Transition density estimates for diagonal systems of SDEs driven by ...

TRANSITION DENSITY ESTIMATES FOR DIAGONAL SYSTEMS OF SDES DRIVEN BY CYLINDRICAL α-STABLE PROCESSES

arXiv:1711.07539v1 [math.PR] 20 Nov 2017

TADEUSZ KULCZYCKI AND MICHAL RYZNAR Abstract. We consider the system of stochastic differential equation dXt = A(Xt− ) dZt , X0 = x, driven by cylindrical α-stable process Zt in Rd . We assume that A(x) = (aij (x)) is diagonal and aii (x) are bounded away from zero, from infinity and H¨ older continuous. We construct transition density pA (t, x, y) of the process Xt and show sharp two-sided estimates of this density. We also prove H¨ older and gradient estimates of x → pA (t, x, y). Our approach is based on the method developed by Chen and Zhang in [11].

1. Introduction Let

(1)

(d)

Zt = (Zt , . . . , Zt ), (i)

be cylindrical α-stable process, that is Zt , i = 1, . . . , d are independent one-dimensional symmetric standard α-stable processes of index α ∈ (0, 2), d ∈ N, d ≥ 2. We consider the system of stochastic differential equation dXt = A(Xt− ) dZt ,

X0 = x,

(1)

where A(x) = (aij (x)) is diagonal and there are constants b1 , b2 , b3 > 0, β ∈ (0, 1] such that for any x, y ∈ Rd , i ∈ {1, . . . , d} b1 ≤ aii (x) ≤ b2 ,

(2)

|aii (x) − aii (y)| ≤ b3 |x − y|β . (3) In the sequel, without loss of generality, we assume that β ∈ (0, α/4]. It is well known that system of SDEs (1) has a unique weak solution [1]. The generator of X is given by (see ([1, (2.3)])) Z d X dwi Aα [f (x + aii (x)wi ei ) + f (x − aii (x)wi ei ) − 2f (x)] , lim Lf (x) = + 2 |wi |>ε |wi |1+α ε→0 i=1

{ej }dj=1

where is the standard basis in Rd and Aα = 2α Γ((1 + α)/2)/(π 1/2 |Γ(−α/2)|). Let us denote the transition density of one-dimensional symmetric standard α-stable process of index α ∈ (0, 2) by gt (x − y), t > 0, x, y ∈ R. Clearly, the transition density of Q Z(t) is given by dj=1 gt (xj − yj ). 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 solution to SDE (1) has a positive jointly continuous transition density function pA (t, x, y) in (t, x, y) ∈ (0, ∞) × Rd × Rd with respect to the Lebesgue measure on Rd . (ii) The transition density solves ∂ A p (t, x, y) = LpA (t, ·, y)(x), ∂t

(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, grant no. 2015/17/B/ST1/01043. 1

2

T. KULCZYCKI AND M. RYZNAR

for all t ∈ (0, ∞) and x, y ∈ Rd . (iii) For any T > 0 there exist c1 = c1 (T, d, α, b1 , b2 , b3 , β) ≥ 1 such that for any x, y ∈ Rd , t ∈ (0, T ] c−1 1

d Y

gt (xi − yi ) ≤ pA (t, x, y) ≤ c1

d Y

gt (xi − yi ).

(5)

i=1

i=1

(iv) For any T > 0 and γ ∈ (0, α ∧ 1) there exists c2 = c2 (T, γ, d, α, b1 , b2 , b3 , β) > 0 such that for any x, x′ , y ∈ Rd , t ∈ (0, T ] ! d d Y Y A p (t, x, y) − pA (t, x′ , y) ≤ c2 |x − x′ |γ t−γ/α gt (x′i − yi ) . (6) gt (xi − yi ) + i=1

i=1

(v) For any T > 0 and α ∈ (1, 2) there exist c3 = c3 (T, d, α, b1 , b2 , b3 , β) > 0 such that for any x, y ∈ Rd , t ∈ (0, T ] ∇x pA (t, x, y) ≤ c3 t−1/α pA (t, x, y). (7)

