Transition density estimates for diagonal systems of SDEs driven by cylindrical α -stable processes
aa r X i v : . [ m a t h . P R ] N ov TRANSITION DENSITY ESTIMATES FOR DIAGONAL SYSTEMS OFSDES DRIVEN BY CYLINDRICAL α -STABLE PROCESSES TADEUSZ KULCZYCKI AND MICHA L RYZNAR
Abstract.
We consider the system of stochastic differential equation dX t = A ( X t − ) dZ t , X = x , driven by cylindrical α -stable process Z t in R d . We assume that A ( x ) = ( a ij ( x ))is diagonal and a ii ( x ) are bounded away from zero, from infinity and H¨older continuous.We construct transition density p A ( t, x, y ) of the process X t and show sharp two-sidedestimates of this density. We also prove H¨older and gradient estimates of x → p A ( t, x, y ).Our approach is based on the method developed by Chen and Zhang in [11]. Introduction
Let Z t = ( Z (1) t , . . . , Z ( d ) t ) , be cylindrical α -stable process, that is Z ( i ) t , i = 1 , . . . , d are independent one-dimensionalsymmetric standard α -stable processes of index α ∈ (0 , d ∈ N , d ≥
2. We consider thesystem of stochastic differential equation dX t = A ( X t − ) dZ t , X = x, (1)where A ( x ) = ( a ij ( x )) is diagonal and there are constants b , b , b > β ∈ (0 ,
1] suchthat for any x, y ∈ R d , i ∈ { , . . . , d } b ≤ a ii ( x ) ≤ b , (2) | a ii ( x ) − a ii ( y ) | ≤ b | x − y | β . (3)In the sequel, without loss of generality, we assume that β ∈ (0 , α/ X is given by (see ([1, (2.3)])) L f ( x ) = d X i =1 lim ε → + A α Z | w i | >ε [ f ( x + a ii ( x ) w i e i ) + f ( x − a ii ( x ) w i e i ) − f ( x )] dw i | w i | α , where { e j } dj =1 is the standard basis in R d and A α = 2 α Γ((1 + α ) / / ( π / | Γ( − α/ | ).Let us denote the transition density of one-dimensional symmetric standard α -stableprocess of index α ∈ (0 ,
2) by g t ( x − y ), t > x, y ∈ R . Clearly, the transition density of Z ( t ) is given by Q dj =1 g t ( x j − y j ).The main result of this paper is the following theorem. Theorem 1.1. (i) The strong Markov process X ( t ) formed by the unique weak solu-tion to SDE (1) has a positive jointly continuous transition density function p A ( t, x, y ) in ( t, x, y ) ∈ (0 , ∞ ) × R d × R d with respect to the Lebesgue measure on R d .(ii) The transition density solves ∂∂t p A ( t, x, y ) = L p A ( t, · , y )( x ) , (4) T. Kulczycki was supported in part by the National Science Centre, Poland, grant no.2015/17/B/ST1/01233, M. Ryznar was supported in part by the National Science Centre, Poland, grantno. 2015/17/B/ST1/01043. for all t ∈ (0 , ∞ ) and x, y ∈ R d .(iii) For any T > there exist c = c ( T, d, α, b , b , b , β ) ≥ such that for any x, y ∈ R d , t ∈ (0 , T ] c − d Y i =1 g t ( x i − y i ) ≤ p A ( t, x, y ) ≤ c d Y i =1 g t ( x i − y i ) . (5) (iv) For any T > and γ ∈ (0 , α ∧ there exists c = c ( T, γ, d, α, b , b , b , β ) > such that for any x, x ′ , y ∈ R d , t ∈ (0 , T ] (cid:12)(cid:12) p A ( t, x, y ) − p A ( t, x ′ , y ) (cid:12)(cid:12) ≤ c | x − x ′ | γ t − γ/α d Y i =1 g t ( x i − y i ) + d Y i =1 g t ( x ′ i − y i ) ! . (6) (v) For any T > and α ∈ (1 , there exist c = c ( T, d, α, b , b , b , β ) > such thatfor any x, y ∈ R d , t ∈ (0 , T ] (cid:12)(cid:12) ∇ x p A ( t, x, y ) (cid:12)(cid:12) ≤ c t − /α p A ( t, x, y ) . (7)Systems of stochastic differential equations driven by cylindrical α -stable processes haveattracted a lot of attention in recent years see e.g. [1, 33, 40, 32, 41, 35]. In [1] Bass andChen proved existence and uniqueness of weak solutions of systems of SDEs (1) undervery mild assumptions on matrices A ( x ) (i.e. they assumed that A ( x ) are continuousand bounded in x and nondegenerate for each x ). Our paper may be treated as the firststep in studying fine properties of transition densities of systems of SDEs driven by L´evyprocesses with singular L´evy measures. Fine properties of such transition densities areof great interest but in the case of singular L´evy measures relatively little is known. Wedecided to study the particular case of diagonal matrices A ( x ) in (1) because in that caseone can obtain sharp two-sided estimates of these densities. It seems that in the caseof general non-diagonal matrices in (1) such sharp two-sided estimates are impossible toobtain. Nevertheless, we believe that our results will help to obtain qualitative estimatesof transition densities also in the case of general matrices in (1).The direct inspiration to study transition densities of solutions to (1) was a question ofZabczyk concerning gradient estimates of these densities. Another source of inspirationwas a recent paper [5] of Bogdan, Knopova and Sztonyk, where they constructed heat ker-nels and obtained upper bounds and H¨older estimates of them for quite general anisotropicspace-inhomogeneous non-local operators. However, the considered jump kernels cannotbe “too singular”. In particular, the results from [5] can be applied for systems (1) onlywhen d = 2 and α ∈ (1 ,
2) (see the condition α + γ > d in the assumption A1 on page 5in [5]). Moreover, even for d = 2 and α ∈ (1 , R d are studied with jumpkernels of the type κ ( x, z ) / | z | d + α , α ∈ (0 , L , there are manydifferences between our paper and [11]. The main new elements, in comparison to [11], arethe proof of crucial Theorem 3.2, the proof of Lemma 4.4, the estimates (55-57) and theproof of lower bound estimates of p A ( t, x, y ). It is worth pointing out that in our paper wehave shown that the transition density p A ( t, x, y ) satisfies the equation (4) for all x, y ∈ R d while in [11] it is shown that the heat kernel p κα ( t, x, y ) satisfies the analogous equationonly when x = y . A similar remark concerns gradient estimates of p A ( t, x, y ), which wemanaged to show for all x, y ∈ R d . On the other hand, we were able to prove gradientestimates of p A ( t, x, y ) only for α ∈ (1 ,
2) (in [11] gradient estimates were obtained for
RANSITION DENSITIES FOR SDES DRIVEN BY CYLINDRICAL STABLE PROCESSES 3 α ∈ [1 , x → a ii ( x ) are C ∞ ( R d ) functions. What is more, even in this case,the upper bound estimates are of the form sup x,y ∈ R d p A ( t, x, y ) ≤ ct − d/α , while the lowerbound estimates of p A ( t, x, y ) are also much less precise than ours. They are precise onlyfor x = y , in which case it follows from [29] that p A ( t, x, x ) ≈ t − d/α . The results from[30, 31, 13] cannot be applied to system (1).The paper is organized as follows. In Section 2 we introduce the notation and collectknown facts needed in the sequel. In Section 3 we construct the function p A ( t, x, y ) in termsof the perturbation series q ( t, x, y ) = P ∞ n =0 q n ( t, x, y ) using Picard’s iteration. In Theorem3.2 we obtain the estimates of q ( t, x, y ) which are absolutely crucial for the rest of the paper.In Section 4 we show that the semigroup defined by P At f ( x ) = R R d p A ( t, x, y ) f ( y ) dy is aFeller semigroup. Next, applying [1], we argue that p A ( t, x, y ) is, in fact, the transitiondensity of the solution of system (1) and we prove most parts of the main theorem. InSection 5 we show lower bound estimates of p A ( t, x, y ) by using probabilistic arguments.2. Preliminaries
All constants appearing in this paper are positive and finite. In the whole paper wefix
T > d ≥ d ∈ N , α ∈ (0 , b , b , b , β , where b , b , b , β appear in (2) and(3). We adopt the convention that constants denoted by c (or c , c , . . . ) may change theirvalue from one use to the next. In the whole paper, unless is explicitly stated otherwise,we understand that constants denoted by c (or c , c , . . . ) depend on T, d, α, b , b , b , β .We also understand that they may depend on the choice of the constant γ ∈ (0 , β ) (or γ ∈ (0 , α ∧ f ( x ) ≈ g ( x ) for x ∈ A if f, g ≥ A and there is a constant c ≥ c − f ( x ) ≤ g ( x ) ≤ cf ( x ) for x ∈ A .Denote σ i ( x ) = a αii ( x ) . Note that there exists c such that for any x, y ∈ R d we have | σ i ( x ) − σ i ( y ) | ≤ c (cid:16) | x − y | β ∧ (cid:17) . (8)By simple change of variable we get L f ( x ) = d X i =1 lim ε → + A α Z | z i | >ε [ f ( x + e i z i ) + f ( x − e i z i ) − f ( x )] σ i ( x ) dz i | z i | α . T. KULCZYCKI AND M. RYZNAR
Let us introduce some notation which was used in [11]. For a function f : R d → R wedenote δ f ( x, z ) = f ( x + z ) + f ( x − z ) − f ( x ) . Similarly, for a function f : R + × R d → R we write δ f ( t, x, z ) = f ( t, x + z ) + f ( t, x − z ) − f ( t, x ) . We also denote ρ βγ ( t, x ) = t γ/α ( | x | β ∧ t /α + | x | ) − − α , t > , x ∈ R . It is well known that g t ( x ) ≈ ρ α ( t, x ) t > , x ∈ R . (9)One of the most important tools used in our paper are convolution estimates [11, (2.3-2.4)]. They are similar to [24, Lemma 1.4] and [39, Lemma 2.3]. In [11] they are stated for t ∈ (0 , t ∈ (0 , T ]. For reader’s conveniencewe collected them in Lemma 2.1. Lemma 2.1. (i)
There is C = C ( α ) such that for any t > and any β ∈ [0 , α/ , γ ∈ R , Z R ρ β γ ( t, z ) dz ≤ Ct γ β − αα . (10)(ii) For
T > there is C = C ( α, T ) such that for any < s < t ≤ T , x ∈ R and any β , β ∈ [0 , α/ , γ , γ ∈ R we have Z R ρ β γ ( t − s, x − z ) ρ β γ ( s, z ) dz ≤ C h ( t − s ) γ β β − αα s γ α + ( t − s ) γ α s γ β β − αα i ρ ( t, x )+ C h ( t − s ) γ β − αα s γ α ρ β ( t, x ) + ( t − s ) γ α s γ β − αα ρ β ( t, x ) i . (11)(iii) For
T > there is C = C ( α, T ) such that for any < t ≤ T , x ∈ R and any β , β ∈ [0 , α/ , γ , γ ∈ R with γ + β > and γ + β > we have Z t Z R ρ β γ ( t − s, x − z ) ρ β γ ( s, z ) dz ds ≤ C B (cid:18) γ + β α , γ + β α (cid:19) (cid:16) ρ γ + γ + β + β + ρ β γ + γ + β + ρ β γ + γ + β (cid:17) ( t, x ) , (12) where B ( u, w ) is the Beta function. Similarly as in [11] we introduce, for y ∈ R d , the freezing operator L y by L y f ( x ) = A α d X i =1 Z R δ f ( x, e i z i ) σ i ( y ) dz i | z i | α and L y f ( t, x ) = A α d X i =1 Z R δ f ( t, x, e i z i ) σ i ( y ) dz i | z i | α . Put p y ( t, x ) = d Y i =1 a ii ( y ) g t (cid:18) x i a ii ( y ) (cid:19) . It is clear that p y ( t, x ) is the heat kernel of the operator L y . In particular, we have ∂∂t p y ( t, x ) = L y p y ( t, x ) , t > , x, y ∈ R d . (13) RANSITION DENSITIES FOR SDES DRIVEN BY CYLINDRICAL STABLE PROCESSES 5
In the sequel we will use the following standard estimate. For any γ ∈ (0 ,
1] there exists c = c ( γ ) such that for any θ ≥ Z t ( t − s ) γ − s θ − ds ≤ cθ γ t ( γ − θ − . (14)We use the notation N = N ∪ { } .3. Upper bound estimates
The main aim of this section is to construct the function p A ( t, x, y ). This is done byusing Levi’s method. Is is worth mentioning that this method was used in the framework ofpseudodifferential operators by Kochubei [23]. In recent years it was used in several papersto study gradient and Schr¨odinger perturbations of fractional Laplacians and relativisticstable operators see e.g. [3, 14, 15, 6, 8, 9, 39]. As we have already mentioned we use theapproach by Chen and Zhang [11]. It is worth adding that in [11], in contrast to previouspapers, a new way of “freezing” coefficient was used.Now, we briefly present the main steps used in this section. We define p A ( t, x, y ) by(16). Heuristically, p A ( t, x, y ) is equal to the transition density p y ( t, x − y ) (of the processwith the “frozen” jump measure corresponding to the generator L y ) plus some correc-tion R t R R d p z ( t − s, x, z ) q ( s, z, y ) dz ds , which is given in terms of the perturbation series q ( t, x, y ) = P ∞ n =0 q n ( t, x, y ). The most difficult result in this section is Theorem 3.2 whichgives upper bound estimates of q ( t, x, y ). Due to a different structure of the generator L incomparison to the L´evy-type operator L κα from [11] there are essential differences betweenour proof and analogous proof in [11], see in particular the definition of the auxiliary func-tion H Lk ( t, x, y ) and the induction proof of (20). The next important step in this sectionis Theorem 3.9 where we derive H¨older type estimates of q ( t, x, y ). We also show crucialLemma 3.14 which is the main step in obtaining gradient estimates of p A ( t, x, y ).For x, y ∈ R d , t >
0, let q ( t, x, y ) = ( L x − L y ) p y ( t, · )( x − y )and for n ∈ N let q n ( t, x, y ) = Z t Z R d q ( t − s, x, z ) q n − ( s, z, y ) dz ds. (15)For x, y ∈ R d , t > q ( t, x, y ) = ∞ X n =0 q n ( t, x, y )and p A ( t, x, y ) = p y ( t, x − y ) + Z t Z R d p z ( t − s, x, z ) q ( s, z, y ) dz ds. (16)By [11, (2.28)] and (9) one easily obtains Lemma 3.1.
