Spatial regularity of semigroups generated by Lévy type operators
aa r X i v : . [ m a t h . P R ] A p r SPATIAL REGULARITY OF SEMIGROUPS GENERATED BYLÉVY TYPE OPERATORS
MINGJIE LIANG JIAN WANG
Abstract.
We apply the probabilistic coupling approach to establish the spatialregularity of semigroups associated with Lévy type operators, by assuming thatthe martingale problem of Lévy type operators is well posed. In particular, wecan prove the Lipschitz continuity of the semigroups under Hölder continuity ofcoefficients, even when the Lévy kernel corresponding to Lévy type operators issingular.
Keywords:
Lévy type operator; coupling; spatial regularity; martingale problem
MSC 2010: Introduction and Main Results
We consider the following Lévy type operator(1.1) Lf ( x ) = Z (cid:16) f ( x + z ) − f ( x ) − h∇ f ( x ) , z i B (0 , ( z ) (cid:17) c ( x, z ) ν ( dz ) , where ν is a Lévy measure, i.e., ν ( { } ) = 0 and R (1 ∧ | z | ) ν ( dz ) < ∞ , and ( x, z ) c ( x, z ) is a continuous function such that c ( x, z ) ∈ ( c ∗ , c ∗ ) for some constants
College of Mathematics and Informatics, Fujian Normal University, 350007 Fuzhou,P.R. China. [email protected] . J. Wang:
College of Mathematics and Informatics & Fujian Key Laboratory of MathematicalAnalysis and Applications (FJKLMAA), Fujian Normal University, 350007 Fuzhou, P.R. China. [email protected] . The aim of this paper is to establish the spatial regularity of semigroups associ-ated with the operator L given by (1.1). We will adopt the probabilistic couplingapproach, which recently has been extensively studied in [20, 21, 22, 33]. To studyanalytic properties for Lévy type operators via probabilistic method, as one of thestanding assumptions, the existence of a strong Markov process associated withLévy type operators was assumed. For example, see [5, 6, 7, 21, 33], where thestrong Markov property played an important role. Similarly, throughout this pa-per we shall assume that the martingale problem for ( L, C b ( R d )) is well posed (seeSection 2.2 below for its definition). In particular, there is a strong Markov process X := (( X t ) t ≥ , ( P x ) x ∈ R d ) , whose generator is just the Lévy type operator L . Below,for any f ∈ B b ( R d ) (the set of bounded measurable functions on R d ), let P t f ( x ) = E x f ( X t ) , x ∈ R d , t ≥ be the semigroups corresponding to the operator L .To state the contribution of our paper, we consider the following Lévy measurepartly motivated by [14]. (Actually, the paper [14] treated Lévy kernel case butwith a fixed order, see [14, (1.2)].) Suppose that there are constants c , c > and < α ≤ α < such that(1.3) c | z | d + α V ξ ( z ) dz ≤ ν ( dz ) ≤ c | z | d + α dz, where(1.4) V ξ = { z ∈ R d : | z | ≤ and h z, ξ i ≥ δ | z |} with ξ ∈ S d − and constant δ ∈ (0 , . Theorem 1.1.
Under the assumption (1.3) , we define for any r > , w ( r ) = sup x,y ∈ R d : | x − y | = r Z {| z |≤ } | z | | c ( x, z ) − c ( y, z ) | ν ( dz ) , α ∈ [1 , x,y ∈ R d : | x − y | = r Z {| z |≤ } | z || c ( x, z ) − c ( y, z ) | ν ( dz ) , α ∈ (0 , . Then the following statements hold. (1) If α ∈ (1 , and (1.5) lim r → w ( r ) r α − log θ (1 /r ) = 0 for some θ > , then there exists a constant C > such that for all f ∈ B b ( R d ) and t > , sup x = y | P t f ( x ) − P t f ( y ) || x − y | ≤ C k f k ∞ ( t ∧ − /α (cid:20) log (1+ θ ) /α (cid:18) t ∧ e (cid:19) (cid:21) . (2) If α ∈ [1 , and (1.6) lim r → w ( r ) r α − log(1 /r ) = 0 , then, for any θ > { α =1 } , there exists a constant C > such that for all f ∈ B b ( R d ) and t > , sup x = y | P t f ( x ) − P t f ( y ) || x − y | (cid:12)(cid:12) log | x − y | (cid:12)(cid:12) θ ≤ C k f k ∞ ( t ∧ − /α (cid:20) log − θ +(1 /α ) (cid:18) t ∧ e (cid:19) (cid:21) . PATIAL REGULARITY OF LÉVY TYPE OPERATORS 3 (3) If α ∈ (0 , and (1.7) lim r → w ( r ) r α − = 0 , then for any θ ∈ (0 , α ) , there exists a constant C > such that for all f ∈ B b ( R d ) and t > , sup x = y | P t f ( x ) − P t f ( y ) || x − y | θ ≤ C k f k ∞ ( t ∧ − θ/α . Note that, the continuity assumptions on w ( r ) in the theorem above are weakerthan those on the function x c ( x, z ) (uniformly with respect to z ). The latter wasused to study the Hölder continuity of solutions to a class of second order non-linearelliptic integro-differential equations in [3, Theorem 3.1]. See also [2, Section 4.2] formore details. We also mention that similar assumptions (but a little stronger) on w ( r ) have been adopted to study the pathwise uniqueness of solution to stochasticdifferential equations driven by the pure jump process ( J t ) t ≥ of the form (1.2), see[17, Theorem 3.1], [34, Theorem 1.2] or [35, Theorem 2.1].1.1. Applications.
A Borel measurable function u on R d is called harmonic withrespect to L , if P t u ( x ) = u ( x ) for all x ∈ R d and t > . The following result is adirect consequence of Theorem 1.1. Corollary 1.2.