Systems of stochastic differential equations driven by cylindrical α-stable processes have attracted a lot of attention in recent years see e.g. [1, 33, 40, 32, 41, 35]. In [1] Bass and Chen proved existence and uniqueness of weak solutions of systems of SDEs (1) under very mild assumptions on matrices A(x) (i.e. they assumed that A(x) are continuous and bounded in x and nondegenerate for each x). Our paper may be treated as the first step in studying fine properties of transition densities of systems of SDEs driven by L´evy processes with singular L´evy measures. Fine properties of such transition densities are of great interest but in the case of singular L´evy measures relatively little is known. We decided to study the particular case of diagonal matrices A(x) in (1) because in that case one can obtain sharp two-sided estimates of these densities. It seems that in the case of general non-diagonal matrices in (1) such sharp two-sided estimates are impossible to obtain. Nevertheless, we believe that our results will help to obtain qualitative estimates of 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 of Zabczyk concerning gradient estimates of these densities. Another source of inspiration was a recent paper [5] of Bogdan, Knopova and Sztonyk, where they constructed heat kernels and obtained upper bounds and H¨older estimates of them for quite general anisotropic space-inhomogeneous non-local operators. However, the considered jump kernels cannot be “too singular”. In particular, the results from [5] can be applied for systems (1) only when d = 2 and α ∈ (1, 2) (see the condition α + γ > d in the assumption A1 on page 5 in [5]). Moreover, even for d = 2 and α ∈ (1, 2), the obtained estimates are far from being optimal. In our paper we use a very elegant and efficient method developed by Chen and Zhang in [11]. Their approach is based on Levi’s freezing coefficient argument (cf. [27, 12, 26]). In [11] the non-local and non-symmetric L´evy type operators on Rd are studied with jump kernels of the type κ(x, z)/|z|d+α , α ∈ (0, 2). It turned out that similar ideas can be applied also in our situation where jump kernels are much more singular. We follow the road-map from [11] however, due to a specific structure of the operator L, there are many differences between our paper and [11]. The main new elements, in comparison to [11], are the proof of crucial Theorem 3.2, the proof of Lemma 4.4, the estimates (55-57) and the proof of lower bound estimates of pA (t, x, y). It is worth pointing out that in our paper we have shown that the transition density pA (t, x, y) satisfies the equation (4) for all x, y ∈ Rd while in [11] it is shown that the heat kernel pκα (t, x, y) satisfies the analogous equation only when x 6= y. A similar remark concerns gradient estimates of pA (t, x, y), which we managed to show for all x, y ∈ Rd . On the other hand, we were able to prove gradient estimates of pA (t, x, y) only for α ∈ (1, 2) (in [11] gradient estimates were obtained for