For any t ∈ (0 , T ] and x, y ∈ R d we have d X k =1 Z R (cid:12)(cid:12) δ p y ( t, x, z k e k ) (cid:12)(cid:12) dz k | z k | α ≤ ct d − d Y i =1 ρ ( t, x i ) . An immediate consequence of the above lemma and (13) is the following estimate (cid:12)(cid:12)(cid:12)(cid:12) ∂∂t p y ( t, x ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ ct d − d Y i =1 ρ ( t, x i ) , (17)for t ∈ (0 , T ], x, y ∈ R d . T. KULCZYCKI AND M. RYZNAR
Theorem 3.2.
The series P ∞ n =0 q n ( t, x, y ) is absolutely and locally uniformly convergenton (0 , T ] × R d × R d . For any x, y ∈ R d , t ∈ (0 , T ] we have | q ( t, x, y ) | ≤ ct d − " d Y i =1 ρ ( t, x i − y i ) t β/α + d X m =1 (cid:16) | x m − y m | β ∧ (cid:17) . (18) Moreover, q ( t, x, y ) is jointly continuous in ( t, x, y ) ∈ (0 , T ] × R d × R d .Proof. By (8) and then Lemma 3.1 we get | q ( t, x, y ) | ≤ A α d X k =1 Z R (cid:12)(cid:12) δ p y ( t, x − y, e k z k ) (cid:12)(cid:12) | σ k ( x ) − σ k ( y ) | dz k | z k | α ≤ c (cid:16) | x − y | β ∧ (cid:17) d X k =1 Z R (cid:12)(cid:12) δ p y ( t, x − y, e k z k ) (cid:12)(cid:12) dz k | z k | α ≤ M t d − " d X m =1 (cid:16) | x m − y m | β ∧ (cid:17) d Y k =1 ρ ( t, x k − y k ) , (19)where M = M ( T, d, α, b , b , b , β ).Put I = { L = ( l , . . . , l d ) : ∀ i ∈ { , . . . , d } l i = 0 or l i = β } . For any L = ( l , . . . , l d ) ∈ I denote | L | = 1 β d X i =1 l i . For k ∈ N and L = ( l , . . . , l d ) ∈ I put H Lk ( t, x, y ) = t d − kβ/α " d Y i =1 ρ ( t, x i − y i ) d Y j =1 (cid:16) | x j − y j | l j ∧ (cid:17) . We will show that there is C = C ( T, d, α, b , b , b , β ) such that for any n ∈ N , x, y ∈ R d , t ∈ (0 , T ], | q n ( t, x, y ) | ≤ M C n (( n + 1)!) β/α X k ∈ N , L ∈ I k + | L | = n +1 H Lk ( t, x, y ) , (20)where M is the constant from (19). Let D ( t, x, y, m, k, L ) = M Z t Z R d H L m ( t − s, x, z ) H Lk ( s, z, y ) dz ds, where L m ∈ I is such that l m = β and | L m | = 1. Observe that (19) can be rewritten as | q ( t, x, y ) | ≤ M d X m =1 H L m ( t, x, y ) . (21)We will prove (20) by induction. The main step consists of proving that for any n ∈ N , d X m =1 X k ∈ N , L ∈ I k + | L | = n +1 D ( t, x, y, m, k, L ) ≤ C ( n + 2) β/α X k ∈ N , L ∈ I k + | L | = n +2 H Lk ( t, x, y ) . (22)For n = 0 the estimate (20) holds by (21). RANSITION DENSITIES FOR SDES DRIVEN BY CYLINDRICAL STABLE PROCESSES 7
Assume that (20) holds for some n ∈ N . By (15), (21) and our induction hypothesiswe obtain | q n +1 ( t, x, y ) | ≤ M C n (( n + 1)!) β/α d X m =1 X k ∈ N , L ∈ I k + | L | = n +1 D ( t, x, y, m, k, L ) . (23)Then, if (22) is true, then (20) holds for n + 1. Hence in order to complete the proof it isenough to show (22).To this end we consider 3 cases. Case 1. L = (0 , . . . , k = n + 1.We have D ( t, x, y, m, k, L ) = M Z t s nβ/α d Y i =1 i = m Z R ρ α ( t − s, x i − z i ) ρ α ( s, z i − y i ) dz i × Z R ρ β ( t − s, x m − z m ) ρ β ( s, z m − y m ) dz m ds. By (11), we obtain Z R ρ α ( t − s, x i − z i ) ρ α ( s, z i − y i ) dz i ≤ cρ α ( t, x i − y i )and Z R ρ β ( t − s, x m − z m ) ρ β ( s, z m − y m ) dz m ≤ c h ( t − s ) ( β − α ) /α s β/α ρ ( t, x m − y m ) + s (2 β − α ) /α ρ ( t, x m − y m )+ s ( β − α ) /α ρ β ( t, x m − y m ) i . Hence D ( t, x, y, m, k, L ) ≤ ct d − " d Y i =1 ρ ( t, x i − y i ) × (cid:20)Z t ( t − s ) ( β − α ) /α s ( n +1) β/α ds + Z t s (( n +2) β − α ) /α ds + Z t s (( n +1) β − α ) /α ds (cid:16) | x m − y m | β ∧ (cid:17)(cid:21) . By (14) this implies that D ( t, x, y, m, k, L ) ≤ c ( n + 1) β/α t d − " d Y i =1 ρ ( t, x i − y i ) × h t ( n +2) β/α + t ( n +1) β/α (cid:16) | x m − y m | β ∧ (cid:17)i . (24) Case 2. L = ( l , . . . , l d ) = (0 , . . . , l m = 0. T. KULCZYCKI AND M. RYZNAR
Put Z ( L ) = { i ∈ { , . . . , d } : l i = β } and i ( L ) = inf Z ( L ). Clearly m / ∈ Z ( L ). We have D ( t, x, y, m, k, L ) = M Z t Z R ρ β ( t − s, x m − z m ) ρ α ( s, z m − y m ) dz m × Z R ρ α ( t − s, x i ( L ) − z i ( L ) ) ρ β ( s, z i ( L ) − y i ( L ) ) dz i ( L ) × Y i ∈ Z ( L ) i = i ( L ) Z R ρ α ( t − s, x i − z i ) ρ βα ( s, z i − y i ) dz i × Y i/ ∈ Z ( L ) i = m Z R ρ α ( t − s, x i − z i ) ρ α ( s, z i − y i ) dz i s kβ/α ds. By (11) this is bounded from above by c Z t h ( t − s ) ( β − α ) /α sρ ( t, x m − y m ) + t β/α ρ ( t, x m − y m ) + ρ β ( t, x m − y m ) i × h t β/α ρ ( t, x i ( L ) − y i ( L ) ) + ts ( β − α ) /α ρ ( t, x i ( L ) − y i ( L ) ) + ρ β ( t, x i ( L ) − y i ( L ) ) i × Y i ∈ Z ( L ) i = i ( L ) h t ( α + β ) /α ρ ( t, x i − y i ) + tρ β ( t, x i − y i ) i × Y i/ ∈ Z ( L ) i = m tρ ( t, x i − y i ) s kβ/α ds. Note that Z ( L ) = | L | . We have Y i ∈ Z ( L ) i = i ( L ) h t ( α + β ) /α ρ ( t, x i − y i ) + tρ β ( t, x i − y i ) i ≤ ct | L |− Y i ∈ Z ( L ) i = i ( L ) ρ ( t, x i − y i ) × X r ≤| L |− r ∈ N X { k ,...,k r }⊂ Z ( L ) \{ i ( L ) } t ( | L |− r − β/α r Y i =1 (cid:16) | x k i − y k i | β ∧ (cid:17) , (25) RANSITION DENSITIES FOR SDES DRIVEN BY CYLINDRICAL STABLE PROCESSES 9 where for r = 0 we understand that Q ri =1 (cid:0) | x k i − y k i | β ∧ (cid:1) = 1 . It follows that D ( t, x, y, m, k, L ) ≤ ct d − " d Y i =1 ρ ( t, x i − y i ) × X r ≤| L |− r ∈ N X { k ,...,k r }⊂ Z ( L ) \{ i ( L ) } t ( | L |− r − β/α " r Y i =1 (cid:16) | x k i − y k i | β ∧ (cid:17) × (cid:20) t ( α + β ) /α Z t ( t − s ) ( β − α ) /α s kβ/α ds + t Z t ( t − s ) ( β − α ) /α s ( β + kβ ) /α ds + t Z t ( t − s ) ( β − α ) /α s kβ/α ds (cid:16) | x i ( L ) − y i ( L ) | β ∧ (cid:17) + t β/α Z t s kβ/α ds + t ( α + β ) /α Z t s ( β − α + kβ ) /α ds + t β/α Z t s kβ/α ds (cid:16) | x i ( L ) − y i ( L ) | β ∧ (cid:17) × (cid:20) t β/α Z t s kβ/α ds + t Z t s ( β − α + kβ ) /α ds + Z t s kβ/α ds (cid:16) | x i ( L ) − y i ( L ) | β ∧ (cid:17)(cid:21) × (cid:16) | x m − y m | β ∧ (cid:17)i . Using this and (14) we get D ( t, x, y, m, k, L ) ≤ ct d − " d Y i =1 ρ ( t, x i − y i ) × X r ≤| L |− r ∈ N X { k ,...,k r }⊂ Z ( L ) \{ i ( L ) } t ( | L |− r − β/α " r Y i =1 (cid:16) | x k i − y k i | β ∧ (cid:17) × k + 1) β/α h t ( α + kβ +2 β ) /α + t ( α + kβ + β ) /α (cid:16) | x i ( L ) − y i ( L ) | β ∧ (cid:17) t ( α + kβ + β ) /α (cid:16) | x m − y m | β ∧ (cid:17) + t ( α + kβ ) /α (cid:16) | x i ( L ) − y i ( L ) | β ∧ (cid:17) (cid:16) | x m − y m | β ∧ (cid:17)i . Note that k + | L | = n + 1. It follows that D ( t, x, y, m, k, L ) ≤ ct d − ( k + 1) β/α " d Y i =1 ρ ( t, x i − y i ) × X r ≤| L | +1 r ∈ N X { k ,...,k r }⊂ Z ( L ) ∪{ m } t ( n +2 − r ) β/α r Y i =1 (cid:16) | x k i − y k i | β ∧ (cid:17) . (26) Case 3. L = ( l , . . . , l d ), l m = β . We have D ( t, x, y, m, k, L ) = M Z t Z R ρ β ( t − s, x m − z m ) ρ β ( s, z m − y m ) dz m × Y i ∈ Z ( L ) i = m Z R ρ α ( t − s, x i − z i ) ρ βα ( s, z i − y i ) dz i × Y i/ ∈ Z ( L ) Z R ρ α ( t − s, x i − z i ) ρ α ( s, z i − y i ) dz i s kβ/α ds. By (11), this is bounded from above by c Z t h ( t − s ) (2 β − α ) /α ρ ( t, x m − y m ) + s (2 β − α ) /α ρ ( t, x m − y m )+( t − s ) ( β − α ) /α ρ β ( t, x m − y m ) + s ( β − α ) /α ρ β ( t, x m − y m ) i × Y i ∈ Z ( L ) i = m h t ( α + β ) /α ρ ( t, x i − y i ) + tρ β ( t, x i − y i ) i × Y i/ ∈ Z ( L ) tρ ( t, x i − y i ) s kβ/α ds. Using similar reasoning as in (25) this is bounded from above by ct d − " d Y i =1 ρ ( t, x i − y i ) × X r ≤| L |− r ∈ N X { k ,...,k r }⊂ Z ( L ) \{ m } t ( | L |− r − β/α r Y i =1 (cid:16) | x k i − y k i | β ∧ (cid:17)(cid:20)Z t ( t − s ) (2 β − α ) /α s kβ/α ds + Z t s (2 β − α + kβ ) /α ds + Z t ( t − s ) ( β − α ) /α s kβ/α ds (cid:16) | x m − y m | β ∧ (cid:17)Z t s ( β − α + kβ ) /α ds (cid:16) | x m − y m | β ∧ (cid:17)(cid:21) By (14) it follows that D ( t, x, y, m, k, L ) ≤ ct d − " d Y i =1 ρ ( t, x i − y i ) × X r ≤| L |− r ∈ N X { k ,...,k r }⊂ Z ( L ) \{ m } t ( | L |− r − β/α r Y i =1 (cid:16) | x k i − y k i | β ∧ (cid:17) k + 1) β/α h t ( kβ +2 β ) /α + t ( kβ + β ) /α (cid:16) | x m − y m | β ∧ (cid:17)i RANSITION DENSITIES FOR SDES DRIVEN BY CYLINDRICAL STABLE PROCESSES 11