Under the setting of Theorem . , we have the following three state-ments. (1) If assumptions in the first assertion of Theorem . hold, then any boundedmeasurable function u is Lipschitz continuous. (2) If assumptions in the second assertion of Theorem . hold, then, for any θ > { α =1 } , any bounded measurable function u is r log θ | r | -order continuous. (3) If assumptions in the third assertion of Theorem . hold, then for any ε > ,any bounded measurable function u is ( α − ε ) -Hölder continuous. Next, we apply Theorem 1.1 to prove the following Liouville theorem. See [29,Theorem 2.1] for the related discussion for general symmetric α -stable operators. Corollary 1.3.
Consider the setting of Theorem . , and let u be a harmonic func-tion respect to the operator L . Suppose that there exists a constant c > such thatfor all r ≥ , k u k L ∞ ( B r (0) ,dx ) ≤ cr β , where B r (0) = { z ∈ R d : | z | < r } and ≤ β < α . If assumptions in any assertionof Theorem . hold, then u is a polynomial of degree at most ⌊ β ⌋ , where ⌊ x ⌋ denotesthe integer part of x . Perturbation result.
To further illustrate the power of the coupling ap-proach, we next give a perturbation result corresponding to Theorem 1.1. Below wewill consider the following Lévy type operator(1.8) L ∗ f ( x ) = Z (cid:16) f ( x + z ) − f ( x ) −h∇ f ( x ) , z i B (0 , ( z ) (cid:17) ( c ( x, z ) ν ( dz ) + µ ( x, dz )) , where c ( x, z ) and ν ( dz ) are assumed same as those in the beginning of this section,and µ ( x, dz ) is a Lévy kernel on R d satisfying that sup x ∈ R d Z (1 ∧ | z | ) µ ( x, dz ) < ∞ MINGJIE LIANG JIAN WANG and for any h ∈ C b ( R d ) , the function x Z R d h ( z ) | z | | z | µ ( x, dz ) is continuous. We assume that the martingale problem for ( L ∗ , C b ( R d )) is well-posed. Clearly, the operator L ∗ is just the operator L perturbed by the Lévy kernel µ ( x, dz ) . We emphasize that, we do not assume that the Lévy kernel µ ( x, dz ) isabsolutely continuous with respect to the Lebesgue measure.For any r > , we define w µ ( r ) = sup x,y ∈ R d : | x − y | = r Z {| z |≤ } | z | | µ ( x, dz ) − µ ( y, dz ) | ; if additionally(1.9) sup x ∈ R d Z {| z |≤ } | z | µ ( x, dz ) < ∞ , then w µ ( r ) above is replaced by w µ ( r ) = sup x,y ∈ R d : | x − y | = r Z {| z |≤ } | z || µ ( x, dz ) − µ ( y, dz ) | . Theorem 1.4.
Under assumptions of Theorem . and notations above, define w ∗ ( r ) = w ( r ) + w µ ( r ) , r > . Then (1) the first assertion of Theorem . holds, if α ∈ (1 , and (1.5) holds with w ∗ ( r ) replacing w ( r ) . (2) the second assertion of Theorem . holds, if α ∈ [1 , and (1.6) holds with w ∗ ( r ) replacing w ( r ) . (3) the third assertion of Theorem . holds, if α ∈ (0 , , (1.9) is satisfied, and (1.7) holds with w ∗ ( r ) replacing w ( r ) . The remainder of this paper is arranged as follows. The next section is devoted tothe construction of a new coupling operator for the Lévy type operator L given by(1.1), and the existence of coupling process on R d associated with the constructedcoupling operator. In Section 3, we first present some preliminary estimates forthe coupling operator, and then give general results for the regularity of associatedsemigroups, by making full use of the coupling operator and the coupling processconstructed in Section 2. Finally, the proofs of all results above are given in the lastpart of Section 3.2. Coupling operator and coupling process for Lévy type operators
This section is split into two parts. We first present a new coupling operator forthe Lévy type operator L given by (1.1), and then prove the existence of couplingprocess on R d associated with the constructed coupling operator. PATIAL REGULARITY OF LÉVY TYPE OPERATORS 5
Coupling operator for Lévy type operator.
The construction below isheavily based on the refined basic coupling for stochastic differential equations drivenby additive Lévy noises first introduced in [22]. See [20] for the recent study onstochastic differential equations driven by multiplicative Lévy noises. However, theLévy type operator L given by (1.1) essentially is different from stochastic differentialequations with jumps, and the main difficulty here is due to that the coefficient c ( x, z ) in the operator L depends on both space variables, which requires a new ideafor the construction of a coupling operator.For any x, y, z, u ∈ R d , define c ( x, y, u, z ) = c ( x, z ) ∧ c ( y, z ) ∧ c ( x, z − u ) ∧ c ( y, z − u ) ,ν u ( dz ) = ν ∧ ( δ u ∗ ν )( dz ) ,µ x,y,u ( dz ) = c ( x, y, u, z ) ν u ( dz ) , ˜ ν x,y ( dz ) = ( c ( x, z ) ∧ c ( y, z )) ν ( dz ) . (2.1)In particular, for any x, y, z, u ∈ R d , c ( x, y, u, z ) = c ( x, y, − u, z − u ) . The function c ( x, y, u, z ) and the kernel µ x,y,u ( dz ) are crucial in the construction ofthe coupling operator below.For any x , y ∈ R d and κ > , let ( x − y ) κ = (cid:18) ∧ κ | x − y | (cid:19) ( x − y ) . We consider the jump system as follows ( x, y ) −→ ( x + z, y + z + ( x − y ) κ ) , µ x,y, ( y − x ) κ ( dz );( x + z, y + z + ( y − x ) κ ) , µ x,y, ( x − y ) κ ( dz );( x + z, y + z ) , (cid:0) ˜ ν x,y − µ x,y, ( y − x ) κ − µ x,y, ( x − y ) κ (cid:1) ( dz );( x + z, y ) , ˜ c ( x, y, z ) ν ( dz );( x, y + z ) , ˜ c ( y, x, z ) ν ( dz ) , where ˜ c ( x, y, z ) = c ( x, z ) − c ( x, z ) ∧ c ( y, z ) . MINGJIE LIANG JIAN WANG
Furthermore, for any h ∈ C b ( R d ) and x, y ∈ R d , we define the following operatorassociated with the jumping system above e Lh ( x, y )= 12 Z (cid:16) h ( x + z, y + z + ( x − y ) κ ) − h ( x, y ) − h∇ x h ( x, y ) , z i {| z |≤ } − h∇ y h ( x, y ) , z + ( x − y ) κ i {| z +( x − y ) κ |≤ } (cid:17) µ x,y, ( y − x ) κ ( dz )+ 12 Z (cid:16) h ( x + z, y + z + ( y − x ) κ ) − h ( x, y ) − h∇ x h ( x, y ) , z i {| z |≤ } − h∇ y h ( x, y ) , z + ( y − x ) κ i {| z +( y − x ) κ |≤ } (cid:17) µ x,y, ( x − y ) κ ( dz )+ Z (cid:16) h ( x + z, y + z ) − h ( x, y ) − h∇ x h ( x, y ) , z i {| z |≤ } − h∇ y h ( x, y ) , z i {| z |≤ } (cid:17) (cid:16) ˜ ν x,y − µ x,y, ( x − y ) κ − µ x,y, ( y − x ) κ (cid:17) ( dz )+ Z (cid:16) h ( x + z, y ) − h ( x, y ) − h∇ x h ( x, y ) , z i {| z |≤ } (cid:17) ˜ c ( x, y, z ) ν ( dz )+ Z (cid:16) h ( x, y + z ) − h ( x, y ) − h∇ y h ( x, y ) , z i {| z |≤ } (cid:17) ˜ c ( y, x, z ) ν ( dz ) , (2.2)where ∇ x h ( x, y ) and ∇ y h ( x, y ) are defined as the gradient of h ( x, y ) with respect to x ∈ R d and y ∈ R d respectively.We will claim that Proposition 2.1.