TRANSITION DENSITIES FOR SDES DRIVEN BY CYLINDRICAL STABLE PROCESSES

3

α ∈ [1, 2)). It is worth mentioning that quite recently a very interesting generalization of the results from [11] appeared in [19]. The problem of estimates of transition densities for jump L´evy and L´evy-type processes has been intensively studied in recent years see e.g. [11, 19, 2, 10, 7, 22, 16, 20, 28, 21]. However, relatively few results concern processes with jump kernels which are not comparable to isotropic ones. We have already mentioned here the paper [5]. One should also mention the papers by Sztonyk et al. [4, 17, 16, 37] but they only concern heat kernels of translation invariant generators and convolution semigroups for which the existence and many properties follow by Fourier methods. There are also known estimates of anisotropic non-convolution heat kernels given in [36, 18] however these are obtained under the assumption that the jump kernel is dominated by that of the rotation invariant stable process. For estimates of derivatives of L´evy densities we refer the reader to [38, 3, 34, 16, 25, 20, 11]. Some estimates of transition densities for processes which are solutions of systems of SDEs driven by L´evy processes with singular L´evy measures were obtained in [29, 30, 31, 13]. However, the results from [29], when applied to system (1), do not imply such sharp estimates which are obtained in Theorem 1.1. In particular, they can be applied to system (1) only when x → aii (x) are C ∞ (Rd ) functions. What is more, even in this case, the upper bound estimates are of the form supx,y∈Rd pA (t, x, y) ≤ ct−d/α , while the lower bound estimates of pA (t, x, y) are also much less precise than ours. They are precise only for x = y, in which case it follows from [29] that pA (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 collect A known facts needed in the sequel. In Section P∞ 3 we construct the function p (t, x, y) in terms of the perturbation series q(t, x, y) = n=0 qn (t, x, y) using Picard’s iteration. In Theorem 3.2 we obtain the estimates of q(t, x, y) which are absolutely crucialR for the rest of the paper. In Section 4 we show that the semigroup defined by PtA f (x) = Rd pA (t, x, y)f (y) dy is a Feller semigroup. Next, applying [1], we argue that pA (t, x, y) is, in fact, the transition density of the solution of system (1) and we prove most parts of the main theorem. In Section 5 we show lower bound estimates of pA (t, x, y) by using probabilistic arguments. 2. Preliminaries All constants appearing in this paper are positive and finite. In the whole paper we fix T > 0, d ≥ 2, d ∈ N, α ∈ (0, 2), b1 , b2 , b3 , β, where b1 , b2 , b3 , β appear in (2) and (3). We adopt the convention that constants denoted by c (or c1 , c2 , . . .) may change their value from one use to the next. In the whole paper, unless is explicitly stated otherwise, we understand that constants denoted by c (or c1 , c2 , . . .) depend on T, d, α, b1 , b2 , b3 , β. We also understand that they may depend on the choice of the constant γ ∈ (0, β) (or γ ∈ (0, α ∧ 1)). We write f (x) ≈ g(x) for x ∈ A if f, g ≥ 0 on A and there is a constant c ≥ 1 such that c−1 f (x) ≤ g(x) ≤ cf (x) for x ∈ A. Denote σi (x) = aαii (x). Note that there exists c such that for any x, y ∈ Rd we have   |σi (x) − σi (y)| ≤ c |x − y|β ∧ 1 .

(8)

By simple change of variable we get Lf (x) =

d X i=1

Aα lim ε→0+ 2

Z

|zi |>ε

[f (x + ei zi ) + f (x − ei zi ) − 2f (x)] σi (x)

dzi . |zi |1+α

4

T. KULCZYCKI AND M. RYZNAR

Let us introduce some notation which was used in [11]. For a function f : Rd → R we denote δf (x, z) = f (x + z) + f (x − z) − 2f (x). Similarly, for a function f : R+ × Rd → R we write δf (t, x, z) = f (t, x + z) + f (t, x − z) − 2f (t, x). We also denote ρβγ (t, x) = tγ/α (|x|β ∧ 1)(t1/α + |x|)−1−α ,

t > 0, x ∈ R.

It is well known that t > 0, x ∈ R. (9) gt (x) ≈ ρ0α (t, x) One of the most important tools used in our paper are convolution estimates [11, (2.32.4)]. They are similar to [24, Lemma 1.4] and [39, Lemma 2.3]. In [11] they are stated for t ∈ (0, 1]. It is easy to check that they hold also for t ∈ (0, T ]. For reader’s convenience we collected them in Lemma 2.1. Lemma 2.1. (i) There is C = C(α) such that for any t > 0 and any β1 ∈ [0, α/2], γ1 ∈ R, Z γ1 +β1 −α ρβγ11 (t, z) dz ≤ Ct α .

(10)

R

(ii) For T > 0 there is C = C(α, T ) such that for any 0 < s < t ≤ T , x ∈ R and any β1 , β2 ∈ [0, α/4], γ1 , γ2 ∈ R we have Z ρβγ11 (t − s, x − z)ρβγ22 (s, z) dz R i h γ1 γ2 +β1 +β2 −α γ1 +β1 +β2 −α γ2 α α ρ00 (t, x) s α + (t − s) α s ≤ C (t − s) i h γ1 γ2 +β2 −α γ1 +β1 −α γ2 (11) +C (t − s) α s α ρβ0 2 (t, x) + (t − s) α s α ρβ0 1 (t, x) .