Hence D ( t, x, y, m, k, L ) ≤ ct d − ( k + 1) β/α " d Y i =1 ρ ( t, x i − y i ) × X r ≤| L | r ∈ N X { k ,...,k r }⊂ Z ( L ) t ( n +2 − r ) β r Y i =1 (cid:16) | x k i − y k i | β ∧ (cid:17) . (27)Recall that n + 1 = k + | L | , | L | ≤ d , so k ≥ n + 1 − d . Hence k +1) β/α ≤ c ( n +2) β/α ,where c = c ( d ). Consequently, (23), (24), (26), (27) gives that (22) holds, which finishesthe induction proof.From (20) we immediately obtain that for any n ∈ N | q n ( t, x, y ) | ≤ cC n (( n + 1)!) β/α t d − " d Y i =1 ρ ( t, x i − y i ) t β/α + d X m =1 (cid:16) | x m − y m | β ∧ (cid:17) . (28)It follows that P ∞ n =0 q n ( t, x, y ) is absolutely and locally uniformly convergent on (0 , T ] × R d × R d and (18) holds.By the properties of p y ( t, x ) it is easy to justify that q ( t, x, y ) is jointly continuousin ( t, x, y ) ∈ (0 , T ] × R d × R d . By (15) and induction method the same property holdsfor q n ( t, x, y ) for each n ∈ N . Since P ∞ n =0 q n ( t, x, y ) is absolutely and locally uniformlyconvergent we finally obtain that q ( t, x, y ) is jointly continuous in ( t, x, y ) ∈ (0 , T ] × R d × R d . (cid:3) By elementary calculations, for any t > u, w ∈ R satisfying | u − w | ≤ t /α , we have ρ ( t, u ) ≈ ρ ( t, w ) . (29) Lemma 3.3.
There exists c = c ( α, d ) such that for any m ∈ { , . . . , d } , t > , x, x ′ ∈ R d we have (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) m Y i =1 g t ( x i ) − m Y i =1 g t (cid:0) x ′ i (cid:1)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ c m Y i =1 g t ( | x i | ∧ | x ′ i | ) ! ∧ m X j =1 | x j − x ′ j | t /α + | x i | ∧ | x ′ i | . (30) If additionally | x − x ′ | ≤ t /α , then (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) m Y i =1 g t ( x i ) − m Y i =1 g t (cid:0) x ′ i (cid:1)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ ct m m Y i =1 ρ ( t, x i ) ! ∧ m X j =1 t − /α | x j − x ′ j | . (31) Proof.
Let g (3) t ( · ) be the radial profile of the transition density of the standard 3-dimensional α -stable isotropic process. Then it is well known (see e.g. [3, (11)]) that dg t ( x ) dx = − πxg (3) t ( | x | ) , x ∈ R . By the standard estimates of transition density of the α -stable isotropic process we have g (3) t ( | x | ) ≤ c g t ( x ) (cid:0) | x | + t /α (cid:1) , which yields (cid:12)(cid:12)(cid:12)(cid:12) dg t ( x ) dx (cid:12)(cid:12)(cid:12)(cid:12) ≤ c | x | (cid:0) | x | + t /α (cid:1) g t ( x ) ≤ c g t ( x ) | x | + t /α , x ∈ R . Next, for any u, w ∈ R , from the above gradient estimate of g t and the fact that g t ( u )is decreasing in | u | , | g t ( u ) − g t ( w ) | ≤ c | u − w || u | ∧ | w | + t /α g t ( | u | ∧ | w | ) . (32)Hence, by monotonicity of g t ( u ) and (32), (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) m Y i =1 g t ( x i ) − m Y i =1 g t (cid:0) x ′ i (cid:1)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ m X i =1 (cid:12)(cid:12) g t ( x i ) − g t (cid:0) x ′ i (cid:1)(cid:12)(cid:12) Y j = i, ≤ j ≤ m g t (cid:0) | x i | ∧ | x ′ i | (cid:1) ≤ c m Y i =1 g t ( | x i | ∧ | x ′ i | ) ! m X j =1 | x i − x ′ i || x i | ∧ | x ′ i | + t /α Combining this with the obvious inequality (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) m Y i =1 g t ( x i ) − m Y i =1 g t (cid:0) x ′ i (cid:1)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ m Y i =1 g t ( | x i | ∧ | x ′ i | )we finish the proof of (30).To get the second inequality we apply (29). (cid:3) As a direct conclusion of Lemma 3.3 we get
Corollary 3.4.
For any k ∈ { , . . . , d } , t > , x, x ′ , y ∈ R d satisfying | x − x ′ | ≤ t /α wehave (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) d Y i =1 i = k a ii ( y ) g t (cid:18) x i a ii ( y ) (cid:19) − d Y i =1 i = k a ii ( y ) g t (cid:18) x ′ i a ii ( y ) (cid:19)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ ct d − d Y i =1 i = k ρ ( t, x i ) d X j =1 ( t − /α | x j − x ′ j | ∧ . Corollary 3.5.
For any t > and x, y, w ∈ R d we have | p x ( t, w ) − p y ( t, w ) | ≤ cp x ( t, w )( | x − y | β ∧ . Proof.
We have | p x ( t, w ) − p y ( t, w ) | = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) d Y i =1 a − ii ( y ) g t (cid:18) w i a ii ( y ) (cid:19) − d Y i =1 a − ii ( x ) g t (cid:18) w i a ii ( x ) (cid:19)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) d Y i =1 a − ii ( y ) − d Y i =1 a − ii ( x ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) d Y i =1 g t (cid:18) w i a ii ( y ) (cid:19) + d Y i =1 a − ii ( x ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) d Y i =1 g t (cid:18) w i a ii ( y ) (cid:19) − d Y i =1 g t (cid:18) w i a ii ( x ) (cid:19)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Next, by (3) and (2), (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) d Y i =1 a − ii ( y ) − d Y i =1 a − ii ( x ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ c ( | x − y | β ∧ , RANSITION DENSITIES FOR SDES DRIVEN BY CYLINDRICAL STABLE PROCESSES 13 and by Lemma 3.3 together with (3) and (2) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) d Y i =1 g t (cid:18) w i a ii ( y ) (cid:19) − d Y i =1 g t (cid:18) w i a ii ( x ) (cid:19)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ c ∧ m X j =1 | w i || a ii ( x ) − a ii ( y ) | t /α + | w i | d Y i =1 g t (cid:18) w i a ii ( x ) (cid:19) ≤ c ( | x − y | β ∧ d Y i =1 g t (cid:18) w i a ii ( x ) (cid:19) . The proof is completed. (cid:3)
Corollary 3.6.
For any x ∈ R d , t > we have Z R d − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) d Y i =2 a ii ( y ) g t (cid:18) x i − y i a ii ( y ) (cid:19) − d Y i =2 a ii (˜ y ) g t (cid:18) x i − y i a ii (˜ y ) (cid:19)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) dy . . . dy d ≤ c ( | x − y | β ∧ , (33) where ˜ y = ( x , y , . . . , y d ) .Proof. By the same arguments as in the proof Corollary 3.5 we have (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) d Y i =2 a ii ( y ) g t (cid:18) x i − y i a ii ( y ) (cid:19) − d Y i =2 a ii (˜ y ) g t (cid:18) x i − y i a ii (˜ y ) (cid:19)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ c " d Y i =2 g t ( x i − y i ) ( | y − ˜ y | β ∧ . Observing that | y − ˜ y | = | x − y | we obtain the conclusion by integration. (cid:3) Lemma 3.7.
For any t ∈ (0 , T ] , x, x ′ , y ∈ R d satisfying | x − x ′ | ≤ t /α we have d X k =1 Z R (cid:12)(cid:12) δ p y ( t, x, z k e k ) − δ p y ( t, x ′ , z k e k ) (cid:12)(cid:12) dz k | z k | α ≤ ct d − d Y i =1 ρ ( t, x i ) ! d X j =1 ( t − /α | x j − x ′ j | ∧ . (34) Proof.
Fix k ∈ { , . . . , d } For t > z ∈ R put h y ( t, z ) = a kk ( y ) g t (cid:18) za kk ( y ) (cid:19) . We have Z R (cid:12)(cid:12) δ p y ( t, x, z k e k ) − δ p y ( t, x ′ , z k e k ) (cid:12)(cid:12) dz k | z k | α = Z R (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) d Y i =1 i = k a ii ( y ) g t (cid:18) x i a ii ( y ) (cid:19) δ h y ( t, x k , z k ) − d Y i =1 i = k a ii ( y ) g t (cid:18) x ′ i a ii ( y ) (cid:19) δ h y ( t, x ′ k , z k ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) dz k | z k | α ≤ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) d Y i =1 i = k a ii ( y ) g t (cid:18) x i a ii ( y ) (cid:19) − d Y i =1 i = k a ii ( y ) g t (cid:18) x ′ i a ii ( y ) (cid:19)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Z R (cid:12)(cid:12) δ h y ( t, x k , z k ) (cid:12)(cid:12) dz k | z k | α + d Y i =1 i = k a ii ( y ) g t (cid:18) x ′ i a ii ( y ) (cid:19) Z R (cid:12)(cid:12) δ h y ( t, x k , z k ) − δ h y ( t, x ′ k , z k ) (cid:12)(cid:12) dz k | z k | α . By [11, (2.28)] we have Z R (cid:12)(cid:12) δ h y ( t, x k , z k ) (cid:12)(cid:12) dz k | z k | α ≤ cρ ( t, x k ) , while, by [11, (2.29)], Z R (cid:12)(cid:12) δ h y ( t, x k , z k ) − δ h y ( t, x ′ k , z k ) (cid:12)(cid:12) dz k | z k | α ≤ cρ ( t, x k )( t − /α | x k − x ′ k | ∧ . Applying the above inequalities and Corollary 3.4 we obtain the desired bound (34). (cid:3)
Lemma 3.8.