The operator e L defined by (2.2) is indeed a coupling operator of L ; that is, for any f, g ∈ C b ( R d ) , letting h ( x, y ) = f ( x ) + g ( y ) for all x, y ∈ R d , itholds that e Lh ( x, y ) = Lf ( x ) + Lg ( y ) . Proof.
The proof is similar to that in [22, Section 2.1] with slight modifications, andfor the sake of completeness we present it here. Let h ( x, y ) = g ( y ) for any x, y ∈ R d ,where g ∈ C b ( R d ) . Then, according to (2.2), e Lh ( x, y ) = 12 Z (cid:16) g ( y + z + ( x − y ) κ ) − g ( y ) − h∇ g ( y ) , z + ( x − y ) κ i {| z +( x − y ) κ |≤ } (cid:17) µ x,y, ( y − x ) κ ( dz )+ 12 Z (cid:16) g ( y + z + ( y − x ) κ ) − g ( y ) − h∇ g ( y ) , z + ( y − x ) κ i {| z +( y − x ) κ |≤ } (cid:17) µ x,y, ( x − y ) κ ( dz )+ Z (cid:16) g ( y + z ) − g ( y ) − h∇ g ( y ) , z i {| z |≤ } (cid:17) × (cid:16) ˜ ν x,y − µ x,y, ( x − y ) κ − µ x,y, ( y − x ) κ (cid:17) ( dz )+ Z (cid:16) g ( y + z ) − g ( y ) − h∇ g ( y ) , z i {| z |≤ } (cid:17) ˜ c ( y, x, z ) ν ( dz ) . PATIAL REGULARITY OF LÉVY TYPE OPERATORS 7
Changing the variables z + ( x − y ) κ → u and z + ( y − x ) κ → u respectively andusing Lemma 2.2 below in the first two terms of the right hand side of the equalityabove lead to e Lh ( x, y )= 12 Z (cid:16) g ( y + u ) − g ( y ) − h∇ g ( y ) , u i {| u |≤ } (cid:17) × c ( x, y, ( y − x ) κ , u − ( x − y ) κ ) ν ( y − x ) κ ( d ( u − ( x − y ) κ ))+ 12 Z (cid:16) g ( y + u ) − g ( y ) − h∇ g ( y ) , u i {| u |≤ } (cid:17) × c ( x, y, ( x − y ) κ , u − ( y − x ) κ ) ν ( x − y ) κ ( d ( u − ( y − x ) κ ))+ Z (cid:16) g ( y + z ) − g ( y ) − h∇ g ( y ) , z i {| z |≤ } (cid:17) × (cid:16) ˜ ν x,y − µ x,y, ( x − y ) κ − µ x,y, ( y − x ) κ (cid:17) ( dz )+ Z (cid:16) g ( y + z ) − g ( y ) − h∇ g ( y ) , z i {| z |≤ } (cid:17) ˜ c ( y, x, z ) ν ( dz )= 12 Z (cid:16) g ( y + u ) − g ( y ) − h∇ g ( y ) , u i {| u |≤ } (cid:17) c ( x, y, ( x − y ) κ , u ) ν ( x − y ) κ ( du )+ 12 Z (cid:16) g ( y + u ) − g ( y ) − h∇ g ( y ) , u i {| u |≤ } (cid:17) c ( x, y, ( y − x ) κ , u ) ν ( y − x ) κ ( du )+ Z (cid:16) g ( y + z ) − g ( y ) − h∇ g ( y ) , z i {| z |≤ } (cid:17) × (cid:16) ˜ ν x,y − µ x,y, ( x − y ) κ − µ x,y, ( y − x ) κ (cid:17) ( dz )+ Z (cid:16) g ( y + z ) − g ( y ) − h∇ g ( y ) , z i {| z |≤ } (cid:17) ˜ c ( y, x, z ) ν ( dz )= 12 Z (cid:16) g ( y + u ) − g ( y ) − h∇ g ( y ) , u i {| u |≤ } (cid:17) µ x,y, ( x − y ) κ ( du )+ 12 Z (cid:16) g ( y + u ) − g ( y ) − h∇ g ( y ) , u i {| u |≤ } (cid:17) µ x,y, ( y − x ) κ ( du )+ Z (cid:16) g ( y + z ) − g ( y ) − h∇ g ( y ) , z i {| z |≤ } (cid:17) × (cid:16) ˜ ν x,y − µ x,y, ( x − y ) κ − µ x,y, ( y − x ) κ (cid:17) ( dz )+ Z (cid:16) g ( y + z ) − g ( y ) − h∇ g ( y ) , z i {| z |≤ } (cid:17) ˜ c ( y, x, z ) ν ( dz )= Lg ( y ) . On the other hand, if h ( x, y ) = f ( x ) for any x, y ∈ R d and any f ∈ C b ( R d ) , thenwe can easily see that e Lh ( x, y ) = Lf ( x ) . Combining with both conclusions aboveyields that the operator e L defined by (2.2) is a coupling operator of L . (cid:3) The following lemma has been used in the proof above.