(iii) For T > 0 there is C = C(α, T ) such that for any 0 < t ≤ T , x ∈ R and any β1 , β2 ∈ [0, α/4], γ1 , γ2 ∈ R with γ1 + β1 > 0 and γ2 + β2 > 0 we have Z tZ ρβγ11 (t − s, x − z)ρβγ22 (s, z) dz ds 0 R    γ1 + β1 γ2 + β2  0 ≤ CB ργ1 +γ2 +β1 +β2 + ρβγ11+γ2 +β2 + ρβγ12+γ2 +β1 (t, x), (12) , α α where B(u, w) is the Beta function.

Similarly as in [11] we introduce, for y ∈ Rd , the freezing operator Ly by d Z Aα X dzi Ly f (x) = δf (x, ei zi ) σi (y) 1+α 2 |zi | R i=1

and

d Z dzi Aα X δf (t, x, ei zi ) σi (y) 1+α . L f (t, x) = 2 |zi | R y

i=1

Put

py (t, x) =

d Y i=1

1 gt aii (y)



xi aii (y)



.

It is clear that py (t, x) is the heat kernel of the operator Ly . In particular, we have ∂ py (t, x) = Ly py (t, x), ∂t

t > 0, x, y ∈ Rd .

(13)

TRANSITION 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 ≥ 1 we have Z t c (14) (t − s)γ−1 sθ−1 ds ≤ γ t(γ−1)+(θ−1)+1 . θ 0 We use the notation N0 = N ∪ {0}.

3. Upper bound estimates The main aim of this section is to construct the function pA (t, x, y). This is done by using Levi’s method. Is is worth mentioning that this method was used in the framework of pseudodifferential operators by Kochubei [23]. In recent years it was used in several papers to study gradient and Schr¨ odinger perturbations of fractional Laplacians and relativistic stable operators see e.g. [3, 14, 15, 6, 8, 9, 39]. As we have already mentioned we use the approach by Chen and Zhang [11]. It is worth adding that in [11], in contrast to previous papers, a new way of “freezing” coefficient was used. Now, we briefly present the main steps used in this section. We define pA (t, x, y) by (16). Heuristically, pA (t, x, y) is equal to the transition density py (t, x − y) (of the process with Rthe “frozen” jump measure corresponding to the generator Ly ) plus some correctR tion 0 Rd pP z (t − s, x, z)q(s, z, y) dz ds, which is given in terms of the perturbation series q(t, x, y) = ∞ n=0 qn (t, x, y). The most difficult result in this section is Theorem 3.2 which gives upper bound estimates of q(t, x, y). Due to a different structure of the generator L in comparison to the L´evy-type operator Lκα from [11] there are essential differences between our proof and analogous proof in [11], see in particular the definition of the auxiliary function HkL (t, x, y) and the induction proof of (20). The next important step in this section is Theorem 3.9 where we derive H¨older type estimates of q(t, x, y). We also show crucial Lemma 3.14 which is the main step in obtaining gradient estimates of pA (t, x, y). For x, y ∈ Rd , t > 0, let q0 (t, x, y) = (Lx − Ly ) py (t, ·)(x − y) and for n ∈ N let qn (t, x, y) =

Z tZ 0

For x, y ∈ Rd , t > 0 we define

Rd

q0 (t − s, x, z)qn−1 (s, z, y) dz ds.

q(t, x, y) =

∞ X

(15)

qn (t, x, y)

n=0

and pA (t, x, y) = py (t, x − y) +

Z tZ 0

Rd

pz (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 ∈ Rd we have d d Z Y X δpy (t, x, zk ek ) dzk ≤ ctd−1 ρ00 (t, xi ). 1+α |z | k R i=1

k=1

An immediate consequence of the above lemma and (13) is the following estimate d Y ∂ py (t, x) ≤ ctd−1 ρ00 (t, xi ), (17) ∂t i=1

for t ∈ (0, T ], x, y ∈

Rd .

6

T. KULCZYCKI AND M. RYZNAR

P Theorem 3.2. The series ∞ n=0 qn (t, x, y) is absolutely and locally uniformly convergent d d on (0, T ] × R × R . For any x, y ∈ Rd , t ∈ (0, T ] we have # #" " d d   X Y β 0 β/α d−1 |xm − ym | ∧ 1 . (18) ρ0 (t, xi − yi ) t + |q(t, x, y)| ≤ ct m=1

i=1

Moreover, q(t, x, y) is jointly continuous in (t, x, y) ∈ (0, T ] × Rd × Rd .

