On the coupling property of Lévy processes

www.imstat.org/aihp 2011, Vol. 47, No. 4, 1147–1159

DOI:10.1214/10-AIHP400

© Association des Publications de l’Institut Henri Poincaré, 2011

On the coupling property of Lévy processes

René L. Schilling

and Jian Wang

aSchool of Mathematics and Computer Science, Fujian Normal University, 350007, Fuzhou, P. R. China. E-mail:jianwang@fjnu.edu.cn bInstitut für Mathematische Stochastik, TU Dresden, 01062 Dresden, Germany. E-mail:rene.schilling@tu-dresden.de

Received 21 July 2010; revised 25 October 2010; accepted 2 November 2010

Abstract. We give necessary and sufficient conditions guaranteeing that the coupling for Lévy processes (with non-degenerate jump part) is successful. Our method relies on explicit formulae for the transition semigroup of a compound Poisson process and earlier results by Mineka and Lindvall–Rogers on couplings of random walks. In particular, we obtain that a Lévy process admits a successful coupling, if it is a strong Feller process or if the Lévy (jump) measure has an absolutely continuous component.

Résumé. Nous donnons les conditions nécessaires et suffisantes pour le succès du couplage entre des processus de Lévy (avec partie de sauts non-dégénérée). Notre méthode est basée sur les formules explicites pour le semigroupe de transition d’un processus de Poisson composé, et les résultats de Mineka et Lindvall–Rogers sur le couplage d’une marche aléatoire. En particulier, nous montrons qu’un processus de Lévy admet un couplage, s’il est un processus fortement fellerien ou si la mesure de Lévy (mesure de sauts) possède une composante absolument continue.

MSC:60G51; 60G50; 60J25; 60J75

Keywords:Coupling property; Lévy processes; Compound Poisson processes; Random walks; Mineka coupling; Strong Feller property

1. Introduction and main results

The coupling method is a powerful tool in the study of Markov processes and interacting particle systems. There are some comprehensive books on this topic now, see e.g. [2,7,13,15]. Let (Xt)t≥0 be a Markov process on R^d with transition probability function{P_t(x,·)}t≥0,x∈R^d. AnR^2d-valued process(X_t, X_t)_t≥0is calleda coupling of the Markov process(Xt)t≥0, if both(X_t)t≥0and(X_t)t≥0are Markov processes which have the same transition functions P_t(x,·)but possibly different initial distributions. In this case,(X_t)_t≥0and(X_t)_t≥0are called themarginal processes of the coupling process; the coupling time is defined byT :=inf{t≥0: X_t=X_t}. The coupling(X_t, X_t)_t≥0is said to besuccessfulifT is a.s. finite. A Markov process(Xt)t≥0admits asuccessful coupling(also: enjoys thecoupling property) if for any two initial distributionsμ₁andμ₂, there exists a successful coupling with marginal processes possessing the same transition probability functionsP_t(x,·)and starting fromμ₁andμ₂, respectively. It is known, see [7,12], that the coupling property is equivalent to the statement that

tlim→∞μ1Pt−μ2PtVar=0 forμ1andμ2∈P R^d

. (1.1)

As usual,μP (A)=

P (x, A)μ(dx)is the left action of the semigroup and · Varstands for the total variation norm.

If a Markov process admits a successful coupling, then it also has the Liouville property, i.e. every bounded harmonic function is constant; in this context a function f is harmonic, if Lf =0 where L is the generator of the Markov process. See [3,4] and references therein for this result and more details on the coupling property.

The aim of this paper is to study the coupling property of Lévy processes by using explicit conditions on Lévy measures. Our work is mainly motivated by the recent paper [14], which contains some interesting results on the

coupling property of Ornstein–Uhlenbeck processes; the paper [14] uses mainly the conditional Girsanov theorem on Poisson space and assumes that the corresponding Lévy measure has a non-trivial absolutely continuous part. Our technique here is completely different from F.-Y. Wang’s paper [14]. We use an explicit expression of the compound Poisson semigroup and combine this with the Mineka and Lindvall–Rogers couplings for random walks, see [8].