For any x, x ′ , y ∈ R d , t ∈ (0 , T ] and γ ∈ (0 , β ) we have | q ( t, x, y ) − q ( t, x ′ , y ) | ≤ c (cid:16) | x − x ′ | β − γ ∧ (cid:17) × d X k =1 d Y i =1 i = k ( tρ ( t, x i − y i )) (cid:16) ρ γ ( t, x k − y k ) + ρ βγ − β ( t, x k − y k ) (cid:17) + d X k =1 d Y i =1 i = k ( tρ ( t, x ′ i − y i )) (cid:16) ρ γ ( t, x ′ k − y k ) + ρ βγ − β ( t, x ′ k − y k ) (cid:17) Proof.
Case 1. | x − x ′ | > t /α .By (19) we get, for z = x or x ′ , | q ( t, z, y ) | ≤ c (cid:16) | x − x ′ | β − γ ∧ (cid:17) t d − d Y i =1 ρ ( t, z i − y i ) ! d X m =1 ( | z m − y m | ∧ β t γ − βα ! . Case 2. | x − x ′ | ≤ t /α . RANSITION DENSITIES FOR SDES DRIVEN BY CYLINDRICAL STABLE PROCESSES 15
We have | q ( t, x, y ) − q ( t, x ′ , y ) | = A (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) d X k =1 Z R δ p y ( t, x − y, z k e k )( a αkk ( x ) − a αkk ( y )) dz k | z k | α − d X k =1 Z R δ p y ( t, x ′ − y, z k e k )( a αkk ( x ′ ) − a αkk ( y )) dz k | z k | α (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ d X k =1 Z R (cid:12)(cid:12) δ p y ( t, x − y, z k e k ) − δ p y ( t, x ′ − y, z k e k ) (cid:12)(cid:12) | a αkk ( x ) − a αkk ( y ) | dz k | z k | α + d X k =1 Z R (cid:12)(cid:12) δ p y ( t, x ′ − y, z k e k ) (cid:12)(cid:12) (cid:12)(cid:12) a αkk ( x ) − a αkk ( x ′ ) (cid:12)(cid:12) dz k | z k | α ≤ c (cid:16) | x − y | β ∧ (cid:17) d X k =1 Z R (cid:12)(cid:12) δ p y ( t, x − y, z k e k ) − δ p y ( t, x ′ − y, z k e k ) (cid:12)(cid:12) dz k | z k | α + c (cid:16) | x − x ′ | β ∧ (cid:17) d X k =1 Z R (cid:12)(cid:12) δ p y ( t, x ′ − y, z k e k ) (cid:12)(cid:12) dz k | z k | α = I + II . By Lemma 3.7 we getI ≤ c (cid:16) | x − y | β ∧ (cid:17) (cid:16) t − /α | x − x ′ | ∧ (cid:17) t d − d Y i =1 ρ ( t, x i − y i ) . (35)Note that t − /α | x − x ′ | ≤ t − /α | x − x ′ | ≤ t γ − βα | x − x ′ | β − γ . Let m ∈ { , . . . , d } be suchthat | x m − y m | = max i ∈{ ,...,d } | x i − y i | . It follows that (35) is bounded from above by ct γ − βα | x − x ′ | β − γ t d − d Y i =1 ρ ( t, x i − y i ) ! (cid:16) | x m − y m | β ∧ (cid:17) ≤ c (cid:16) | x − x ′ | β − γ ∧ (cid:17) d Y i =1 i = m tρ ( t, x i − y i ) ρ βγ − β ( t, x m − y m ) . We have | x − x ′ | ≤ t /α so 1 ≤ | x − x ′ | − γ t γ/α . It follows that (cid:0) | x − x ′ | β ∧ (cid:1) ≤ (cid:0) | x − x ′ | β − γ ∧ (cid:1) t γ/α . Using this and Lemma 3.1 we getII ≤ c (cid:16) | x − x ′ | β ∧ (cid:17) t d − d Y i =1 ρ ( t, x ′ i − y i ) ≤ c (cid:16) | x − x ′ | β − γ ∧ (cid:17) t d − d X k =1 d Y i =1 i = k ρ ( t, x ′ i − y i ) ρ γ ( t, x ′ k − y k ) . (cid:3) Theorem 3.9.
For any x, x ′ , y ∈ R d , t ∈ (0 , T ] and γ ∈ (0 , β ) we have | q ( t, x, y ) − q ( t, x ′ , y ) | ≤ c (cid:16) | x − x ′ | β − γ ∧ (cid:17) × d X k =1 d Y i =1 i = k ( tρ ( t, x i − y i )) (cid:16) ρ γ ( t, x k − y k ) + ρ βγ − β ( t, x k − y k ) (cid:17) + d X k =1 d Y i =1 i = k ( tρ ( t, x ′ i − y i )) (cid:16) ρ γ ( t, x ′ k − y k ) + ρ βγ − β ( t, x ′ k − y k ) (cid:17) . (36) Proof.
By the definition of q n , (28) and Lemma 3.8 we get for n ∈ N | q n ( t, x, y ) − q n ( t, x ′ , y ) |≤ Z t Z R d | q ( t − s, x, z ) − q ( t − s, x ′ , z ) || q n − ( s, z, y ) | dz ds ≤ cC n − ( n !) β/α (cid:16) | x − x ′ | β − γ ∧ (cid:17) (cid:0) A ( t, x, y ) + A ( t, x ′ , y ) (cid:1) , (37)where C is the constant from (28) and A ( t, x, y )= Z t Z R d d X k =1 d Y i =1 i = k ρ α ( t − s, x i − z i ) (cid:16) ρ γ ( t − s, x k − z k ) + ρ βγ − β ( t − s, x k − z k ) (cid:17) × d X m =1 d Y j =1 j = m ρ α ( s, z j − y j ) (cid:16) ρ β ( s, z m − y m ) + ρ β ( s, z m − y m ) (cid:17) dz . . . dz d ds. We have A ( t, x, y ) = d X k =1 d Y i =1 i = k tρ ( t, x i − y i ) B k ( t, x, y )+ d X k =1 d X m =1 m = k d Y i =1 i = k, i = m tρ ( t, x i − y i ) × [ D k,m ( t, x, y ) + E k,m ( t, x, y ) + F k,m ( t, x, y ) + G k,m ( t, x, y )] , (38)where B k ( t, x, y ) = Z t Z R (cid:16) ρ γ ( t − s, x k − z k ) + ρ βγ − β ( t − s, x k − z k ) (cid:17) × (cid:16) ρ β ( s, z k − y k ) + ρ β ( s, z k − y k ) (cid:17) dz k ds,D k,m ( t, x, y ) = Z t Z R ρ γ ( t − s, x k − z k ) ρ α ( s, z k − y k ) dz k × Z R ρ α ( t − s, x m − z m ) ρ β ( s, z m − y m ) dz m ds, RANSITION DENSITIES FOR SDES DRIVEN BY CYLINDRICAL STABLE PROCESSES 17 E k,m ( t, x, y ) = Z t Z R ρ γ ( t − s, x k − z k ) ρ α ( s, z k − y k ) dz k × Z R ρ α ( t − s, x m − z m ) ρ β ( s, z m − y m ) dz m ds,F k,m ( t, x, y ) = Z t Z R ρ βγ − β ( t − s, x k − z k ) ρ α ( s, z k − y k ) dz k × Z R ρ α ( t − s, x m − z m ) ρ β ( s, z m − y m ) dz m ds,G k,m ( t, x, y ) = Z t Z R ρ βγ − β ( t − s, x k − z k ) ρ α ( s, z k − y k ) dz k × Z R ρ α ( t − s, x m − z m ) ρ β ( s, z m − y m ) dz m ds. By (12) we get B k ( t, x, y ) ≤ c (cid:16) ρ γ ( t, x k − y k ) + ρ βγ − β ( t, x k − y k ) (cid:17) . (39)Using (11) we obtain D k,m ( t, x, y ) ≤ c Z t (cid:16) ( t − s ) γ − αα s + ( t − s ) γα (cid:17) (cid:16) s βα + ( t − s ) s β − αα (cid:17) dsρ ( t, x k − y k ) ρ ( t, x m − y m ) ≤ ct γα + βα ρ ( t, x k − y k ) ρ ( t, x m − y m ) ≤ ctρ γ ( t, x k − y k ) ρ ( t, x m − y m ) , (40) E k,m ( t, x, y ) ≤ c Z t (cid:16) ( t − s ) γ − αα s + ( t − s ) γα (cid:17) (cid:16) ( t − s ) β/α + ( t − s ) s β − αα (cid:17) ds × ρ ( t, x k − y k ) ρ ( t, x m − y m )+ c Z t (cid:16) ( t − s ) γ − αα s + ( t − s ) γα (cid:17) dsρ ( t, x k − y k ) ρ β ( t, x m − y m ) ≤ ct γα + βα ρ ( t, x k − y k ) ρ ( t, x m − y m ) + ct γα ρ ( t, x k − y k ) ρ β ( t, x m − y m ) ≤ ctρ γ ( t, x k − y k ) ρ ( t, x m − y m ) , (41) F k,m ( t, x, y ) ≤ c Z t (cid:16) ( t − s ) γ − αα s + ( t − s ) γ − βα s βα (cid:17) (cid:16) s βα + ( t − s ) s β − αα (cid:17) ds × ρ ( t, x k − y k ) ρ ( t, x m − y m )+ c Z t (cid:16) ( t − s ) γ − βα (cid:16) s βα + ( t − s ) s β − αα (cid:17)(cid:17) dsρ β ( t, x k − y k ) ρ ( t, x m − y m ) ≤ ct γα + βα ρ ( t, x k − y k ) ρ ( t, x m − y m ) + ct γα ρ β ( t, x k − y k ) ρ ( t, x m − y m ) ≤ ctρ γ ( t, x k − y k ) ρ ( t, x m − y m ) , (42) G k,m ( t, x, y ) ≤ c Z t (cid:16) ( t − s ) γ − αα s + ( t − s ) γ − βα s βα (cid:17) (cid:16) ( t − s ) βα + ( t − s ) s β − αα (cid:17) ds × ρ ( t, x k − y k ) ρ ( t, x m − y m )+ c Z t (cid:16) ( t − s ) γ − αα s + ( t − s ) γ − βα s βα (cid:17) dsρ ( t, x k − y k ) ρ β ( t, x m − y m )+ c Z t (cid:16) ( t − s ) γ − βα (cid:16) ( t − s ) βα + ( t − s ) s β − αα (cid:17)(cid:17) dsρ β ( t, x k − y k ) ρ ( t, x m − y m )+ c Z t ( t − s ) γ − βα dsρ β ( t, x k − y k ) ρ β ( t, x m − y m ) ≤ ct γα + βα ρ ( t, x k − y k ) ρ ( t, x m − y m ) + ct γα ρ ( t, x k − y k ) ρ β ( t, x m − y m )+ ct γα ρ β ( t, x k − y k ) ρ ( t, x m − y m ) + ct γα − βα ρ β ( t, x k − y k ) ρ β ( t, x m − y m ) ≤ ctρ γ ( t, x k − y k ) ρ ( t, x m − y m ) + ctρ βγ − β ( t, x k − y k ) ρ ( t, x m − y m ) . (43)By (38-43) we obtain A ( t, x, y ) ≤ c d X k =1 d Y i =1 i = k tρ ( t, x i − y i ) (cid:16) ρ γ ( t, x k − y k ) + ρ βγ − β ( t, x k − y k ) (cid:17) . Using this and (37) we obtain that for any n ∈ N , x, x ′ , y ∈ R d , t ∈ (0 , T ] and γ ∈ (0 , β ) | q n ( t, x, y ) − q n ( t, x ′ , y ) | ≤ cC n − ( n !) β/α (cid:16) | x − x ′ | β − γ ∧ (cid:17) × d X k =1 d Y i =1 i = k ( tρ ( t, x i − y i )) (cid:16) ρ γ ( t, x k − y k ) + ρ βγ − β ( t, x k − y k ) (cid:17) + d X k =1 d Y i =1 i = k ( tρ ( t, x ′ i − y i )) (cid:16) ρ γ ( t, x ′ k − y k ) + ρ βγ − β ( t, x ′ k − y k ) (cid:17) . This, Lemma 3.8 and the definition of q imply the assertion of the theorem. (cid:3) Lemma 3.10.