MINGJIE LIANG JIAN WANG
Lemma 2.2.
For any z ∈ R d with z = 0 , ν z ( du ) is a finite measure on ( R d , B ( R d )) such that µ z ( d ( u + z )) = µ − z ( du ) In particular, µ z ( R d ) = µ − z ( R d ) . Proof.
The proof has been given in [22, Remark 2.1 and Corollary 6.2]. We omit ithere. (cid:3)
Coupling process for Lévy type operators.
The purpose of this part is toconstruct a coupling process associated with the coupling operator e L given by (2.2).Though the following argument is standard (see [21, Section 2] or [33, Section 2.2]for example), we still would like to present some details here.Let D ([0 , ∞ ); R d ) be the space of right continuous R d -valued functions having leftlimits on [0 , ∞ ) and equipped with the Skorokhod topology. For t ≥ , denote by X t the projection coordinate map on D ([0 , ∞ ); R d ) . A probability measure P x on theSkorokhod space D ([0 , ∞ ); R d ) is said to be a solution to the martingale problem for ( L, C b ( R d )) with initial value x ∈ R d , if P x ( X = x ) = 1 and, for every f ∈ C b ( R d ) , (cid:26) f ( X t ) − f ( x ) − Z t Lf ( X s ) ds, t ≥ (cid:27) is a P x -martingale. The martingale problem for ( L, C b ( R d )) is said to be well-posed if it has a unique solution for every initial value x ∈ R d . The definitionsabove are well adapted to the martingale problem for ( e L, C b ( R d )) with necessarymodifications. We can refer to [1, 4, 8, 13, 15, 17, 19, 24, 25, 26, 27, 28, 31] and thereferences therein for more details about martingale problem for non-local operators.In order to prove the existence of the martingale problem for the coupling operator ( e L, C b ( R d )) , we will write the coupling operator e L into the form as the expressionof Lévy type operator on C b ( R d ) . For any x, y ∈ R d , and A ∈ B ( R d ) , set e ν ( x, y, A ) := 12 Z { ( z,z − ( x − y ) κ ∈ A } µ x,y, ( y − x ) κ ( dz ) + 12 Z { ( z,z +( x − y ) κ ) ∈ A } µ x,y, ( x − y ) κ ( dz )+ Z { ( z,z ) ∈ A } (cid:16) ˜ ν x,y − µ x,y, ( y − x ) κ − µ x,y, ( x − y ) κ (cid:17) ( dz )+ Z { ( z, ∈ A } ˜ c ( x, y, z ) ν ( dz ) + Z { (0 ,z ) ∈ A } ˜ c ( y, x, z ) ν ( dz ) . Then, for any x, y ∈ R d and f ∈ C b ( R d ) , e Lf ( x, y ) = Z R d × R d (cid:16) f (cid:0) ( x, y ) + ( u , u ) (cid:1) − f ( x, y ) − h∇ x f ( x, y ) , u i {| u |≤ } − h∇ y f ( x, y ) , u i {| u |≤ } (cid:17) e ν ( x, y, du , du ) . For any h ∈ C b ( R d ) and x , y ∈ R d , we have Z R d h ( u ) | u | | u | e ν ( x, y, du )= Z R d × R d h (( u , u )) | u | + | u | | u | + | u | e ν ( x, y, du , du ) PATIAL REGULARITY OF LÉVY TYPE OPERATORS 9 = 12 Z R d h (( z, z + ( x − y ) κ )) | z | + | z + ( x − y ) κ | | z | + | z + ( x − y ) κ | µ x,y, ( y − x ) κ ( dz )+ 12 Z R d h (( z, z − ( x − y ) κ )) | z | + | z − ( x − y ) κ | | z | + | z − ( x − y ) κ | µ x,y, ( x − y ) κ ( dz )+ Z R d h (( z, z )) | z | + | z | | z | + | z | (cid:16) ˜ ν x,y − µ x,y, ( x − y ) κ − µ x,y, ( y − x ) κ (cid:17) ( dz )+ Z R d h (( z, | z | | z | (cid:0) c ( x, z ) − c ( x, z ) ∧ c ( y, z ) (cid:1) ν ( dz )+ Z R d h ((0 , z )) | z | | z | (cid:0) c ( y, z ) − c ( x, z ) ∧ c ( y, z ) (cid:1) ν ( dz ) . Since c ( x, z ) is bounded and ( x, z ) c ( x, z ) is continuous, the function ( x, y ) R R d h ( u ) | u | | u | e ν ( x, y, du ) is continuous too. Therefore, by [31, Theorem 2.2], thereis a solution to the martingale problem for ( e L, C b ( R d )) , i.e., there are a probabilityspace ( e Ω , f F , ( f F t ) t ≥ , e P ) and an R d := ( R ∪ {∞} ) d -valued process ( e X t ) t ≥ :=( X ′ t , X ′′ t ) t ≥ such that ( e X t ) t ≥ is ( f F t ) t ≥ -progressively measurable, and for every f ∈ C b ( R d ) , (cid:26) f ( e X t ∧ e ζ ) − Z t ∧ e ζ e Lf ( e X s ) ds, t ≥ (cid:27) is an ( f F t ) t ≥ -local martingale, where e ζ is the explosion time of ( e X t ) t ≥ , i.e., e ζ = lim n →∞ inf n t ≥ | X ′ t | + | X ′′ t | ≥ n o . By Proposition 2.1, e L is the coupling operator of L , and so both distributions of theprocesses ( X ′ t ) t ≥ and ( X ′′ t ) t ≥ are solutions to the martingale problem of L . Sincewe assume that the martingale problem for ( L, C b ( R d )) is well-posed, the processes ( X ′ t ) t ≥ and ( X ′′ t ) t ≥ are non-explosive, and so we have e ζ = ∞ a.e. That is, thecoupling operator e L generates a non-explosive process ( e X t ) t ≥ .Let T be the coupling time of ( X ′ t ) t ≥ and ( X ′′ t ) t ≥ , i.e., T = inf { t ≥ X ′ t = X ′′ t } . Then T is an ( f F t ) t ≥ -stopping time. Construct a new process ( Y ′ t ) t ≥ as follows Y ′ t = ( X ′′ t , t < T ; X ′ t , t ≥ T. We can verify that ( Y ′ t ) t ≥ is a solution to the martingale problem of L , see [33,Section 2.2]. Since the martingale problem for the operator L is well posed, ( Y ′ t ) t ≥ and ( X ′′ t ) t ≥ are equal in the distribution. Therefore, we conclude that ( X ′ t , Y ′ t ) t ≥ isalso a non-explosive coupling process of ( X t ) t ≥ such that X ′ t = Y ′ t for any t ≥ T andthe generator of ( X ′ t , Y ′ t ) t ≥ before the coupling time T is just the coupling operator e L . 3. Coupling approach for regularity of semigroups
Preliminary calculations.