A Lévy process(X_t)_t≥0onR^dis a stochastic process with stationary and independent increments and càdlàg (right continuous with finite left limits) paths. It is well known thatX_t is a (strong) Markov process whose infinitesimal generator is, forf∈C_b(R^d), of the form

Lf (x)=1 2

d i,j=1

q_i,j∂²f (x)

∂x_i∂x_j + f (x+z)−f (x)−z· ∇f (x) 1+ |z|²

ν(dz)+b· ∇f (x),

whereQ=(q_i,j)^d_i,j=1is a positive semi-definite matrix,b∈R^dis the drift vector andνis the Lévy or jump measure;

the Lévy measureνis aσ-finite measure on(R^d,B(R^d))such that

(1∧ |z|²)ν(dz) <∞. Note that the Lévy triplet (b, Q, ν)characterizes, up to indistinguishability, the process (X_t)_t≥0 uniquely. Our standard reference for Lévy processes is the monograph [11]. We writePt(x, A)=Pt(A−x),A∈B(R^d), for the transition probability ofXt.

Letμandν be two bounded measures on(R^d,B(R^d)). We defineμ∧ν:=μ−(μ−ν)⁺, where(μ−ν)^±is the Jordan–Hahn decomposition of the signed measureμ−ν. In particular,μ∧ν=ν∧μand it is easy to see that μ∧ν(R^d)=¹₂[μ(R^d)+ν(R^d)− μ−νVar],cf. [2]. We can now state our main result.

Theorem 1.1. Let(X_t)_t≥0be ad-dimensional Lévy process with Lévy triplet(b, Q, ν).For everyε >0,defineν_εby

ν_ε(B)= ν(B), ν

R^d

<∞;

ν

z∈B: |z| ≥ε , ν

R^d

= ∞. (1.2)

Assume that there existε, δ >0such that inf

x∈R^d,|x|≤δ

ν_ε∧(δ_x∗ν_ε) R^d

>0. (1.3)

Then,there exists a constantC=C(ε, δ, ν) >0such that for allx, y∈R^dandt >0, P_t(x,·)−P_t(y,·)

Var≤C(1+ |x−y|)

√t ∧2.

In particular,the Lévy processXtadmits a successful coupling.

Remark 1.2. Condition(1.3)guarantees that P_t(x,·)−P_t(y,·)

Var=^O t^−1/2

ast→ ∞

holds locally uniformly for allx, y∈R^d.This order of convergence is known to be optimal for compound Poisson processes,see[14],Remark3.1.In[14]it is pointed out that a pure jump Lévy process admits a successful coupling only if the Lévy measure has a non-discrete support,in order to make the process more active.Condition(1.3)is one possibility to guarantees sufficient jump activity;intuitively it will hold if for sufficiently small values ofε, δ >0and allx∈R^dwith|x| ≤δwe havex+supp(νε)∩supp(νε)=∅;heresupp(νε)is the support of the measureν_ε.

In order to see that(1.3)is sharp,we consider an one-dimensional compound Poisson process with Lévy measure ν supported on Z.Then,for anyδ∈(0,1)and x∈R^d with|x| ≤δ, ν∧(δ_x∗ν)(Z)=0. On the other hand,all functions satisfying f (x+n)=f (x)for x∈R^d and n∈Z are harmonic.By[3,4],this process cannot have the coupling property.

Theorem 3.1 of [14] establishes the coupling property for Lévy processes whose jump measureνhas a non-trivial absolutely continuous part. It seems to us that this condition is not directly comparable with (1.3). In fact, based on the

Lindvall–Rogers ‘zero–two law’ for random walks [8], Propsotion 1, we give in Section4a necessary and sufficient condition guaranteeing that a Lévy process has the coupling property. In this section we will also find the connection between (1.3) and the existence of a non-trivial absolutely continuous component of the Lévy measure. In particular, we obtain some extensions of Theorem1.1and [14], Theorem 3.1.

Once we know that a Lévy process admits the coupling property, many interesting new questions arise which are, however, beyond the scope of the present paper. For example, it would be interesting to construct explicitly the corresponding successful Markov coupling process and to determine its infinitesimal operator. There are a number of applications of optimal Markov processes and operators; we refer to [2,15] for background material and a more detailed account on diffusions and q-processes. We will discuss those topics for Lévy processes in a forthcoming paper [1].

