))), (M_r²(x0))),t1, ε, γ ) >0, such that

P^x0

sups≤t1

|Xs−ψ(s)|< εand|Xt₁−ψ(t1)|< γ

≥c1.

Proof Note thatεis chosen such that B_ε(x₀) ⊂ M_r(x₀). Letρ ∈ (0,1)be such that the conclusion of Lemma3.3holds for all matrices A = A(x)with x ∈ M_r²(x).Take γ ∈ (0, ξ ∧ρ)such that$ρ := γ +ρ < 1 andn ≥ 2 sufficiently large, such that($ρ)ⁿ < γ.

Letv0 :=ψ(t1)−ψ(0)=ψ(t1)−x0,which has lengthξ.By Lemma3.2, there exists a k0 ∈ {1, . . . ,d}such that ifp0is the projection ofv0ontoA(x0)ek₀, then|v0−p0| ≤ρ|v0|.

Note, that|p₀| ≤ |v0| =ξ.By Lemma3.1there existsc₂ >0 and a stopping timeT₀≤t₁/n such that for

D1:=

(

s<Tsup0

|Xs−x0|< γⁿ⁺¹and sup

T0≤s≤t1/n|Xs−(x0+p0)|< γⁿ⁺¹ )

.

the estimate

P^x0(D₁)≥c₂

holds. Sinceγ <1 andγⁿ ≤γ for alln∈N, we have forT0≤s≤t1/n

|ψ(t1)−Xs| ≤ |ψ(t1)−(x0+p0)| + |(x0+p0)−Xs|

≤ |v0−p0| +γⁿ⁺¹=ρξ+γⁿ⁺¹≤$ρξ (3.6) onD₁. Takings=t₁/n, we have

|ψ(t1)−Xt₁/n| ≤$ρξ.

Since$ρ <1 and|ψ(t₁)−x₀| = |v0| =ξ,then (3.6) shows that onD₁ Xs ∈B(x0,2ξ)⊂B(x0, ε/2) ifT0 ≤s≤t1/n.

If 0≤s<T₀,then|X_s−x₀|< γⁿ⁺¹< ξ,and so we have onD₁ {Xs,s∈ [0,t1/n]} ⊂B(x0,2ξ)⊂B(x0, ε/2).

Now letv1 :=ψ(t₁)−X_t1/n.WhenX_t1/n ∈ B(x₀, ε/2),then by Lemma3.3, there exists k1∈ {1, . . . ,d}such that ifp1is the projection ofv1ontoA(Xt1/n)ek1,then|v1−p1| ≤ρ|v1|.

LetT1∈ [t1/n,2t1/n]be a stopping time, determined by Lemma3.1, and D₂:=

( sup

t1/n≤s<T1

|X_s−X_t1/n|< γⁿ⁺¹and sup

T1≤s≤2t1/n|X_s−(X_t1/n+p₁)|< γⁿ⁺¹ )

.

By the Markov property at the timet₁/nand Lemma3.1, there exists the samec₂ >0 such that

P^x0(D₂|Ft1/n)≥c₂