In the following, we assume that κ ∈ (0 , . Let e L be the coupling operator given above. We will estimate e Lf ( | x − y | ) for any ≤ f ∈ C b ([0 , ∞ )) ∩ C ((0 , ∞ )) such that f (0) = 0 , and f ′ ≥ and f ′′ ≤ on (0 , . For any h ∈ C b ( R d ) , define e L R h ( x, y ) = Z (cid:16) h ( x + z, y ) − h ( x, y ) − h∇ x h ( x, y ) , z i {| z |≤ } (cid:17) ˜ c ( x, y, z ) ν ( dz )+ Z (cid:16) h ( x, y + z ) − h ( x, y ) − h∇ y h ( x, y ) , z i {| z |≤ } (cid:17) ˜ c ( y, x, z ) ν ( dz ) . Set e L C := e L − e L R . Note that, if we define the following operator L C f ( x ) := Z (cid:0) f ( x + z ) − f ( x ) − h∇ f ( x ) , z i B (0 , ( z ) (cid:1) ν x,y ( dz )= Z (cid:0) f ( x + z ) − f ( x ) − h∇ f ( x ) , z i B (0 , ( z ) (cid:1) ( c ( x, z ) ∧ c ( y, z )) ν ( dz ) , then, following the proof of Proposition 2.1, we can see that e L C is a coupling operatorof L C .First, according to Lemma 2.2, ν ( y − x ) κ ( dz ) is a finite measure on R d for any x, y ∈ R d with x = y . Then, for any f ∈ C b ([0 , ∞ )) ∩ C ((0 , ∞ )) and x, y ∈ R d with x = y , Z (cid:0) h∇ x f ( | x − y | ) , z i {| z |≤ } + h∇ y f ( | x − y | ) , z + ( x − y ) κ i {| z +( x − y ) κ |≤ } (cid:1) µ x,y, ( y − x ) κ ( dz )= f ′ ( | x − y | ) | x − y | (cid:18) Z {| z |≤ } h x − y, z i µ x,y, ( y − x ) κ ( dz ) − Z {| z +( x − y ) κ |≤ } h x − y, z + ( x − y ) κ i µ x,y, ( y − x ) κ ( dz ) (cid:19) = f ′ ( | x − y | ) | x − y | (cid:18) Z {| z |≤ } h x − y, z i µ x,y, ( y − x ) κ ( dz ) − Z {| z |≤ } h x − y, z i µ x,y, ( x − y ) κ ( dz ) (cid:19) , where in the last equality we used the fact that c ( x, y, ( y − x ) κ , u − ( x − y ) κ ) = c ( x, y, ( x − y ) κ , u ) , x, y, u ∈ R d and Lemma 2.2. Similarly, it holds that Z (cid:0) h∇ x f ( | x − y | ) , z i {| z |≤ } + h∇ y f ( | x − y | ) , z + ( x − y ) κ i {| z +( x − y ) κ |≤ } (cid:1) µ x,y, ( x − y ) κ ( dz )= f ′ ( | x − y | ) | x − y | (cid:18) Z {| z |≤ } h x − y, z i µ x,y, ( x − y ) κ ( dz ) − Z {| z |≤ } h x − y, z i µ x,y, ( y − x ) κ ( dz ) (cid:19) . PATIAL REGULARITY OF LÉVY TYPE OPERATORS 11
Therefore, for any x , y ∈ R d with x = y , e L C f ( | x − y | ) = 12 µ x,y, ( x − y ) κ ( R d ) (cid:0) f (cid:0) | x − y | + κ ∧ | x − y | (cid:1) − f ( | x − y | ) (cid:1) + 12 µ x,y, ( y − x ) κ ( R d ) (cid:0) f (cid:0) | x − y | − κ ∧ | x − y | (cid:1) − f ( | x − y | ) (cid:1) = 12 µ x,y, ( x − y ) κ ( R d ) h f (cid:0) | x − y | + κ ∧ | x − y | (cid:1) + f (cid:0) | x − y | − κ ∧ | x − y | (cid:1) − f ( | x − y | ) i , where in the last equality we have used the fact that µ x,y, ( x − y ) κ ( R d ) = µ x,y, ( y − x ) κ ( R d ) . Next, we assume that f ≥ with f (0) = 0 on [0 , ∞ ) , and f ′ ≥ and f ′′ ≤ on (0 , . Let ε ∈ (0 , κ ] . Then, for any ε ∈ (0 , ε ] and any x, y ∈ R d with | x − y | ≤ ε ,we have(3.1) e L C f ( | x − y | ) ≤ J ( | x − y | ) (cid:16) f (2 | x − y | ) − f ( | x − y | ) (cid:17) , where in the inequality above(3.2) J ( r ) := inf x,y ∈ R d : | x − y | = r µ x,y, ( x − y ) ( R d ) and we have used the fact that f (2 r ) = f ( r ) + Z rr f ′ ( s ) ds = f ( r ) + Z r f ′ ( s + r ) ds ≤ f ( r ) + Z r f ′ ( s ) ds = 2 f ( r ) for any r ∈ (0 , ε ] .We will give estimates for e L R f ( | x − y | ) . For any f ∈ C b ([0 , ∞ )) ∩ C ((0 , ∞ )) with f ≥ , f ′ ≥ and f ′′ ≤ on (0 , . Then, for