2. The coupling property of compound Poisson processes

In this section, we consider the coupling property of compound Poisson processes. Let(Lt)t≥0be a compound Poisson process onR^dsuch thatL0=xand with Lévy measureν. Then,Lt can be written as

L_t=x+

N_t

i=1

ξ_i, t≥0,

whereN_t is the Poisson process with rateλ:=ν(R^d)and(ξ_i)_i≥1is a sequence of i.i.d. random variables onR^dwith distributionν(·)/λ; moreover, we assume that theξ_i’s are independent of N_t. As usual we use the convention that ₀

i=1ξ_i=0.

The transition semigroup for a compound Poisson process is explicitly known. This allows us to reduce the coupling problem for a compound Poisson processes to that of a random walk. LetP_tandLbe the semigroup and the generator forL_t, respectively. Then, it is well known that for anyf∈B_b(R^d),

Lf (x)= f (x+z)−f (x) ν(dz)

=λ f (x+z)−f (x) ν₀(dz)

and

P_t=e^{t L}=^∞

n=0

tⁿLⁿ n! =e^−λt

∞ n=0

(t λ)ⁿν₀^∗n

n! , t≥0; (2.1)

hereν₀^∗nis then-fold convolution ofν₀andν₀^∗0:=δ₀.

The following result explains the relationship of transition probabilities of compound Poisson processes and of random walks.

Proposition 2.1. Let P_t(x,·)be the transition probability of the compound Poisson process L=(L_t)_t≥0 and let S=(S_n)_n≥1,S_n=ξ₁+· · ·+ξ_n,be a random walk where(ξ_i)_i≥1are i.i.d.random variables withξ₁∼ν₀:=ν/ν(R^d).

Then,for allx, y∈R^d P_t(x,·)−P_t(y,·)

Var≤e^−λt ∞ n=0

(λt )ⁿP(x+S_n∈ ·)−P(y+S_n∈ ·)Var

n!

=e^−λt

2(1−δ_x,y)+^∞

n=1

(λt )ⁿP(x+S_n∈ ·)−P(y+S_n∈ ·)Var

n!

,

whereδ_x,yis a Kronecker delta function,i.e.δ_x,y=1ifx=y,and0otherwise.

Proof. LetPt andP_n^S be the semigroups of the compound Poisson processLand the random walkS, respectively.

Because of (2.1) we find P_t(x,·)−P_t(y,·)

Var= sup

f∞≤1

P_tf (x)−P_tf (y)

=e^−λt sup

f∞≤1

∞ n=0

(λt )ⁿ(δx∗ν₀^∗n(f )−δy∗ν₀^∗n(f )) n!

≤e^−λt sup

f∞≤1

∞ n=0

(λt )ⁿ(P_n^Sf (x)−P_n^Sf (y)) n!

≤e^−λt ∞ n=0

(λt )ⁿsup_f∞≤1|P_n^Sf (x)−P_n^Sf (y)| n!

=e^−λt ∞ n=0

(λt )ⁿP(x+S_n∈ ·)−P(y+S_n∈ ·)Var

n! ,

which proves the first assertion.

Forn=0 we haveS0=0; thus P(x+S₀∈ ·)−P(y+S₀∈ ·)

Var= δ_x−δ_yVar=2(1−δ_x,y)

and the second assertion follows.

An immediate of Proposition2.1is the following estimate forP_t(x,·)−P_t(y,·)Varwhich is based on a similar inequality forP(x+S_n∈ ·)−P(y+S_n∈ ·)Var.

Proposition 2.2. Assume that for allx, y∈R^d there is a constantC(x, y) >0such that P(x+S_n∈ ·)−P(y+S_n∈ ·)

Var≤ C(x, y)

√n forn≥1. (2.2)

Then,

P_t(x,·)−P_t(y,·)

Var≤2e^−λt(1−δ_x,y)+

√2C(x, y)(1−e^−λt)

√λt .

Proof. A combination of Proposition2.1and (2.2) yields for allx, y∈R^d P_t(x,·)−P_t(y,·)

Var≤e^−λt

2(1−δ_x,y)+C(x, y) ∞ n=1

(λt )ⁿ

√nn!

.

Jensen’s inequality for the concave functionx^1/2gives ∞

n=1

1 n

(λt )ⁿ

n! ≤ e^λt−1 (e^λt−1)^1/2

_∞

n=1

(λt )ⁿ n·n!

1/2

=

e^λt−1 λt

1/2_∞

n=1

(λt )ⁿ⁺¹ (n+1)!·n+1

n 1/2

≤

e^λt−1 λt

1/2√ 2

e^λt−1−λt1/2

≤

√2(e^λt−1)

√λt .