For all γ ∈ (0 , , x, x ′ , y ∈ R d , t > , we have | p y ( t, x ) − p y ( t, x ′ ) | ≤ c | x − x ′ | γ t − γ/α " d Y i =1 g t ( x i ) + " d Y i =1 g t ( x ′ i ) . Proof.
By Lemma 3.3 we get | p y ( t, x ) − p y ( t, x ′ ) | ≤ c (cid:16) | x − x ′ | t − /α ∧ (cid:17) " d Y i =1 g t ( x i ) + " d Y i =1 g t ( x ′ i ) . Since (cid:0) | x − x ′ | t − /α ∧ (cid:1) ≤ | x − x ′ | γ t − γ/α we obtain the assertion of the lemma. (cid:3) By Lemma 3.3 and the formula for p y ( t, x ) we obtain Lemma 3.11.
For any x, y ∈ R d and t > we have |∇ p y ( t, · )( x − y ) | ≤ ct − α d Y i =1 ρ α ( t, x i − y i ) . RANSITION DENSITIES FOR SDES DRIVEN BY CYLINDRICAL STABLE PROCESSES 19
Lemma 3.12.
For any x ∈ R d and t ∈ (0 , T ] we have (cid:12)(cid:12)(cid:12)(cid:12)Z R d ∇ p y ( t, · )( x − y ) dy (cid:12)(cid:12)(cid:12)(cid:12) ≤ ct β − α . Proof.
Let D p y ( t, · )( x − y ) = lim h → ( p y ( t, x − y + he ) − p y ( t, x − y )) /h . It is enough to provethe estimate for I = R R d D p y ( t, · )( x − y ) dy . Let γ ∈ (0 , β ) and put ˜ y = ( x , y , . . . , y d ).We have | I | = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z R d " a ( y ) g ′ t (cid:18) x − y a ( y ) (cid:19) " d Y i =2 a ii ( y ) g t (cid:18) x i − y i a ii ( y ) (cid:19) − a (˜ y ) g ′ t (cid:18) x − y a (˜ y ) (cid:19) " d Y i =2 a ii (˜ y ) g t (cid:18) x i − y i a ii (˜ y ) (cid:19) dy (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:12)(cid:12)(cid:12)(cid:12)Z R d − (cid:20)Z R (cid:20) a ( y ) g ′ t (cid:18) x − y a ( y ) (cid:19) − a (˜ y ) g ′ t (cid:18) x − y a (˜ y ) (cid:19)(cid:21) dy (cid:21) × " d Y i =2 a ii (˜ y ) g t (cid:18) x i − y i a ii (˜ y ) (cid:19) dy , . . . dy d (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)(cid:12)Z R d a ( y ) g ′ t (cid:18) x − y a ( y ) (cid:19) × " d Y i =2 a ii ( y ) g t (cid:18) x i − y i a ii ( y ) (cid:19) − d Y i =2 a ii (˜ y ) g t (cid:18) x i − y i a ii (˜ y ) (cid:19) dy (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = II + III . By [11, (2.31)] we get (cid:12)(cid:12)(cid:12)(cid:12) a ( y ) g ′ t (cid:18) x − y a ( y ) (cid:19) − a (˜ y ) g ′ t (cid:18) x − y a (˜ y ) (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) ≤ c ( | x − y | β ∧ t − /α ( ρ α ( t, x − y )+ ρ γα − γ ( t, x − y )) . Using this and (10) we obtain II ≤ ct β − α . Note also that (cid:12)(cid:12)(cid:12)(cid:12) a ( y ) g ′ t (cid:18) x − y a ( y ) (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) ≤ cρ α − ( t, x − y ) . Using this, Corollary 3.6 and (10) we get III ≤ ct β − α . (cid:3) Similarly as in [11] we denote φ y ( t, x, s ) = Z R d p z ( t − s, x − z ) q ( s, z, y ) dz and ϕ y ( t, x ) = Z t φ y ( t, x, s ) ds. Clearly we have p A ( t, x, y ) = p y ( t, x − y ) + ϕ y ( t, x ) . (44)By well known estimates of ∇ p z ( t − s, · )( x − z ) and Theorem 3.2 we easily obtain thefollowing result. Lemma 3.13.
For any x, y ∈ R d , t > and s ∈ (0 , t ) we have ∇ x φ y ( t, x, s ) = Z R d ∇ p z ( t − s, · )( x − z ) q ( s, z, y ) dz. The next result is the most important step in proving gradient estimates of p A ( t, x, y ). Lemma 3.14.
For any α ∈ (1 , , x, y ∈ R d and t ∈ (0 , T ] we have ∇ x ϕ y ( t, x ) = Z t Z R d ∇ p z ( t − s, · )( x − z ) q ( s, z, y ) dz ds. (45) and |∇ x ϕ y ( t, x ) | ≤ ct − α d Y i =1 ρ α ( t, x i − y i ) . (46) Proof.
Let x, y ∈ R d , t ∈ (0 , T ] and s ∈ (0 , t ). The main tool used in this case is Theorem3.2. Using this theorem, Lemmas 3.11, 3.13 and (11) we obtain |∇ x φ y ( t, x, s ) | ≤ Z R d |∇ p z ( t − s, · )( x − z ) | | q ( s, z, y ) | dz ≤ c Z R d ( t − s ) − /α p z ( t − s, x − z ) × d X m =1 d Y i =1 i = m ρ α ( s, z i − y i ) [ ρ β ( s, z m − y m ) + ρ β ( s, z m − y m )] dz ≤ c d X m =1 d Y i =1 i = m ρ α ( t, x i − y i ) Z R ρ α − ( t − s, x m − z m ) × [ ρ β ( s, z m − y m ) + ρ β ( s, z m − y m )] dz ≤ c d X m =1 d Y i =1 i = m ρ α ( t, x i − y i ) h ( t − s ) β − α ρ ( t, x m − y m )+( t − s ) α − α s β − αα ρ ( t, x m − y m ) + ( t − s ) − α ρ β ( t, x m − y m )+( t − s ) − α s βα ρ ( t, x m − y m ) i . It follows that ∇ x (cid:20)Z t φ y ( t, x, s ) ds (cid:21) = Z t ∇ x φ y ( t, x, s ) ds, which implies (45). We also obtain (cid:12)(cid:12)(cid:12)(cid:12)Z t ∇ x φ y ( t, x, s ) ds (cid:12)(cid:12)(cid:12)(cid:12) ≤ c d X m =1 d Y i =1 i = m ρ α ( t, x i − y i ) Z t h ( t − s ) β − α ρ ( t, x m − y m )+( t − s ) α − α s β − αα ρ ( t, x m − y m ) + ( t − s ) − α ρ β ( t, x m − y m )+ ( t − s ) − α s βα ρ ( t, x m − y m ) i ds. ≤ c d X m =1 d Y i =1 i = m ρ α ( t, x i − y i ) [ ρ α + β − ( t, x m − y m ) + ρ βα − ( t, x m − y m )] , which implies (46). (cid:3) RANSITION DENSITIES FOR SDES DRIVEN BY CYLINDRICAL STABLE PROCESSES 21
Proposition 3.15.
For any α ∈ (1 , , t ∈ (0 , T ] and x, y ∈ R d we have |∇ x p A ( t, x, y ) | ≤ ct − α d Y i =1 g t ( x i − y i ) . Proof.
The assertion follows from formula (44) and Lemmas 3.11, 3.14. (cid:3) Feller semigroup
For any bounded Borel f : R d → R , t ∈ (0 , ∞ ) and x ∈ R d we define P At f ( x ) = Z R d p A ( t, x, y ) f ( y ) dy. The main aim of this section is to show that { P At } is a Feller semigroup.For any ε ≥ x ∈ R d we put L ε f ( x ) = A α d X i =1 Z { z i : | z i | >ε } δ f ( x, e i z i ) σ i ( x ) dz i | z i | α , L yε f ( x ) = A α d X i =1 Z { z i : | z i | >ε } δ f ( x, e i z i ) σ i ( y ) dz i | z i | α . Using [11, (3.13)] and the same arguments as in the proof of Lemma 3.12 we obtain
Lemma 4.1.
For any ε > , x, y ∈ R d and t ∈ (0 , T ] we have (cid:12)(cid:12)(cid:12)(cid:12)Z R d L xε p y ( t, · )( x − y ) dy (cid:12)(cid:12)(cid:12)(cid:12) ≤ ct β − αα . Lemma 4.2.