any x , y ∈ R d with x = y , e L R f ( | x − y | )= Z (cid:18) f ( | x − y + z | ) − f ( | x − y | ) − f ′ ( | x − y | ) | x − y | h x − y, z i {| z |≤ } (cid:19) × ( c ( x, z ) − c ( x, z ) ∧ c ( y, z )) ν ( dz )+ Z (cid:18) f ( | x − y − z | ) − f ( | x − y | ) + f ′ ( | x − y | ) | x − y | h x − y, z i {| z |≤ } (cid:19) × ( c ( y, z ) − c ( x, z ) ∧ c ( y, z )) ν ( dz )= Z {| z |≤ } (cid:18) f ( | x − y + z | ) − f ( | x − y | ) − f ′ ( | x − y | ) | x − y | h x − y, z i (cid:19) × ( c ( x, z ) − c ( x, z ) ∧ c ( y, z )) ν ( dz )+ Z {| z |≤ } (cid:18) f ( | x − y − z | ) − f ( | x − y | ) + f ′ ( | x − y | ) | x − y | h x − y, z i (cid:19) × ( c ( y, z ) − c ( x, z ) ∧ c ( y, z )) ν ( dz )+ Z {| z | > } (cid:18) f ( | x − y + z | ) − f ( | x − y | ) (cid:19) ( c ( x, z ) − c ( x, z ) ∧ c ( y, z )) ν ( dz )+ Z {| z | > } (cid:18) f ( | x − y − z | ) − f ( | x − y | ) (cid:19) ( c ( y, z ) − c ( x, z ) ∧ c ( y, z )) ν ( dz ) . We further consider the following two cases. (i) Since for any a, b ∈ (0 , , f ( b ) − f ( a ) ≤ f ′ ( a )( b − a ) , we have that for any x, y, z ∈ R d with < | x − y | ≤ ε and | z | ≤ , f ( | x − y + z | ) − f ( | x − y | ) − f ′ ( | x − y | ) | x − y | h x − y, z i≤ f ′ ( | x − y | ) | x − y | (cid:16) | x − y + z || x − y | − | x − y | − h x − y, z i (cid:17) ≤ f ′ ( | x − y | ) | x − y | | z | , where the last inequality follows from the fact that h x − y, z i = 12 (cid:0) | x − y + z | − | x − y | − | z | (cid:1) , x, y, z ∈ R d . This yields that for any x, y ∈ R d with < | x − y | ≤ ε , e L R f ( | x − y | ) ≤ (cid:18) Z {| z |≤ } | c ( x, z ) − c ( y, z ) || z | ν ( dz ) (cid:19) f ′ ( | x − y | ) | x − y | + 2 (cid:18) Z {| z | > } ν ( dz ) (cid:19)(cid:20) sup x,z ∈ R d : | z | > c ( x, z ) (cid:21) k f k ∞ . (3.3)(ii) If(3.4) Z {| z |≤ } | z | ν ( dz ) < ∞ , then for any x, y, z ∈ R d with < | x − y | ≤ ε and | z | ≤ , f ( | x − y + z | ) − f ( | x − y | ) − f ′ ( | x − y | ) | x − y | h x − y, z i ≤ f ′ ( | x − y | ) | z | . Following the same argument as that in (i), we can arrive at for any x, y ∈ R d with < | x − y | ≤ ε e L R f ( | x − y | ) ≤ (cid:18) Z {| z |≤ } | c ( x, z ) − c ( y, z ) || z | ν ( dz ) (cid:19) f ′ ( | x − y | )+ 2 (cid:18) Z {| z | > } ν ( dz ) (cid:19)(cid:20) sup x,z ∈ R d : | z | > c ( x, z ) (cid:21) k f k ∞ . (3.5)Combining all the estimates above, we can get that Proposition 3.1.
Let ≤ f ∈ C b ([0 , ∞ )) ∩ C ((0 , ∞ )) such that f (0) = 0 , and f ′ ≥ and f ′′ ≤ on (0 , . Let < ε ≤ κ ≤ and J ( r ) be defined by (3.2) . Then,for any x, y ∈ R d with < | x − y | ≤ ε , (1) it holds that e Lf ( | x − y | ) ≤ J ( | x − y | ) (cid:16) f (2 | x − y | ) − f ( | x − y | ) (cid:17) + (cid:18) Z {| z |≤ } | c ( x, z ) − c ( y, z ) || z | ν ( dz ) (cid:19) f ′ ( | x − y | ) | x − y | PATIAL REGULARITY OF LÉVY TYPE OPERATORS 13 + 2 (cid:18) Z {| z | > } ν ( dz ) (cid:19)(cid:20) sup x,z ∈ R d : | z | > c ( x, z ) (cid:21) k f k ∞ . (2) if additionally (3.4) is satisfied, then e Lf ( | x − y | ) ≤ J ( | x − y | ) (cid:16) f (2 | x − y | ) − f ( | x − y | ) (cid:17) + 4 (cid:18) Z {| z |≤ } | c ( x, z ) − c ( y, z ) || z | ν ( dz ) (cid:19) f ′ ( | x − y | )+ 2 (cid:18) Z {| z | > } ν ( dz ) (cid:19)(cid:20) sup x,z ∈ R d : | z | > c ( x, z ) (cid:21) k f k ∞ . Remark 3.2.