The required assertion follows form the estimates above.

We will now show thatLhas the coupling property wheneverShas.

Proposition 2.3. Let(L_t)_t≥0be the compound Poisson process with Lévy measureν,and let(S_n)_n≥0,be a random walk whereS_n=S₀+ξ₁+ · · · +ξ_n,and(ξ_i)_i≥1are i.i.d.random variables withξ₁∼ν₀:=ν/ν(R^d).IfS_nadmits a successful coupling,so doesL_t.

Proof. Forx, y∈R^d, letL_tbe a compound Poisson process starting fromx∈R^d. ThenL_t=N_t

i=0ξ_i, whereξ₀=x and where(ξ_i)_i≥1are i.i.d. random variables with common distributionν₀:=ν/ν(R^d);(N_t)_t≥0is a Poisson process with rateλ:=ν(R^d). Moreover,(N_t)_t≥0and(ξ_i)_i≥1are independent.

Set S₀=x and S_n=_n

i=0ξ_i for n≥1. Since S has the coupling property, there exists another random walk S_n =_n

i=0ξ_i such thatS−S₀andS−S₀ have the same law and such that for any starting pointξ₀=y ofSthe coupling time

T_x,y^S :=inf

k≥1: S_k=S_k

is a.s. finite.

Without loss of generality, we can assume thatS_k=S_k fork≥T_x,y^S , and that(ξ_i)_i≥1is independent of(N_t)_t≥0. Define

L_t=

N_t

i=0

ξ_i, t≥0.

ThenL=(L_t)_t≥0is also a compound Poisson process with Lévy measureν and starting pointL₀=y. In order to show thatLhas the coupling property, it is enough to verify that

T_x,y^L :=inf

t >0:L_t=L_t

<∞. (2.3)

We claim that

T_x,y^L =K_x,y withK_x,y:=inf

t >0:N_t≥T_x,y^S

. (2.4)

This implies (2.3). By assumption we know that for almost everyω,T_x,y^S (ω) <∞. Since the Poisson processN_ttends to infinity ast→ ∞, there existsτ0(ω) <∞such thatN_t≥T_x,y^S (ω)for allt≥τ0(ω). Therefore, (2.4) tells us that T_x,y^L ≤τ0<∞.

Let us finally prove (2.4). For this argument we assume thatω is fixed. Let t >0 be such thatN_t ≥T_x,y^S , i.e.

t≥K_x,y. SinceS_k=S_k fork≥T_x,y^S ,S_Nt =S_N

t and, by construction,L_t=L_t; thus,T_x,y^L ≤tand sincet≥K_x,ywas arbitrary, we haveT_x,y^L ≤Kx,y. On the other hand, assume thatKx,y>0. Then, by the very definition ofKx,y, for any ε >0, there existst_ε>0 such thatt_ε> K_x,y−εandN_tε≤T_x,y^S −1. Hence,S_Ntε=S_N

tε, i.e.L_tε =L_t

ε. Therefore, T_x,y^L ≥t_ε> K_x,y−ε. Lettingε→0, we getT_x,y^L ≥K_x,y and the proof is complete.

Note that the proof of Proposition2.3already gives an estimate for the rate at which coupling occurs: for allt >0 P

T_x,y^L > t

=P

Nt< T_x,y^S

=e^−λt

1+ ∞ k=1

P

T_x,y^S > k(λt )^k k!

.

What remains to be done is to get estimates for the coupling time of the random walkS,P(T_x,y^S > k), k≥1. This requires concrete coupling constructions forS; the most interesting random walk couplings rely on a suitable coupling of the steps ξ_j andξ_j of S andS, respectively. In the next section we will, therefore, consider the Mineka and Lindvall–Rogers couplings.

We close this section with two comments on Proposition2.3.

Remark 2.4. (1)Theorem4.3below will show,that the converse of Proposition2.3is also true: LetL=(L_t)_t≥0be a compound Poisson process with Lévy measureν, andS=(S_n)_n≥0,S_n=S₀+ξ₁+ · · · +ξ_n, be a random walk where (ξ_i)_i≥1are i.i.d. random variables withξ₁∼ν₀:=ν/ν(R^d). IfLhas the coupling property so doesS.