For any x, y ∈ R d and t > L x ϕ y ( t, x ) is well defined and we have L x ϕ y ( t, x ) = Z t Z R d L x p z ( t − s, · )( x − z ) q ( s, z, y ) dz ds. (47) Fix γ ∈ (0 , β ) . There exists c such that for any ε > , t ∈ (0 , T ] and x, y ∈ R d we have | L ε p A ( t, · , y )( x ) | ≤ ct − α + γ − βα d Y i =1 ρ α ( t, x i − y i ) . (48) Moreover, t → L x ϕ y ( t, x ) is continuous on (0 , T ) for any x, y ∈ R d .Proof. Let ε >
0. We have L ε ϕ y ( t, x ) = Z t Z R d L xε p z ( t − s, · )( x − z ) q ( s, z, y ) dz ds. By Lemma 3.1 and Theorem 3.2 one easily getslim ε → + Z R d L xε p z ( t − s, · )( x − z ) q ( s, z, y ) dz = Z R d L x p z ( t − s, · )( x − z ) q ( s, z, y ) dz. (49)The most difficult part of the proof is to justifylim ε → + Z t Z R d L xε p z ( t − s, · )( x − z ) q ( s, z, y ) dz ds = Z t lim ε → + Z R d L xε p z ( t − s, · )( x − z ) q ( s, z, y ) dz ds. (50) We have L ε ϕ y ( t, x ) = Z t/ Z R d L xε p z ( t − s, · )( x − z ) q ( s, z, y ) dz ds + Z tt/ Z R d L xε p z ( t − s, · )( x − z ) dzq ( s, x, y ) ds + Z tt/ Z R d L xε p z ( t − s, · )( x − z )( q ( s, z, y ) − q ( s, x, y )) dz ds = D ( t, x, y ) + E ( t, x, y ) + F ( t, x, y ) . For s ∈ (0 , t/
2) by Theorem 3.2, Lemma 3.1 and (11) we obtain Z R d | L xε p z ( t − s, · )( x − z ) q ( s, z, y ) | dz ≤ c d X m =1 d Y i =1 i = m ρ α ( t, x i − y i ) × Z R ρ ( t − s, x m − z m )[ ρ β ( s, z m − y m ) + ρ β ( s, z m − y m )] dz ≤ c d X m =1 d Y i =1 i = m ρ α ( t, x i − y i ) × h ( t − s ) β − αα ρ ( t, x m − y m ) + s β − αα ρ ( t, x m − y m ) + ( t − s ) − ρ β ( t, x m − y m ) i . It follows that lim ε → + Z t/ Z R d L xε p z ( t − s, · )( x − z ) q ( s, z, y ) dz ds = Z t/ lim ε → + Z R d L xε p z ( t − s, · )( x − z ) q ( s, z, y ) dz ds. (51)and D ( t, x, y ) ≤ ct − d Y i =1 ρ α ( t, x i − y i ) . (52)For s ∈ ( t/ , t ) by Theorem 3.2 and Lemma 4.1 we obtain (cid:12)(cid:12)(cid:12)(cid:12)Z R d L xε p z ( t − s, · )( x − z ) dzq ( s, x, y ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ c ( t − s ) β − αα t − d Y i =1 ρ α ( t, x i − y i ) . It follows that lim ε → + Z tt/ Z R d L xε p z ( t − s, · )( x − z ) dzq ( s, x, y ) ds = Z tt/ lim ε → + Z R d L xε p z ( t − s, · )( x − z ) dzq ( s, x, y ) ds. (53)and E ( t, x, y ) ≤ ct − d Y i =1 ρ α ( t, x i − y i ) . (54) RANSITION DENSITIES FOR SDES DRIVEN BY CYLINDRICAL STABLE PROCESSES 23
Now, we need to obtain some estimates which will be crucial in studying the mostdifficult term F ( t, x, y ). By Lemma 3.1 and Theorem 3.9 there exists c (not depending on ε ) such that for any t ∈ (0 , T ] and x, y ∈ R d we have Z R d d X i =1 "Z | w i | >ε | δ p z ( t − s, x − z, w i e i ) | | w i | − − α dw i | q ( s, z, y ) − q ( s, x, y ) | dz ≤ c ( t − s ) − Z R d ( | x − z | β − γ ∧ " d Y i =1 ρ α ( t − s, x i − z i ) dz d X m =1 d Y i =1 i = m ρ α ( s, x i − y i ) × [ ρ βγ − β ( s, x m − y m ) + ρ γ ( s, x m − y m )]+ c ( t − s ) − Z R d ( | x − z | β − γ ∧ " d Y i =1 ρ α ( t − s, x i − z i ) d X m =1 d Y i =1 i = m ρ α ( s, z i − y i ) × [ ρ βγ − β ( s, z m − y m ) + ρ γ ( s, z m − y m )] dz = B ( s, t, x, y ) + B ( s, t, x, y ) . Clearly ( | x − z | β − γ ∧ ≤ P dk =1 ( | x k − z k | β − γ ∧ t − s ) − Z R d ( | x − z | β − γ ∧ " d Y i =1 ρ α ( t − s, x i − z i ) dz ≤ c d X k =1 Z R ρ β − γ ( t − s, x k − z k ) dz k ≤ c ( t − s ) β − γ − αα . Hence B ( s, t, x, y ) ≤ c ( t − s ) β − γ − αα d X m =1 d Y i =1 i = m ρ α ( s, x i − y i ) × [ ρ βγ − β ( s, x m − y m ) + ρ γ ( s, x m − y m )] . (55)We also have B ( s, t, x, y ) ≤ c d X m =1 Z R d d Y i =1 i = m ρ α ( t − s, x i − z i ) ρ α ( s, z i − y i ) × ρ β − γ ( t − s, x m − z m )[ ρ βγ − β ( s, z m − y m ) + ρ γ ( s, z m − y m )] dz + c d X m =1 d X k =1 k = m Z R d d Y i =1 i = m,k ρ α ( t − s, x i − z i ) ρ α ( s, z i − y i ) × ρ α ( t − s, x m − z m )[ ρ βγ − β ( s, z m − y m ) + ρ γ ( s, z m − y m )] × ρ β − γ ( t − s, x k − z k ) ρ α ( s, z k − y k ) dz = B ( s, t, x, y ) + B ( s, t, x, y ) . By (11), we have B ( s, t, x, y ) ≤ c d X m =1 d Y i =1 i = m ρ α ( t, x i − y i ) h ( t − s ) β − γ − αα s γ − βα ρ ( t, x m − y m ) + s β − αα ρ ( t, x m − y m )+( t − s ) β − γ − αα s γ − βα ρ β ( t, x m − y m ) + s γ − αα ρ β − γ ( t, x m − y m )+( t − s ) β − γ − αα s γα ρ ( t, x m − y m ) i . (56)By (11), we also have B ( s, t, x, y ) ≤ c d X m =1 d X k =1 k = m d Y i =1 i = m,k ρ α ( t, x i − y i ) × hh ( t − s ) βα s γ − βα + ( t − s ) αα s γ − αα + s γα i ρ ( t, x m − y m ) + s γ − βα ρ β ( t, x m − y m ) i × hh ( t − s ) β − γ − αα s αα + s β − γα i ρ ( t, x k − y k ) + ρ β − γ ( t, x k − y k ) i (57)By (55-57) and (10), we getlim ε → + Z tt/ Z R d L xε p z ( t − s, · )( x − z )( q ( s, z, y ) − q ( s, x, y )) dz ds = Z tt/ lim ε → + Z R d L xε p z ( t − s, · )( x − z )( q ( s, z, y ) − q ( s, x, y )) dz ds. (58)and F ( t, x, y ) ≤ ct − α + γ − βα d Y i =1 ρ α ( t, x i − y i ) . (59)By (51), (53), (58) we get (50). We also get continuity t → L x ϕ y ( t, x ). By (49) and(50) we obtain (47). Using (52), (54), (59), Lemma 3.1 and formula (44) we get (48). (cid:3) The next result is an analogue of [11, Theorem 4.1]. Its proof is almost the same as theproof of [11, Theorem 4.1] and is omitted.
Proposition 4.3.
Let u ( t, x ) ∈ C b ([0 , T ] × R d ) with lim t → + sup x ∈ R d | u ( t, x ) − u (0 , x ) | = 0 . (60) Assume that t → L u ( t, x ) is continuous on (0 , T ] for each x ∈ R d (61) and for any ε ∈ (0 , and some γ ∈ (( α − ∨ , t ∈ ( ε,T ) | u ( t, x ) − u ( t, x ′ ) | ≤ K ε | x − x ′ | γ , x, x ′ ∈ R d . (62) If u satisfies ∂∂t u ( t, x ) = L u ( t, x ) , t ∈ (0 , T ] , x ∈ R d , (63) then sup t ∈ (0 ,T ) sup x ∈ R d u ( t, x ) ≤ sup x ∈ R d u (0 , x ) . (64) RANSITION DENSITIES FOR SDES DRIVEN BY CYLINDRICAL STABLE PROCESSES 25
Lemma 4.4.
Let x, y ∈ R d . Then the mapping t → ϕ y ( t, x ) is absolutely continuous on (0 , T ] . For any t ∈ (0 , T ) we have ∂ϕ y ∂t ( t, x ) = q ( t, x, y ) + Z t Z R d L z p z ( t − s, · )( x − z ) q ( s, z, y ) dz ds. (65) Proof.
Let h > t + h < T . We have ϕ y ( t + h, x ) − ϕ y ( t, x ) h = 1 h Z t + h Z R d p z ( t + h − s, x − z ) q ( s, z, y ) dz ds − h Z t Z R d p z ( t − s, x − z ) q ( s, z, y ) dz ds = 1 h Z t + h Z R d p z ( t + h − s, x − z ) q ( s, z, y ) dz ds − h Z t Z R d p z ( t + h − s, x − z ) q ( s, z, y ) dz ds + 1 h Z t Z R d p z ( t + h − s, x − z ) q ( s, z, y ) dz ds − h Z t Z R d p z ( t − s, x − z ) q ( s, z, y ) dz ds = 1 h Z t + ht Z R d p z ( t + h − s, x − z ) q ( s, z, y ) dz ds + Z t Z R d p z ( t + h − s, x − z ) − p z ( t − s, x − z ) h q ( s, z, y ) dz ds = I + II . After change of variables t + h − s = u we haveI = 1 h Z h Z R d p z ( u, x − z ) q ( t + h − u, z, y ) dz du = 1 h Z h Z R d ( p z ( u, x − z ) − p x ( u, x − z )) q ( t + h − u, z, y ) dz du + 1 h Z h Z R d p x ( u, x − z ) q ( t + h − u, z, y ) dz du = I + I . By Theorem 3.2 , sup u ≤ h,z ∈ R d q ( t + h − u, z, y ) ≤ M < ∞ . Moreover, from Corollary3.5, | p z ( u, x − z ) − p x ( u, x − z ) | ≤ cp x ( u, x − z )( | x − z | β ∧ . Hence, lim sup h → + | I | ≤ cM lim sup h → + h Z h Z R d p x ( u, x − z )( | x − z | β ∧ dz du = 0 . (66)Next, by Theorem 3.2, the function q ( s, z, y ) is continuous and bounded on [ t, T ] × R d , asa function of s and z . Since the measures µ u ( dz ) = p x ( u, x − z ) dz converge weakly to δ x as u → + , we obtain lim h → + I = q ( t, x, y ) . (67) We have II = Z t/ Z R d p z ( t + h − s, x − z ) − p z ( t − s, x − z ) h q ( s, z, y ) dz ds + Z tt/ Z R d p z ( t + h − s, x − z ) − p z ( t − s, x − z ) h q ( s, z, y ) dz ds = III + IV . By estimates of ∂∂t p z ( t − s, x − z ) following from (17) and Theorem 3.2 we getlim h → + III = Z t/ Z R d ∂∂t p z ( t − s, x − z ) q ( s, z, y ) dz ds. (68)We haveIV = Z tt/ Z R d p z ( t + h − s, x − z ) − p z ( t − s, x − z ) h ( q ( s, z, y ) − q ( s, x, y )) dz ds + Z tt/ Z R d p z ( t + h − s, x − z ) − p z ( t − s, x − z ) h dzq ( s, x, y ) ds = V + VI . Note that for h > s ∈ ( t/ , t ), γ ∈ (0 , β ), x, y, z ∈ R d , by Theorem 3.9 and theestimates of ∂∂t p z ( t − s, x − z ), we obtain (cid:12)(cid:12)(cid:12)(cid:12) p z ( t + h − s, x − z ) − p z ( t − s, x − z ) h (cid:12)(cid:12)(cid:12)(cid:12) | ( q ( s, z, y ) − q ( s, x, y )) |≤ B ( | x − z | β − γ ∧ (cid:12)(cid:12)(cid:12)(cid:12) ∂∂t p z ( t + θh − s, x − z ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ cB ( | x − z | β − γ ∧ t − s ) − d Y i =1 ρ α ( t − s, x i − z i ) , where B = B ( t, α, d, b , b , b , β, γ ) ∈ (0 , ∞ ) and θ = θ ( s, t, h, α, d, b , b , b , β, γ, x, z ) ∈ (0 , Z R d ( | x − z | β − γ ∧ t − s ) − d Y i =1 ρ α ( t − s, x i − z i ) dz ≤ c d X i =1 Z R ρ β − γ ( t − s, x i − z i ) dz i ≤ c ( t − s ) β − γ − αα . It follows thatlim h → + V = Z tt/ Z R d ∂∂t p z ( t − s, x − z )( q ( s, z, y ) − q ( s, x, y )) dz ds. (69)Note that for h > s ∈ ( t/ , t ), x, z ∈ R d we have1 h (cid:18)Z R d p z ( t + h − s, x − z ) dz − Z R d p z ( t − s, x − z ) dz (cid:19) = ∂∂t Z R d p z ( t + θh − s, x − z ) dz = Z R d ∂∂t p z ( t + θh − s, x − z ) dz, where θ = θ ( s, t, h, α, d, b , b , b , β, γ, x ) ∈ (0 , RANSITION DENSITIES FOR SDES DRIVEN BY CYLINDRICAL STABLE PROCESSES 27
Using this, (13) and the definition of q ( t, x, y ) we get Z R d p z ( t + h − s, x − z ) − p z ( t − s, x − z ) h dz = Z R d L z p z ( t + θh − s, · )( x − z ) dz = − Z R d q ( t + θh − s, x, z ) dz + Z R d L x p z ( t + θh − s, · )( x − z ) dz. By (19) and (10), we have (cid:12)(cid:12)(cid:12)(cid:12)Z R d q ( t + θh − s, x, z ) dz (cid:12)(cid:12)(cid:12)(cid:12) ≤ c ( t − s ) β − αα . By Lemma 4.1 we get (cid:12)(cid:12)(cid:12)(cid:12)Z R d L x p z ( t + θh − s, · )( x − z ) dz (cid:12)(cid:12)(cid:12)(cid:12) ≤ c ( t − s ) β − αα . It follows that lim h → + VI = Z tt/ Z R d ∂∂t p z ( t − s, x − z ) dzq ( s, x, y ) ds. (70)By (66-70) we obtainlim h → + ϕ y ( t + h, x ) − ϕ y ( t, x ) h = q ( t, x, y ) + Z t Z R d L z p z ( t − s, · )( x − z ) q ( s, z, y ) dz ds. The proof of the analogous result for lim h → − is very similar and it is omitted. (cid:3) Proposition 4.5.