The estimates for e Lf ( | x − y | ) consist three terms. The first one comesfrom the operator e L C , which is a leading part for our purpose. Other two terms aredue to the operator e L R .3.2. General results.
In the following, we present general results concerning thespatial regularity of semigroups.
Theorem 3.3.
Assume that there is a nonnegative and C b ([0 , ∞ )) ∩ C ((0 , ∞ )) -function ψ such that (i) ψ (0) = 0 , ψ ′ ≥ , ψ ′′ ≤ and ψ ′′′ ≥ on (0 , ; (ii) For any constants c , c > , (3.6) lim sup r → (cid:20) J ν ( r ) r ψ ′′ (2 r ) + c w ( r ) ψ ′ ( r ) r − + c (cid:21) < , where (3.7) J ν ( r ) = inf z ∈ R d : | z | = r ν z ( R d ) with ν z ( dz ) defined in (2.1) , and w ( r ) = sup x,y ∈ R d : | x − y | = r (cid:18) Z {| z |≤ } | c ( x, z ) − c ( y, z ) || z | ν ( dz ) (cid:19) . Then, there are constants
C, ε > such that for all f ∈ B b ( R d ) and t > , (3.8) sup x = y | P t f ( x ) − P t f ( y ) | ψ ( | x − y | ) ≤ C k f k ∞ inf ε ∈ (0 ,ε ] (cid:20) ψ ( ε ) + 1 tλ ψ ( ε ) (cid:21) , where λ ψ ( ε ) := − sup First, by (3.6), we have lim sup r → J ν ( r ) r ψ ′′ (2 r ) < . Due to ψ ′′′ ≥ on (0 , , it holds for any < r ≤ that ψ ( r ) − ψ (2 r ) = − Z r Z r + ss ψ ′′ ( u ) du ds ≥ − ψ ′′ (2 r ) r . Let < ε < κ ≤ and ε ∈ (0 , ε ] . For any x, y ∈ R d with < | x − y | ≤ ε , accordingto Proposition 3.1(1), we find that e Lψ ( | x − y | ) ≤ J ( | x − y | ) ψ ′′ (2 | x − y | ) | x − y | + c w ( | x − y | ) ψ ′ ( | x − y | ) | x − y | + c , where in the inequality above we used the facts that c ( x, z ) is bounded from aboveand ψ is bounded.Since c ( x, z ) is bounded from below, there is a constant c > such that for all r > , J ( r ) ≥ c J ν ( r ) . This further yields that for any x, y ∈ R d with < | x − y | ≤ ε e Lψ ( | x − y | ) ≤ c J ν ( | x − y | ) ψ ′′ (2 | x − y | ) | x − y | + c w ( | x − y | ) ψ ′ ( | x − y | ) | x − y | + c ≤ c J ν ( | x − y | ) ψ ′′ (2 | x − y | ) | x − y | ≤ − c λ ψ ( ε ) , where in the second inequality we used (3.6).Having the inequality above at hand, we can obtain the first desired assertion by[20, Proposition 4.1]. The second desired assertion follows from the argument aboveand Proposition 3.1(2). (cid:3) From Theorem 3.3, we can further deduce the time-space regularity of semigroups.In details, let ( X xt ) t ≥ be the strong Markov process associated with the operator L starting from x . Under assumptions of Theorem 3.3, for any < s < t , x, y ∈ R d and f ∈ B b ( R d ) , | P s f ( x ) − P t f ( y ) | = | E x f ( X s ) − E y f ( X t ) | = | E x f ( X s ) − E y E X yt − s f ( X s ) |≤ E y | E x f ( X s ) − E X yt − s f ( X s ) |≤ C k f k ∞ inf ε ∈ (0 ,ε ] (cid:20) ψ ( ε ) + 1 tλ ψ ( ε ) (cid:21) E y ψ ( | x − X yt − s | ) , where in the second equality we used the Markov property, and the last inequalityfollows from (3.8). In order to estimate E y ψ ( | x − X yt − s | ) , one can refer to [16] for therecent study of moments estimates for Lévy-type processes. The details are omittedhere.3.3. Proofs. To prove Theorem 1.1, we also need the following lemma. Lemma 3.4. Suppose that there are constants α ∈ (0 , and c > such that ν ( dz ) ≥ c | z | d + α V ξ ( dz ) , where V ξ is defined by (1.4) . Then, there are constants c > and r ∈ (0 , suchthat for all < r ≤ r , J ν ( r ) ≥ c r − α , PATIAL REGULARITY OF LÉVY TYPE OPERATORS 15 where J ν is defined by (3.7) .Proof. For any z ∈ R d , let z = ( z , z , · · · , z d ) . Without loss of generality, we mayand can assume that ξ = e = (1 , , · · · , . Denote by q ( z ) = c | z | d + α { z ≥ δ | z | , | z |≤ } ( z ) , z ∈ R d . Then, for any x, z ∈ R d , q ( z ) ∧ q ( x + z ) = { z ≥ δ | z | ,z + x ≥ δ | x + z | , | z |≤ , | x + z |≤ } (cid:18) c | z | d + α ∧ c | z + x | d + α (cid:19) . In the following, we first suppose that x ≥ . Hence, for any x ∈ R d with | x | small enough, Z q ( z ) ∧ q ( x + z ) dz ≥ Z { z ≥ (1+ δ ) | z | / , δ | x | / (1 − δ ) ≤| z |≤ δ/ (1+ δ ) } c ( | x | + | z | ) d + α dz ≥ c Z { z ≥ (1+ δ ) | z | / , δ | x | / (1 − δ ) ≤| z |≤ δ/ (1+ δ ) } | z | d + α dz ≥ c Z { (1+ δ )( z + ··· + z d ) / / (1 − δ ) ≤ z ≤ δ/ [2(1+ δ )] , δ | x | / (1 − δ ) ≤ ( z + ··· + z d ) / ≤ δ/ [2(1+ δ )] } | z | d + α dz ≥ c Z c δ | x | / (1 − δ ) dr Z c (1+ δ ) r/ (1 − δ ) r d − ( z + r ) d + α dz ≥ c | x | − α . If x ≤ , then, following the argument above, we have Z q ( z ) ∧ q ( x + z ) dz ≥ Z { z ≥ (1+ δ ) | z | / , δ ) | x | / (1 − δ ) ≤| z |≤ (1+ δ ) / (3+ δ ) } c ( | x | + | z | ) d + α dz ≥ c | x | − α . Combining all the estimates above, we have obtained the desired assertion. (cid:3) We are now in a position to give Proof of Theorem . . We will apply Theorem 3.3. For (1), noticing that α ∈ (1 , too, we choose φ ( r ) = r (1 − log − θ (1 /r )) for r > small enough, where θ is givenin (1.5). For (2), we take φ ( r ) = r log θ (1 /r ) for r > small enough, where θ > if α > , and θ > if θ = 1 . For (3), since α ∈ (0 , , (3.4) holds. Then, wecan take φ ( r ) = r θ for r > small enough. Therefore, with functions φ above, thedesired assertion follows from Theorem 3.3. (cid:3) Proof of Corollary . . Let u be a bounded harmonic function on R d . Then, for any x, y ∈ R d , u ( x ) − u ( y ) = P u ( x ) − P u ( y ) . This along with Theorem 1.1 immediately yields the desired assertion. (cid:3) Proof of Corollary . . Since the proof is mainly based on that of [29, Theorem 2.1],we only point out necessary modifications here. For simplicity, we just consider thecase (3) in Theorem 1.1. For any ρ ≥ , let v ( x ) = ρ − β u ( ρx ) . Then, it is obviousthat P t v ( x ) = v ( x ) for all x ∈ R d and t > ; moreover, k v k L ∞ ( B R (0) ,dx ) ≤ cR β withthe same constant as u .For any M > , let v M ( x ) = v ( x ) {| x |≤ M } . Then, for any x, y ∈ R d , according toTheorem 1.1(3), we have | v ( x ) − v ( y ) | = | P v ( x ) − P v ( y ) |≤| P v M ( x ) − P v M ( y ) | + | P ( v − v M )( x ) | + | P ( v − v M )( y ) |≤ c | x − y | θ M β + | P v M ( x ) − P v M ( y ) | + | P ( v − v M )( x ) | + | P ( v − v M )( y ) | . On the other hand, under (1.3) (in particular, ν ( dz ) ≤ c | z | d + α dz ), for any ε > and the function h ( x ) = (1 + | x | ) ( α − ε ) / , we can check that there is a constant c > such that for all x ∈ R d , Lh ( x ) ≤ c h ( x ) , which yields that P h ( x ) ≤ c h ( x ) . Thus, for all x ∈ B (0) | P ( v − v M )( x ) | ≤ Z {| z | >M } | v | ( z ) P ( x, dz ) ≤ (cid:18)Z {| z | >M } h ( z ) P ( x, dz ) (cid:19) (cid:18) sup z ∈ M | v | ( z ) h ( z ) (cid:19) ≤ c M − ( α − β − ε ) . In particular, taking ε = ( α − β ) / , we arrive at that for all x ∈ B (0) , | P ( v − v M )( x ) | ≤ c M − ε . Combining with all the estimates above, for any x, y ∈ B (0) , | v ( x ) − v ( y ) | ≤ c (cid:0) | x − y | θ M β − M − ε (cid:1) . Letting M = | x − y | − θ/ ( β + ε ) , we get that for any x, y ∈ B (0) , | v ( x ) − v ( y ) | ≤ c | x − y | γ , where γ = εθ/ ( β + ε ) . This shows the same conclusion as [29, (2.9)]. Furthermore,one can follow the argument of [29, Theorem 2.1] to prove the desired assertion. (cid:3) Next, we present the Proof of Theorem . . Let L be the operator given by (1.1), and let L µ := L ∗ − L .Since L ∗ is a linear operator, we can split the construction of coupling operator L ∗ into those of L and L µ . For L , we still use the coupling operator e L defined by (2.2).For L µ , we define a coupling operator e L µ as follows: for any h ∈ C b ( R d ) , e L µ h ( x, y ) = Z (cid:16) h ( x + z, y + z ) − h ( x, y ) − h∇ x h ( x, y ) , z i {| z |≤ } − h∇ y h ( x, y ) , z i {| z |≤ } (cid:17) µ ( x, dz ) ∧ µ ( y, dz )+ Z (cid:16) h ( x + z, y ) − h ( x, y ) − h∇ x h ( x, y ) , z i {| z |≤ } (cid:17) PATIAL REGULARITY OF LÉVY TYPE OPERATORS 17 × ( µ ( x, dz ) − µ ( x, dz ) ∧ µ ( y, dz ))+ Z (cid:16) h ( x, y + z ) − h ( x, y ) − h∇ y h ( x, y ) , z i {| z |≤ } (cid:17) × ( µ ( y, dz ) − µ ( x, dz ) ∧ µ ( y, dz )) . Then, we can follow the proof of Proposition 2.1 and obtain that e L ∗ := e L + e L µ is acoupling operator of L . Furthermore, using the assumption that for any h ∈ C b ( R d ) ,the function x R R d h ( z ) | z | | z | µ ( x, dz ) is continuous, and repeating the argumentin Section 2.2, we can prove the existence of coupling process associated with thecoupling operator e L ∗ above.Next, for any ≤ f ∈ C b ([0 , ∞ )) ∩ C ((0 , ∞ )) such that f (0) = 0 , and f ′ ≥ and f ′′ ≤ on (0 , , we will adopt the arguments of (3.3) and (3.5) to obtain somesimilar estimates about e L µ f ( | x − y | ) . With these at hand and estimates for e L inProposition 3.1, one can obtain the desired conclusion by using Theorem 3.3 andfollowing the proof of Theorem 1.1 line by line. (cid:3) Acknowledgements. 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