Proof. By (8) and then Lemma 3.1 we get d Z Aα X δpy (t, x − y, ek zk ) |σk (x) − σk (y)| dzk |q0 (t, x, y)| ≤ 2 |zk |1+α R k=1

d Z X δpy (t, x − y, ek zk ) ≤ c |x − y| ∧ 1



≤ M td−1

β

k=1

"

d X

m=1



R

 |xm − ym |β ∧ 1

#"

d Y

dzk |zk |1+α

#

ρ00 (t, xk − yk ) ,

k=1

(19)

where M = M (T, d, α, b1 , b2 , b3 , β). Put I = {L = (l1 , . . . , ld ) : ∀i ∈ {1, . . . , d} li = 0 or li = β}. For any L = (l1 , . . . , ld ) ∈ I denote d 1X |L| = li . β i=1

For k ∈ N0 and L = (l1 , . . . , ld ) ∈ I put  # d " d  Y Y |xj − yj |lj ∧ 1  . ρ00 (t, xi − yi )  HkL (t, x, y) = td−1+kβ/α j=1

i=1

We will show that there is C = C(T, d, α, b1 , b2 , b3 , β) such that for any n ∈ N0 , x, y ∈ Rd , t ∈ (0, T ], X M Cn HkL (t, x, y), (20) |qn (t, x, y)| ≤ β/α ((n + 1)!) k∈N0 , L∈I k+|L|=n+1

where M is the constant from (19). Let Z tZ D(t, x, y, m, k, L) = M 0

Rd

H0Lm (t − s, x, z)HkL (s, z, y) dz ds,

where Lm ∈ I is such that lm = β and |Lm | = 1. Observe that (19) can be rewritten as

|q0 (t, x, y)| ≤ M

d X

H0Lm (t, x, y).

(21)

m=1

We will prove (20) by induction. The main step consists of proving that for any n ∈ N, d X

X

D(t, x, y, m, k, L) ≤

m=1 k∈N0 , L∈I k+|L|=n+1

For n = 0 the estimate (20) holds by (21).

C (n + 2)β/α

X

k∈N0 , L∈I k+|L|=n+2

HkL (t, x, y).

(22)

TRANSITION DENSITIES FOR SDES DRIVEN BY CYLINDRICAL STABLE PROCESSES

7

Assume that (20) holds for some n ∈ N0 . By (15), (21) and our induction hypothesis we obtain |qn+1 (t, x, y)| ≤

M Cn ((n + 1)!)β/α

d X

X

D(t, x, y, m, k, L).

(23)

m=1 k∈N0 , L∈I k+|L|=n+1

Then, if (22) is true, then (20) holds for n + 1. Hence in order to complete the proof it is enough to show (22). To this end we consider 3 cases. Case 1. L = (0, . . . , 0), k = n + 1. We have Z

D(t, x, y, m, k, L) = M

t

0

×

Z

R

By (11), we obtain Z R

 d Z Y   ρ0α (t − s, xi − zi )ρ0α (s, zi − yi ) dzi  snβ/α  

i=1 R i6=m

ρβ0 (t − s, xm − zm )ρ0β (s, zm − ym ) dzm ds.

ρ0α (t − s, xi − zi )ρ0α (s, zi − yi ) dzi ≤ cρ0α (t, xi − yi )

and Z

ρβ0 (t − s, xm − zm )ρ0β (s, zm − ym ) dzm h ≤ c (t − s)(β−α)/α sβ/α ρ00 (t, xm − ym ) + s(2β−α)/α ρ00 (t, xm − ym ) i (β−α)/α β +s ρ0 (t, xm − ym ) . R

Hence

D(t, x, y, m, k, L) ≤ ct ×

d−1

Z

" t

d Y

ρ00 (t, xi

#

− yi )

i=1

Z

t

s((n+2)β−α)/α ds 0 0 Z t   ((n+1)β−α)/α β s ds |xm − ym | ∧ 1 . + (β−α)/α (n+1)β/α

(t − s)

s

ds +

0

By (14) this implies that # " d Y c 0 d−1 D(t, x, y, m, k, L) ≤ ρ0 (t, xi − yi ) t (n + 1)β/α i=1  i h × t(n+2)β/α + t(n+1)β/α |xm − ym |β ∧ 1 . Case 2. L = (l1 , . . . , ld ) 6= (0, . . . , 0), lm = 0.