(2)Proposition2.3can be easily generalized to more general settings,see also[1].More precisely,let(Xt)t≥0be a Markov process onR^dand let(S_t)_t≥0be a subordinator(i.e.an increasing Lévy process)which is independent of X_t.IfX_thas the coupling property and ifS_ttends to infinity ast→ ∞,then the subordinate processX_St also has the coupling property.

3. The Mineka and Lindvall–Rogers couplings – A review

LetS=(Sn)n≥1,Sn=ξ1+ · · · +ξn, be a random walk onR^d with i.i.d. steps(ξi)i≥0such thatξ1∼ν0. The main result of this section is

Theorem 3.1. Suppose that for someδ >0, η0=η0(δ):= inf

x∈R^d,|x|≤δ

ν0∧(δ_x∗ν0) R^d

>0. (3.1)

Then there exists a constantC:=C(δ, η0) >0such that for allx, y∈R^dandn≥1, P(x+Sn∈ ·)−P(y+Sn∈ ·)

Var≤ C(1+ |x−y|)

√n . (3.2)

The proof of Theorem3.1is mainly based on Mineka’s coupling [9], see also [7], Chapter II, Section 14, pp. 44–

47, and the coupling argument of the zero–two law for random walks proved in [8], Proposition 1, by Lindvall and Rogers. These papers do not contain an estimate as explicit as (3.2). Therefore we decided to include a detailed proof on our own which again highlights the role of the sufficient condition (3.1).

We begin with an auxiliary result which describes the total variation norm of a signed measure under a non- degenerate linear transformation.

Lemma 3.2. Letμbe a probability measureμonR^d.Then we have for allx, y∈R^d δx∗μ−δy∗μVar= δx−y∗μ−μVar=δ_|x−y|e₁∗

μ◦R_x⁻₋¹_y

−μ◦R⁻_x−¹_y

Var,

where e₁=(1,0, . . . ,0)∈R^d and R_a is a non-degenerate rotation such thatR_aa= |a|e₁.In particular, for any a∈R^d,

δ_a∗μ−μVar= δ_−a∗μ−μVar.

Proof. Using the definition of the total variation norm we get δ_x∗μ−δ_y∗μVar=2 sup

A∈B(R^d)

δ_x∗μ(A)−δ_y∗μ(A)

=2 sup

A∈B(R^d)

μ(A−x)−μ(A−y)

=2 sup

B∈B(R^d)

μ

B−(x−y)

−μ(B)

= δx−y∗μ−μVar.

Now leta∈R^dand denote byR_athe rotation such thatR_aa= |a|e₁. Clearly, μ◦Ra(A)=μ

Rax∈R^d: x∈A

=μ

y∈R^d: R⁻_a¹(y)∈A

, A∈B R^d

.

Then,

δ_a∗μ−μVar=2 sup

A∈B(R^d)

μ(A−a)−μ(A)

=2 sup

A∈B(R^d)

μ◦R⁻_a¹

R_a(A−a)

−μ◦R_a⁻¹(R_aA)

=2 sup

A∈B(R^d)

μ◦R⁻_a¹

R_aA− |a|e₁

−μ◦R⁻_a¹(R_aA)

=2 sup

B∈B(R^d)

μ◦R_a⁻¹

B− |a|e₁

−μ◦R_a⁻¹(B)

=δ_|a|e

1∗

μ◦R_a⁻¹

−μ◦R⁻_a¹

Var.

Proposition 3.3. Under(3.1),there exists a constantC=C(η₀) >0such that sup

|x−y|≤δ

P(x+S_n∈ ·)−P(y+Sn∈ ·)

Var≤ C

√n. (3.3)

Proof. Step 1.It is easy to see that for anya∈R^d and any probability measureμ, μ^∗n◦R_a⁻¹=

μ◦R⁻_a¹_∗n

.

Lemma3.2shows that (3.3) is equivalent to the following estimate sup

|a|≤δ

P

|a|e1+Sa,n∈ ·

−P(Sa,n∈ ·)

Var≤ C

√n. (3.4)

Here,S_a,nis a random walk inR^dwith i.i.d. stepsξa,1, ξa,2, . . .andξa,1∼ν0◦R_a⁻¹. On the other hand, Lemma3.2also shows that

ν0◦R⁻_a¹

∧ δ_|a|e₁∗

ν0◦R⁻_a¹ R^d

=1−1

2ν₀◦R_a⁻¹− δ_|a|e1∗

ν₀◦R⁻_a¹

Var

=1−1

2ν0−(δa∗ν0)

Var

=ν0∧(δa∗ν0) R^d

.