For all t ∈ (0 , ∞ ) and x, y ∈ R d we have ∂∂t p A ( t, x, y ) = L p A ( t, · , y )( x ) . Proof.
By the definition of q ( t, x, y ) we obtain q ( t, x, y ) = q ( t, x, y ) + Z t Z R d q ( t − s, x, z ) q ( s, z, y ) dz ds. (71)Using (44), (13), Lemma 4.4 and the definition of q ( t, x, y ) we obtain ∂p A ∂t ( t, x, y ) = ∂p y ∂t ( t, x − y ) + ∂ϕ y ∂t ( t, x )= L y p y ( t, x − y ) + q ( t, x, y ) + Z t Z R d L z p z ( t − s, · )( x − z ) q ( s, z, y ) dz ds = L x p y ( t, x − y ) − q ( t, x, y ) + q ( t, x, y ) + Z t Z R d L z p z ( t − s, · )( x − z ) q ( s, z, y ) dz ds. By (71) this is equal to L x p y ( t, x − y ) + Z t Z R d ( q ( t − s, x, z ) + L z p z ( t − s, · )( x − z )) q ( s, z, y ) dz ds = L x p y ( t, x − y ) + Z t Z R d L x p z ( t − s, · )( x − z ) q ( s, z, y ) dz ds. By (47) and (44) this is equal to L x p A ( t, · , y )( x ), which completes the proof. (cid:3) Lemma 4.6.
For any bounded Borel f : R d → R , t ∈ (0 , ∞ ) and x ∈ R d we have L ( P At f )( x ) = ∂∂t P At f ( x ) . Proof.
We have L ( P At f )( x ) = lim ε → + L ε ( P At f )( x ) = lim ε → + Z R d L ε p A ( t, · , y )( x ) f ( y ) dy. By Lemma 4.2 this is equal to Z R d lim ε → + L ε p A ( t, · , y )( x ) f ( y ) dy = Z R d L p A ( t, · , y )( x ) f ( y ) dy. (72)By Proposition 4.5 this is equal to Z R d ∂∂t p A ( t, x, y ) f ( y ) dy = ∂∂t Z R d p A ( t, x, y ) f ( y ) dy. (cid:3) Proposition 4.7.
For t ∈ (0 , T ] and x, y ∈ R d we have p A ( t, x, y ) ≤ c d Y i =1 g t ( x i − y i ) . Proof.
By Theorem 3.2, estimates of p z and (12) we obtain | ϕ y ( t, x ) | = (cid:12)(cid:12)(cid:12)(cid:12)Z t Z R d p z ( t − s, x − z ) q ( s, z, y ) dz ds (cid:12)(cid:12)(cid:12)(cid:12) ≤ c Z t Z R d d Y i =1 ρ α ( t − s, x i − z i ) × d X m =1 d Y i =1 i = m ρ α ( s, z i − y i ) [ ρ β ( s, z m − y m ) + ρ β ( s, z m − y m )] dz ds ≤ c d X m =1 d Y i =1 i = m ρ α ( t, x i − y i ) Z t Z R ρ α ( t − s, x m − z m ) × [ ρ β ( s, z m − y m ) + ρ β ( s, z m − y m )] dz ds ≤ c d X m =1 d Y i =1 i = m ρ α ( t, x i − y i ) [ ρ α + β ( t, x m − y m ) + ρ βα ( t, x m − y m )] ≤ c " d Y i =1 ρ α ( t, x i − y i ) t β/α + d X m =1 (cid:16) | x m − y m | β ∧ (cid:17) . (73)Now the conlusion follows from (16) and estimates of p y . (cid:3) The following result shows that { P At } is a Feller semigroup. Theorem 4.8.
We have:(i) P At : C ( R d ) → C ( R d ) for any t ∈ (0 , ∞ ) ,(ii) lim t → + || P At f − f || ∞ = 0 for any f ∈ C ( R d ) .(iii) p A ( t, x, y ) ≥ for any ( t, x, y ) ∈ (0 , ∞ ) × R d × R d ,(iv) R R d p A ( t, x, y ) dy = 1 for any ( t, x ) ∈ (0 , ∞ ) × R d ,(v) R R d p A ( t, x, z ) p A ( s, z, y ) dz = p A ( s + t, x, y ) for any ( s, t, x, y ) ∈ (0 , ∞ ) × (0 , ∞ ) × R d × R d . RANSITION DENSITIES FOR SDES DRIVEN BY CYLINDRICAL STABLE PROCESSES 29
Proof. (i) follows by the fact that x → p A ( t, x, y ) is continuous and by Proposition 4.7.It is shown in the proof of Proposition 4.7 that | ϕ y ( t, x ) | ≤ c d X m =1 d Y i =1 i = m ρ α ( t, x i − y i ) [ ρ α + β ( t, x m − y m ) + ρ βα ( t, x m − y m )] . Let f ∈ C ( R d ). It follows that lim t → + sup x ∈ R d (cid:12)(cid:12)R R d ϕ y ( t, x ) f ( y ) dy (cid:12)(cid:12) = 0 for any f ∈ C ( R d ). It is clear that lim t → + sup x ∈ R d (cid:12)(cid:12)R R d p y ( t, x − y ) f ( y ) dy − f ( x ) (cid:12)(cid:12) = 0 for any f ∈ C ( R d ). Hence we obtain (ii).For any ( t, x ) ∈ (0 , T ] × R d put u ( t, x ) = P At f ( x ), u (0 , x ) = f ( x ). Note that u ( t, x )satisfies the assumptions of Proposition 4.3. Indeed, (ii) gives (60). By Lemma 4.2 weget (61). By Theorem 1.1 (iv) and Proposition 3.15 we obtain (62). Lemma 4.6 gives(63). Applying Proposition 4.3 to f ∈ C ∞ c , f ≤ u ( t, x ) = − P At x ), u (0 , x ) = 0 also satisfies the assumptions of Proposition 4.3. Using thisproposition we get that P At ≡ s ∈ (0 , T ], f ∈ C ∞ c , f ≥ u ( t, x ) = P At + s f ( x ), u ( t, x ) = P At P As f ( x ), u (0 , x ) = u (0 , x ) = P As f ( x ), u ( t, x ) = u ( t, x ) − u ( t, x ). By Proposition 4.3 applied to u ( t, x ) we get u ≡ u whichimplies (v). (cid:3) Using similar ideas as in the proof of (47) one can easily obtain the following result.
Lemma 4.9.
For any t ∈ (0 , ∞ ) , x ∈ R d and any bounded, H¨older continuous function f we have L (cid:20)Z t P As f ( · ) ds (cid:21) ( x ) = Z t L P As f ( x ) ds. (74) Proposition 4.10.
For any t ∈ (0 , ∞ ) , x ∈ R d and f ∈ C b ( R d ) we have P At f ( x ) = f ( x ) + Z t P As L f ( x ) ds. (75) Proof.
Put u ( t, x ) = f ( x ) + R t P As L f ( x ) ds . By (74) we get L u ( t, x ) = L f ( x ) + Z t L (cid:0) P As L f (cid:1) ( x ) ds. By Lemma 4.6 this is equal to L f ( x ) + Z t ∂∂s (cid:0) P As L f (cid:1) ( x ) ds = P At L f ( x ) = ∂∂t u ( t, x ) . It is easy to check that u ( t, x ) satisfies (60-62). Put ˜ u ( t, x ) = P At f ( x ), ˜ u (0 , x ) = f ( x ) and v ( t, x ) = u ( t, x ) − ˜ u ( t, x ). By the arguments from the proof of Theorem 4.8 we obtain that˜ u ( t, x ) satisfies (60-63). Using Proposition 4.3 for v ( t, x ) we get v ≡ (cid:3) The next theorem gives that L is a generator of the semigroup { P At } . Theorem 4.11.
For any f ∈ C b ( R d ) we have lim t → + P At f ( x ) − f ( x ) t = L f ( x ) , x ∈ R d and the convergence is uniform.Proof. By Proposition 4.10 we havelim t → + P At f ( x ) − f ( x ) t = lim t → + t Z t P As L f ( x ) ds. By Theorem 4.8 (ii) this is equal to L f ( x ) and the convergence is uniform. (cid:3) We are now in a position to provide the proofs of most of the parts of Theorem 1.1. proof of Theorem 1.1 (i), (ii) and the upper bound estimate in (iii).
From Theorem 4.8 andTheorem 4.11 we conclude that there is a Feller process ˜ X t with the transition kernel p A ( t, x, y ) and the generator L . Let P x , E x be the distribution and expectation for theprocess starting from x ∈ R d . First, note that for any function f ∈ C b ( R d ), the process M ˜ X,ft = f ( ˜ X t ) − f ( ˜ X ) − Z t L f ( ˜ X s ) ds is a ( P x , F t ) martingale, where F t is a natural filtration. That is P x solves the martingaleproblem for ( L , C b ( R d )). On the other hand, according to [1, Theorem 6.3], the uniqueweak solution X to the stochastic equation (1) has the law which is the unique solutionto the martingale problem for ( L , C b ( R d )). It follows that that ˜ X and X have the samelaw and p A ( t, x, y ) is the transition kernel of X .The continuity of p A ( t, x, y ) with respect to all variables follows from Theorem 3.2.Positivity is a consequence of the lower bound in (5) which will be proved in the nextsection. Finally, (ii) follows from Proposition 4.5. The upper bound estimate in (iii)follows from Proposition 4.7. (cid:3) proof of Theorem 1.1 (iv). The main tool used in this proof is Theorem 3.2. By Lemma3.10 and Theorem 3.2 we get (cid:12)(cid:12) ϕ y ( t, x ) − ϕ y ( t, x ′ ) (cid:12)(cid:12) ≤ Z t Z R d (cid:12)(cid:12) p z ( t − s, x − z ) − p z ( t − s, x ′ − z ) (cid:12)(cid:12) | q ( s, z, y ) | dz ds ≤ c (cid:0) A ( t, x, y ) + A ( t, x ′ , y ) (cid:1) , (76)where A ( t, x, y ) = | x − x ′ | γ Z t Z R d ( t − s ) − γ/α " d Y i =1 g t − s ( x i − z i ) × s d − " d Y i =1 ρ ( s, z i − y i ) s β/α + d X m =1 (cid:16) | z m − y m | β ∧ (cid:17) dz ds. We have A ( t, x, y ) ≤ c | x − x ′ | γ d X m =1 d Y i =1 i = m g t ( x i − y i ) × Z t Z R ρ α − γ ( t − s, x m − z m ) ρ β ( s, z m − y m ) dz m ds + c | x − x ′ | γ " d Y i =2 g t ( x i − y i ) t Z R ρ α − γ ( t − s, x − z ) ρ β ( s, z − y ) dz ds. By (12) we have Z t Z R ρ α − γ ( t − s, x m − z m ) ρ β ( s, z m − y m ) dz m ds ≤ ρ α − γ + β ( t, x m − y m ) + ρ βα − γ ( t, x m − y m ) , and Z t Z R ρ α − γ ( t − s, x − z ) ρ β ( s, z − y ) dz ds ≤ ρ α − γ + β ( t, x − y ) . RANSITION DENSITIES FOR SDES DRIVEN BY CYLINDRICAL STABLE PROCESSES 31
It follows that A ( t, x, y ) ≤ c | x − x ′ | γ t − γ/α " d Y i =1 g t ( x i − y i ) . Using this, (76), Lemma 3.10 and (44) we get the assertion of Theorem 1.1 (iv). (cid:3) Lower bound estimates
L´evy system.