(24)

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T. KULCZYCKI AND M. RYZNAR

Put Z(L) = {i ∈ {1, . . . , d} : li = β} and i(L) = inf Z(L). Clearly m ∈ / Z(L). We have

Z tZ

ρβ0 (t − s, xm − zm )ρ0α (s, zm − ym ) dzm D(t, x, y, m, k, L) = M 0 R Z 0 ρα (t − s, xi(L) − zi(L) )ρβ0 (s, zi(L) − yi(L) ) dzi(L) ×  R Z   Y ρ0α (t − s, xi − zi )ρβα (s, zi − yi ) dzi  ×   i∈Z(L) R i6=i(L)



 Z  Y  kβ/α × ρ0α (t − s, xi − zi )ρ0α (s, zi − yi ) dzi  ds.  s R i∈Z(L) / i6=m

By (11) this is bounded from above by

c

Z th

i (t − s)(β−α)/α sρ00 (t, xm − ym ) + tβ/α ρ00 (t, xm − ym ) + ρβ0 (t, xm − ym )

h0 i × tβ/α ρ00 (t, xi(L) − yi(L) ) + ts(β−α)/α ρ00 (t, xi(L) − yi(L) ) + ρβ0 (t, xi(L) − yi(L) )   i  Y h (α+β)/α 0 β  t ρ (t, x − y ) + tρ (t, x − y ) × i i i i 0 0   i∈Z(L) i6=i(L)





 Y  kβ/α × tρ00 (t, xi − yi ) ds.  s i∈Z(L) / i6=m

Note that #Z(L) = |L|. We have

Y h

t(α+β)/α ρ00 (t, xi

− yi ) +

i∈Z(L) i6=i(L)

×

X

tρβ0 (t, xi





i  Y 0   − yi ) ≤ ct|L|−1  ρ (t, x − y ) i i 0   i∈Z(L) i6=i(L)

X

r≤|L|−1 {k1 ,...,kr }⊂Z(L)\{i(L)} r∈N0

t

(|L|−r−1)β/α

r  Y i=1

 |xki − yki |β ∧ 1 ,

(25)

TRANSITION DENSITIES FOR SDES DRIVEN BY CYLINDRICAL STABLE PROCESSES

Qr

where for r = 0 we understand that

D(t, x, y, m, k, L) ≤ ct

d−2

" d Y

i=1

ρ00 (t, xi


|xki − yki |β ∧ 1 = 1. It follows that #

− yi )

i=1

×

X

9

# " r  Y |xki − yki |β ∧ 1 t(|L|−r−1)β/α

X

r≤|L|−1 {k1 ,...,kr }⊂Z(L)\{i(L)} r∈N0  Z t (α+β)/α (β−α)/α kβ/α

i=1

Z

t

× t (t − s) s ds + t (t − s)(β−α)/α s(β+kβ)/α ds 0 0 Z t   +t (t − s)(β−α)/α skβ/α ds |xi(L) − yi(L) |β ∧ 1 0 Z t Z t Z t   2β/α kβ/α (α+β)/α (β−α+kβ)/α β/α +t s ds + t s ds + t skβ/α ds |xi(L) − yi(L) |β ∧ 1 0 0 0  Z t Z t Z t   β/α kβ/α β (β−α+kβ)/α kβ/α × t s ds |xi(L) − yi(L) | ∧ 1 s ds + s ds + t 0 0 0  i × |xm − ym |β ∧ 1 . Using this and (14) we get

D(t, x, y, m, k, L) ≤ ctd−2 ×

X

X

"

d Y

#

ρ00 (t, xi − yi )

i=1

t

(|L|−r−1)β/α

r≤|L|−1 {k1 ,...,kr }⊂Z(L)\{i(L)} r∈N0

" r Y i=1

#  |xki − yki | ∧ 1 β

  h 1 β (α+kβ+2β)/α (α+kβ+β)/α |x − y | ∧ 1 t + t i(L) i(L) (k + 1)β/α   i   t(α+kβ+β)/α |xm − ym |β ∧ 1 + t(α+kβ)/α |xi(L) − yi(L) |β ∧ 1 |xm − ym |β ∧ 1 .