Therefore, (3.1) implies that for anya∈R^d

|ainf|≤δ

ν₀◦R_a⁻¹

∧ δ_|a|e1∗

ν₀◦R_a⁻¹ R^d

= inf

|a|≤δ

ν₀∧(δ_a∗ν₀) R^d

>0. (3.5)

In order to simplify the notation, we useν:=ν₀◦R_a⁻¹andS_n:=S_a,n. With this notation (3.4) becomes

|supa|≤δ

P

|a|e1+Sn∈ ·

−P(Sn∈ ·)

Var≤ C

√n. (3.6)

Step 2.Assume that|a| ∈(0, δ]and setν_|a|=δ_|a|e1∗νandν_−|a|=δ_−|a|e1∗ν. Let(ξ, ξ )∈R^d×R^d be a pair of random variables with the following distribution

P

(ξ, ξ )∈A×D

=

⎧⎪

⎨

⎪⎩

1

2(ν∧ν_−|a|)(A), D=

|a|e1

;

1

2(ν∧ν_|a|)(A), D=

−|a|e1

; ν−¹₂(ν∧ν_−|a|+ν∧ν_|a|)

(A), D= {0};

whereA∈B(R^d)andD∈ {{−|a|e₁},{0},{|a|e₁}}. We see from (3.5) that P

ξ= |a|e₁

=1 2

ν∧(δ_−|a|e1∗ν) R^d

≥1 2 inf

|a|≤δν₀∧(δ_a∗ν₀) R^d

.

By Lemma3.2, P

ξ= −|a|e₁

=1 2

ν∧(δ_|a|e1∗ν) R^d

=1 2

ν∧(δ_−|a|e1∗ν) R^d

=P

ξ= |a|e1

.

It is clear that the distribution ofξ isν. Letξ=ξ+ξ. We claim that the distribution ofξis alsoν. Indeed, for anyA∈B(R^d),

P ξ∈A

=P

ξ− |a|e₁∈A, ξ= −|a|e₁ +P

ξ+ |a|e₁∈A, ξ= |a|e₁

+P(ξ∈A, ξ=0)

=δ_−|a|e1∗(ν∧ν_|a|)(A)

2 +δ_|a|e1∗(ν∧ν_−|a|)(A)

2 +

ν−ν∧ν_−|a|+ν∧ν_|a| 2

(A)

=μ(A),

where we have used thatδ_−|a|e₁ ∗(ν∧ν_|a|)=ν∧ν_−|a| andδ_|a|e₁ ∗(ν∧ν_−|a|)=ν∧ν_|a|. Now we construct the coupling(Sn, S_n)ofSnwith the i.i.d. pairs(ξi, ξ_i), i≥1, where(ξ1, ξ₁)∼(ξ, ξ). Sinceξ_i−ξi=ξ, we know that ξ_i−ξ_iis, for alli≥1, symmetrically distributed, takes only the values−|a|e₁, 0 and|a|e₁. Because of (3.5), we have

P (a):=P

ξ_i¹−ξ_i¹=0

=

ν−1

2(ν∧ν_−|a|+ν∧ν_|a|) R^d

= 1−ν∧ν_−|a| R^d

≤ 1− inf

|a|≤δν0∧(δa∗ν0) R^d

=:γ (δ) <1,

whereξi =(ξ_i¹, ξ_i², . . . , ξ_i^d) andξ_i =(ξ_i¹, ξ_i², . . . , ξ_i^d). Set S^j_n =n

i=1ξ_i^j andS_n^j=n

i=1ξ_i^j. We observe that (S¹−S¹)is a random walk, whose step sizes are−|a|, 0 and|a|with probability¹₂(1−P (a)),P (a)and¹₂(1−P (a)), respectively. SinceS_n^j=S_n^jfor 2≤j≤d, we get

P

|a|e₁+S_n∈ ·

−P(S_n∈ ·)

Var≤2P T^S> n

, (3.7)

where

T^S=inf

k≥1:S_k¹=S_k¹+ |a| .