Let P x , E x be the distribution and expectation for the process X t starting from x ∈ R d . By F t we denote a natural filtration. For x ∈ R d and Borel A ⊂ R d we define the jumping measure J ( x, A ) = A α d X i =1 Z A ⊗ k = i δ x k ( dy k ) a αii ( x ) dy i | y i − x i | α , where δ x k is a Dirac measure on R concentrated at x k .The purpose of this subsection is to provide arguments for the L´evy system formula.Namely, we will show that for any x ∈ R d and any non-negative measurable function f on R + × R d × R d vanishing on { ( s, x, y ) ∈ R + × R d × R d ; x = y } and F t stopping time T ,we have E x X s ≤ T f ( s, X s − , X s ) = E x Z T Z R d f ( s, X s , y ) J ( X s , dy ) ds. (77)Since we exactly follow the approach of [11] we only briefly sketch the arguments.It is well known that for f ∈ C b ( R d ), L f ( x ) = A α d X i =1 Z R [ f ( x + a ii ( x ) w i e i ) + f ( x − a ii ( x ) w i e i ) − f ( x )] dw i | w i | α . For y ∈ R d we denote | y | ∞ = max i {| y i |} the sup-norm in R d . For x ∈ R d and r > B ( x, r ) = { y ∈ R d , | y − x | ∞ < r } . Then for f ∈ C b ( R d ), we can rewrite theformula of the generator as L f ( x ) = lim r ց Z B c ( x,r ) ( f ( y ) − f ( x )) J ( x, dy ) . As it has been already observed in the last section, for any function f ∈ C b ( R d ), theprocess M ft = f ( X t ) − f ( X ) − Z t L f ( X s ) ds is a ( P x , F t ) martingale. Suppose that A and B are two bounded closed subsets of R d having a positive distance from each other. Let f ∈ C b ( R d ) be such that f ( x ) = 0 , x ∈ A and f ( x ) = 1 , x ∈ B . We consider a martingale transform of M ft , N ft = Z t A ( X s − ) dM fs . By the Ito formula, if X s − ∈ A , we have dM fs = f ( X s ) − f ( X s − ) − L f ( X s ) ds = f ( X s ) − L f ( X s ) ds. This implies that N ft = X s ≤ t A ( X s − ) f ( X s ) − Z t A ( X s ) L f ( X s ) ds = X s ≤ t A ( X s − ) f ( X s ) − Z t A ( X s ) Z f ( y ) J ( X s , dy ) ds Approximating B by a decreasing sequence of smooth functions we show that X s ≤ t A ( X s − ) B ( X s ) − Z t A ( X s ) Z B J ( X s , dy ) ds is a martingale, hence E x X s ≤ t A ( X s − ) B ( X s ) = E x Z t A ( X s ) Z B J ( X s , dy ) ds. Using this and a routine measure theoretic argument, we get E x X s ≤ t f ( X s − , X s ) = E x Z t Z R d f ( X s , y ) J ( X s , dy ) ds for any x ∈ R d and any non-negative measurable function f on R d × R d vanishing on thediagonal.Finally, following the same arguments as in [10, Appendix A], we obtain (77).5.2. Lower bound of p A . We essentially follow the approach from [11], where an argu-ment relied on certain exit and hitting times estimates was applied, but the singularityof the jumping measure forces us to use an induction argument. We start with the neardiagonal estimate of the transition kernel.
Lemma 5.1.
For any a > there is c = c ( a, d, α, b , b , b , β ) and < t ≤ , t = t ( a, d, α, b , b , b , β ) such that for t ≤ t and x, y ∈ R d with | y − x | ∞ ≤ at /α , p A ( t, x, y ) ≥ ct − d/α . (78) Proof.
By (73), if | y − x | ∞ ≤ at /α , we have | ϕ y ( t, x ) | ≤ c d Y i =1 ρ α ( t, x i − y i ) " t β/α + d X m =1 (cid:16) | x m − y m | β ∧ (cid:17) ≤ c t − d/α t β/α . Hence, we can find t ≤ t ≤ t and | y − x | ∞ ≤ at /α we have p A ( t, x, y ) = p y ( t, x − y ) + ϕ y ( t, x ) ≥ p y ( t, x − y ) − | ϕ y ( t, x ) | ≥ c t − d/α − c t − d/α t β/α ≥ ct − d/α . (cid:3) Let for a Borel D ⊂ R d , τ D = inf { t > X t / ∈ D } and T D = inf { t > X t ∈ D } be the first exit and hitting time of D , respectively. Lemma 5.2.
There is c such that, for t ≤ , R > , x ∈ R d , P x ( τ B ( x,R ) ≤ t ) ≤ c tR α . RANSITION DENSITIES FOR SDES DRIVEN BY CYLINDRICAL STABLE PROCESSES 33
Proof.
Applying the strong Markow property, we obtain P x ( τ B ( x,R ) ≤ t ) ≤ P x ( τ B ( x,R ) ≤ t ; | X ( t ) − x | ∞ ≤ R/
8) + P x ( | X ( t ) − x | ∞ ≥ R/ ≤ P x ( τ B ( x,R ) ≤ t ; | X ( t ) − X ( τ B ( x,R ) ) | ∞ ≥ R/ P x ( | X ( t ) − x | ∞ ≥ R/ E x ( τ B ( x,R ) ≤ t ; P X ( τ B ( x,R ) ) ( | X ( t − τ B ( x,R ) ) − X ( τ B ( x,R ) ) | ∞ ≥ R/ P x ( | X ( t ) − x | ∞ ≥ R/ ≤ z sup s ≤ t P z ( | X ( s ) − z | ∞ ≥ R/ ≤ c tR α . The last step follows from the upper estimate (5) of the heat kernel p A ( t, x, y ). (cid:3) Lemma 5.3.
Let r > and x, y ∈ R d . Assume that | x − y | ≥ r and max ≤ i ≤ d | x i − y i | ≤ r . Then for t > , P x ( X ( t ) ∈ B ( y, r )) ≥ c rt | y − x | α P x ( τ B ( x,r ) ≥ t ) inf z P z ( τ B ( z, r ) > t ) Proof.
Let σ = T B ( y, r ) be the first hitting time. By the strong Markow property P x ( X ( t ) ∈ B ( y, r )) ≥ P x ( σ ≤ t ; sup σ ≤ s ≤ σ + t | X ( s ) − X ( σ ) | ∞ ≤ r )= E x (cid:18) σ ≤ t ; P X ( σ ) (cid:18) sup s ≤ t | X ( s ) − X ( σ ) | ∞ ≤ r (cid:19)(cid:19) ≥ P x ( σ ≤ t ) inf z P z (sup s ≤ t | X ( s ) − z | ∞ ≤ r ) ≥ inf z P z ( τ B ( z, r ) > t ) P x ( X ( t ∧ τ B ( x,r ) ) ∈ B ( y, r )) . By the L´evy system formula (77), we have P x (( X ( t ∧ τ B ( x,r ) ) ∈ B ( y, r )) = E x Z t ∧ τ B ( x,r ) Z B ( y, r ) J ( X s , du ) ds. We may assume x < y . Since | x − y | ≥ r and max ≤ i ≤ d | x i − y i | ≤ r , for z ∈ B ( x, r ),we have Z B ( y, r ) J ( z, du ) = Z y +2 ry − r a α ( z ) dw | w − z | α ≥ c r | y − x | α . Hence, P x (( X ( t ∧ τ B ( x,r ) ) ∈ B ( y, r )) ≥ c r | y − x | α E x [ t ∧ τ B ( x,r ) ] ≥ c rt | y − x | α P x ( τ B ( x,r ) ≥ t ) . (cid:3) Lemma 5.4.
There is t > , t = t ( d, α, b , b , b , β ) , such that for < t ≤ t , x, y ∈ R d satisfying max ≤ i ≤ d | x i − y i | ≤ t /α we have p A ( t, x, y ) ≥ c d Y i =1 g t ( x i − y i ) . Proof.
We pick t > a = 12 in Lemma 5.1. Due to the near diagonalestimate (78), it is enough to consider | x − y | ≥ t /α and max ≤ i ≤ d | x i − y i | ≤ t /α .Applying Lemma 5.3 with r = 2 t /α , we obtain p A ( t, x, y ) ≥ Z B ( y, r ) p A ( t , x, z ) p A ( t , z, y ) dz ≥ inf z ∈ B ( y, r ) p A ( t , z, y ) P x (( X ( t ) ∈ B ( y, r )) ≥ c inf z ∈ B ( y, r ) p A ( t , z, y ) rt | y − x | α P x ( τ B ( x,r ) ≥ t ) × inf z P z ( τ B ( z, r ) > t ) , (79)where t i > t + t = t .Now, due to Lemma 5.2, we can pick 0 < λ <
1, independently of t , such thatinf z P z ( τ B ( z,r ) ≥ λt ) ≥ / . Moreover, we can select λ so small that 8 ≤ − λ ) /α . Then for | z − y | ∞ ≤ r =8 t /α ≤ − λ ) t ) /α , by Lemma 5.1, we have p A ((1 − λ ) t, z, y ) ≥ ct − d/α . Taking t = λt, t = (1 − λ ) t and applying (79) we arrive at p A ( t, x, y ) ≥ ct − ( d − /α t | y − x | α ≥ c d Y i =1 g t ( x i − y i ) . (cid:3) Proof of the lower bound estimates in Theorem 1.1 (iii).
For a natural k ≤ d − V k ( t ) = { ( x, y ) ∈ R d ; min ≤ i ≤ k | x i − y i | ≥ t /α and max k +1 ≤ i ≤ d | x i − y i | ≤ t /α } . We set V ( t ) = { ( x, y ) ∈ R d ; max ≤ i ≤ d | x i − y i | ≤ t /α } and V d ( t ) = { ( x, y ) ∈ R d ; min ≤ i ≤ d | x i − y i | ≥ t /α } . By a renumeration argument it is enough to prove the corresponding lower bound on V k ( t ) , k = 0 , . . . , d . At first, we assume that t ≤ t , where t was found in Lemma 5.4. Wehave already proved the lower bound on V ( t ) and V ( t ). We show how to extend it to V ( t ).Thus , we consider the case | x − y | ≥ t /α , | x − y | ≥ t /α and max ≤ i ≤ d | x i − y i | ≤ t /α .Let x ′ = ( y , x , . . . , x d ). If z ∈ B ( x ′ , t /α /
4) then | x − z | ≥ (3 / t /α , max ≤ i ≤ d | x i − z i | ≤ t /α and | y − z | ≥ (3 / t /α , max i =2 | y i − z i | ≤ t /α . Hence, by Lemma 5.4, p A ( t, x, z ) ≥ ct − ( d − /α t | z − x | α ≥ ct − ( d − /α t | y − x | α ,p A ( t, z, y ) ≥ ct − ( d − /α t | z − y | α ≥ ct − ( d − /α t | y − x | α . RANSITION DENSITIES FOR SDES DRIVEN BY CYLINDRICAL STABLE PROCESSES 35
Finally, p A (2 t, x, y ) ≥ Z B ( x ′ ,t /α / p A ( t, x, z ) p A ( t, z, y ) dz ≥ ct − ( d − /α t | y − x | α t − ( d − /α t | y − x | α Z B ( x ′ ,t /α / dz ≥ c t | y − x | α t | y − x | α t − ( d − /α ≥ c d Y i =1 g t ( x i − y i ) . This concludes the proof of the lower bound on V ( t ). In a similar fashion, by inductionargument, we show that, if ( x, y ) ∈ V k ( t ) and t ≤ t , then p A ( t, x, y ) ≥ c t | y − x | α × · · · × t | y k − x k | α t − ( d − k ) /α ≥ c d Y i =1 g t ( x i − y i ) ≥ cp ( t, x − y ) , which ends the proof for the case t ≤ t . If t > t then we can write t = nt + s , with s < t and n ∈ N . Then by already proved lower bound p A ( t, x, y ) = Z R d · · · Z R d p A ( t , x, z ) . . . p A ( t , z n , z n +1 ) p A ( s, z n +1 , y ) dz . . . dz n +1 ≥ c n +1 Z R d · · · Z R d p ( t , x − z ) . . . p ( t , z n − z n +1 ) p ( s, z n +1 − y ) dz . . . dz n +1 = c n +1 p ( t, x − y ) . The proof is completed. (cid:3) proof of Theorem 1.1 (v).
The assertion follows from Proposition 3.15 and the lower boundestimate in Theorem 1.1 (iii). (cid:3)
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Faculty of Pure and Applied Mathematics, Wroc law University of Science and Technol-ogy, Wyb. Wyspia´nskiego 27, 50-370 Wroc law, Poland.
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