×

Note that k + |L| = n + 1. It follows that

# " d Y ctd−1 ρ0 (t, xi − yi ) D(t, x, y, m, k, L) ≤ (k + 1)β/α i=1 0 ×

X

X

r≤|L|+1 {k1 ,...,kr }⊂Z(L)∪{m} r∈N0

Case 3. L = (l1 , . . . , ld ), lm = β.

t(n+2−r)β/α

r   Y |xki − yki |β ∧ 1 . i=1

(26)

10

T. KULCZYCKI AND M. RYZNAR

We have Z tZ

ρβ0 (t − s, xm − zm )ρβ0 (s, zm − ym ) dzm D(t, x, y, m, k, L) = M 0 R   Z  Y   × ρ0α (t − s, xi − zi )ρβα (s, zi − yi ) dzi   i∈Z(L) R i6=m



Y Z

×

R i∈Z(L) /



ρ0α (t − s, xi − zi )ρ0α (s, zi − yi ) dzi  skβ/α ds.

By (11), this is bounded from above by Z th (t − s)(2β−α)/α ρ00 (t, xm − ym ) + s(2β−α)/α ρ00 (t, xm − ym ) c 0 i +(t − s)(β−α)/α ρβ0 (t, xm − ym ) + s(β−α)/α ρβ0 (t, xm − ym )   i  Y h (α+β)/α 0 β  t ρ (t, x − y ) + tρ (t, x − y ) × i i i i  0 0  i∈Z(L) i6=m



×

Y

i∈Z(L) /



tρ00 (t, xi − yi ) skβ/α ds.

Using similar reasoning as in (25) this is bounded from above by # " d Y ctd−1 ρ00 (t, xi − yi ) i=1

X

×

X

t(|L|−r−1)β/α

r≤|L|−1 {k1 ,...,kr }⊂Z(L)\{m} r∈N0 Z t (2β−α)/α kβ/α

(t − s)

s

i=1

ds +

0

Z

t

r   Y |xki − yki |β ∧ 1

Z

t

s(2β−α+kβ)/α ds

0

  + (t − s)(β−α)/α skβ/α ds |xm − ym |β ∧ 1 0 Z t   (β−α+kβ)/α β s ds |xm − ym | ∧ 1 0

By (14) it follows that

D(t, x, y, m, k, L) ≤ ctd−1 ×

X

X

"

d Y

ρ00 (t, xi − yi )

i=1

r≤|L|−1 {k1 ,...,kr }⊂Z(L)\{m} r∈N0

#

t(|L|−r−1)β/α

r   Y |xki − yki |β ∧ 1 i=1

 i h 1 β (kβ+2β)/α (kβ+β)/α |x − y | ∧ 1 t + t m m (k + 1)β/α

TRANSITION DENSITIES FOR SDES DRIVEN BY CYLINDRICAL STABLE PROCESSES

11

Hence # " d Y ctd−1 0 ρ (t, xi − yi ) D(t, x, y, m, k, L) ≤ (k + 1)β/α i=1 0 ×

X

X

t

(n+2−r)β

r   Y |xki − yki |β ∧ 1 .