Step 3.We will now estimate P(T^S> n). Let V₁, V₂, . . .be i.i.d. symmetric random variables, whose common distribution is given by

P(Vi=x)=

⎧⎪

⎨

⎪⎩

1 2

1−P (a)

, x= −|a|;

1 2

1−P (a)

, x= |a|;

P (a), x=0.

DefineZ_n=_n

i=1V_i. We have seen in Step 2 thatT^S=inf{n≥1: Z_n= |a|}. Then, by the reflection principle, P

T^S> n

=P

maxk≤nZ_k<|a|

≤2P

0≤Z_n≤ |a| .

SinceZis the sum of i.i.d. random variables with mean 0 and varianceσ²= |a|²(1−P (a)), we can use the central limit theorem to deduce, for sufficiently large values ofn,

P

T^S> n

=2P

0≤ Zn

|a|√

1−P (a)√

n≤ 1

√1−P (a)√ n

≤2P

0≤ Z_n

|a|√

1−P (a)√

n≤ 1

√1−γ (δ)√ n

≤ C

√2π 1/√

1−γ (δ)√ n 0

e^−x2/2dx

≤C1,γ (δ)

√n .

In the first inequality above we have used the fact that 1−P (a)≥1−γ (δ) >0.Therefore we find from (3.7) for all largen≥1

P

|a|e₁+S_n∈ ·

−P(S_n∈ ·)

Var≤ 2C2,γ (δ)

√n .

Since the right-hand side is bounded by 2, this estimate actually holds for alln≥1.

We can now use Lemma3.2to get P

±|a|e₁+S_n∈ ·

−P(S_n∈ ·)

Var≤2C2,γ (δ)

√n

which immediately yields (3.6), since|a| ≤δwas arbitrary.

We close this section with the proofs of Theorems3.1and1.1.

Proof of Theorem3.1. For anyx, y∈R^d, setk= [^|x−_δ^y|] +1. Pickx₀, x₁, . . . , x_k∈R^dsuch thatx₀=x,x_k=yand

|x_i−x_i−1| ≤δfor 1≤i≤k. By Proposition3.3, P(x+S_n∈ ·)−P(y+S_n∈ ·)

Var≤ k i=1

P(x_i+S_n∈ ·)−P(x_i−1+S_n∈ ·)

Var

≤ C(δ, η₀)(1+ |x−y|)

√n ,

which is what we claimed.

Proof of Theorem1.1. Step 1.Assume first that the Lévy triplet is of the form(0,0, ν)and that the Lévy measure satisfiesλ=ν(R^d) <∞. This means thatX_t is compound Poisson process. We use the notations from Section2. For allx∈R^d,

ν∧(δ_x∗ν) R^d

=λ

ν₀∧(δ_x∗ν₀) R^d

>0.

Therefore, we can apply Proposition2.2and Theorem3.1to get Theorem1.1in this case.

Step 2.If(Xt)t≥0is a general Lévy process with Lévy triplet(b, Q, ν), we splitXt into two independent parts Xt=X_t+X_t,

whereX_t is the compound Poisson process with Lévy triplet(0,0, νε)–νεis defined in (1.2) – andX_t is the Lévy process with triplet(b, Q, ν−νε). Denote byP_t, P_t,P_t(x,dy), P_t(x,dy)the transition semigroups and transition functions of the processesX_t andX_t, respectively. Then,P_t=P_tP_t. Observe thatP_t is a contraction semigroup, i.e.P_tf_∞≤1 wheneverf_∞≤1. Therefore,

P_t(x,·)−P_t(y,·)

Var= sup

f∞≤1

P_tf (x)−P_tf (y)

= sup

f∞≤1

P_tP_tf (x)−P_tP_tf (y)

≤ sup

h∞≤1

P_th(x)−P_th(y)

=P_t(x,·)−P_t(y,·)

Var,

which reduces the general case to the compound Poisson setting considered in the first part.

4. Extensions: The Lindvall–Rogers ‘zero–two’ law

Motivated by Lindvall–Rogers’s zero–two law for random walks [8], Propsotion 1, we present a necessary and suf- ficient condition for the coupling property of a Lévy process. We will add a few simple sufficient criteria in terms of the Lévy measure which are easy to verify. Throughout this section we assume that(X_t)_t≥0 is ad-dimensional Lévy process with Lévy measureν≡0; as usual,X0=0. ByPt(x,·)andPt we denote the transition probability and transition semigroup, respectively.

Theorem 4.1 (Criterion for successful couplings). The following statements are equivalent:

(1) The Lévy process(X_t)_t≥0has the coupling property.

(2) There existst₀>0 such that for any t ≥t₀, the transition probability P_t(x,·)has (with respect to Lebesgue measure)an absolutely continuous component.

In either case,for everyx, y∈R^d,there exists a constantC(x, y) >0such that P_t(x,·)−P_t(y,·)

Var≤C(x, y)

√t , t >0. (4.1)

If the Lévy process has the strong Feller property, i.e. the corresponding semigroup mapsB_b(R^d)intoC_b(R^d), Theorem4.1becomes particularly simple.

Corollary 4.2. Suppose that there exists somet₀>0such that the semigroupP_t0mapsB_b(R^d)intoC_b(R^d).Then,the Lévy processX_t has the coupling property.In particular,every Lévy process which enjoys the strong Feller property has the coupling property.

Proof. By assumptionP_t0 is a convolution operator which mapsB_b(R^d)intoC_b(R^d). Due to a result by Hawkes, cf.

[5] or [6], Lemma 4.8.20,P_t0 and allP_t witht≥t₀are of the formP_t(x)=p_t∗f (x), wherep_t(x)is the transition density of the process. Therefore condition (2) of Theorem4.1is satisfied.

Now, we turn to the proof of Theorem4.1.

Proof of Theorem4.1. As mentioned in Section1, the coupling property is equivalent to (1.1). Observe that μ1P_t−μ2P_tVar≤ μ1P_t−P_tVar+ μ2P_t−P_tVar

≤

δ_x∗P_t−δ₀∗P_tVarμ₁(dx)+

δ_x∗P_t−δ₀∗P_tVarμ₂(dx).

Thus, if

tlim→∞δ_x∗P_t−δ₀∗P_tVar=0 forx∈R^d, (4.2)

then we can use the dominated convergence theorem to see that (1.1) holds. Therefore, the assertions (1.1) and (4.2) are equivalent.

Sinceδ_x∗P_t−δ₀∗P_tVaris decreasing int, we see that (4.2) is also equivalent to

nlim→∞δ_x∗P_n−δ₀∗P_nVar=0 forx∈R^d. (4.3)

Letξ1, ξ2, . . .be i.i.d. random variables onR^dwithξ1∼μ1:=P(X1∈ ·)and setSn=n

i=1ξiforn≥1 andS0=0.

Since the increments of a Lévy process are independent and stationary, (4.3) is the same as

nlim→∞P(x+Sn∈ ·)−P(Sn∈ ·)

Var=0 forx∈R^d. (4.4)

According to [8], Remark (ii), pp. 124–125, or [10], Chapter 3, Section 3, Theorem 3.9, (4.4) holds if, and only if, μ₁is spread out, i.e. for somem≥1,μ^∗₁^m=P_m(0,·):=P(X_m∈ ·)has an absolutely continuous component. Since the semigroup of a Lévy process is a convolution semigroup, it is easy to see that for every t ≥m, the transition probabilityP_t(x,·)has an absolutely continuous part. Combining all the assertions above, we have proved that the statements (1) and (2) are equivalent.

Moreover, the arguments used in the proof of Proposition3.3together with [8], Proposition 1, show that P(x+Sn∈ ·)−P(Sn∈ ·)

Var=^O n^−1/2

asn→ ∞

whenever the random walkSnhas the coupling property. Therefore, (4.1) follows from the arguments used in the first part of the proof, in particular sincet→ δx∗Pt−δ0∗PtVaris decreasing and

P(x+Sn∈ ·)−P(Sn∈ ·)

Var= δx∗Pn−δ0∗PnVar.

Let us finally derive some sufficient conditions in terms of the Lévy measure, which extend Theorem1.1and [14], Theorem 3.1.

Theorem 4.3 (Sufficient criteria for successful couplings). Let(Xt)t≥0be a Lévy process onR^dwith Lévy measure ν≡0and defineε >0as in(1.2),i.e.forB∈B(R^d)

ν_ε(B)= ν(B), ν

R^d

<∞;

ν

z∈B: |z| ≥ε , ν

R^d

= ∞.

If there exists someε >0such that one of the following conditions is satisfied (1) For somel≥1,ν_ε^∗lhas an absolutely continuous component.