(27)

i=1

r≤|L| {k1 ,...,kr }⊂Z(L) r∈N0

1 c Recall that n + 1 = k + |L|, |L| ≤ d, so k ≥ n + 1 − d. Hence (k+1) β/α ≤ (n+2)β/α , where c = c(d). Consequently, (23), (24), (26), (27) gives that (22) holds, which finishes the induction proof. From (20) we immediately obtain that for any n ∈ N0 # #" " d d   X Y cC n β 0 β/α d−1 |xm − ym | ∧ 1 . (28) ρ0 (t, xi − yi ) t + |qn (t, x, y)| ≤ t ((n + 1)!)β/α m=1 i=1 P∞ It follows that n=0 qn (t, x, y) is absolutely and locally uniformly convergent on (0, T ] × Rd × Rd and (18) holds. By the properties of py (t, x) it is easy to justify that q0 (t, x, y) is jointly continuous in (t, x, y) ∈ (0, T ] × Rd × Rd . By (15) P and induction method the same property holds for qn (t, x, y) for each n ∈ N . Since ∞ n=0 qn (t, x, y) is absolutely and locally uniformly convergent we finally obtain that q(t, x, y) is jointly continuous in (t, x, y) ∈ (0, T ] × Rd × Rd . 

By elementary calculations, for any t > 0, u, w ∈ R satisfying |u − w| ≤ t1/α , we have ρ00 (t, u) ≈ ρ00 (t, w).

(29)

Lemma 3.3. There exists c = c(α, d) such that for any m ∈ {1, . . . , d}, t > 0, x, x′ ∈ Rd we have  m ! m m m ′| Y Y X Y |x − x
j j . (30) gt (|xi | ∧ |x′i |) 1 ∧ gt x′i ≤ c gt (xi ) − t1/α + |xi | ∧ |x′i | i=1

i=1

i=1

j=1

If additionally |x − x′ | ≤ t1/α , then

m m Y Y
gt x′i ≤ ctm gt (xi ) − i=1

i=1

m Y i=1

(3)

 ! m X t−1/α |xj − x′j | . ρ00 (t, xi ) 1 ∧

(31)

j=1

Proof. Let gt (·) 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 dgt (x) (3) = −4πxgt (|x|), x ∈ R. dx By the standard estimates of transition density of the α-stable isotropic process we have (3)

gt (|x|) ≤ c which yields

gt (x) |x| + t1/α


2 ,

dgt (x) |x| gt (x) ,
2 gt (x) ≤ c dx ≤ c |x| + t1/α |x| + t1/α

x ∈ R.

12

T. KULCZYCKI AND M. RYZNAR

Next, for any u, w ∈ R, from the above gradient estimate of gt and the fact that gt (u) is decreasing in |u|, |gt (u) − gt (w)| ≤ c

|u − w| gt (|u| ∧ |w|) . |u| ∧ |w| + t1/α

Hence, by monotonicity of gt (u) and (32), m m Y Y
gt x′i gt (xi ) − i=1 i=1  m X
 gt (xi ) − gt x′i ≤ i=1

≤c

m Y

!

gt (|xi | ∧ |x′i |)

i=1

Y

j6=i,1≤j≤m

m X j=1


gt |xi | ∧ |x′i | 

|xi − x′i | |xi | ∧ |x′i | + t1/α

Combining this with the obvious inequality m m m Y Y Y
′ gt (|xi | ∧ |x′i |) gt xi ≤ gt (xi ) − i=1

i=1

i=1

(32)

we finish the proof of (30). To get the second inequality we apply (29).



As a direct conclusion of Lemma 3.3 we get Corollary 3.4. For any k ∈ {1, . . . , d}, t > 0, x, x′ , y ∈ Rd satisfying |x − x′ | ≤ t1/α we have d  Y  ′   d Y xi xi − aii (y)gt aii (y)gt aii (y) aii (y) i=1 i=1 i6=k i6=k   d d Y  X −1/α ≤ ctd−1  ρ00 (t, xi ) (t |xj − x′j | ∧ 1). i=1 i6=k

j=1

Corollary 3.5. For any t > 0 and x, y, w ∈ Rd we have

|px (t, w) − py (t, w)| ≤ cpx (t, w)(|x − y|β ∧ 1). Proof. We have d    Y  d Y wi wi −1 −1 |px (t, w) − py (t, w)| = aii (y)gt aii (x)gt − aii (y) aii (x) i=1 i=1 d d d Y  w  Y Y i gt a−1 (x) (y) − ≤ a−1 ii ii aii (y) i=1 i=1 i=1     d d d Y Y Y wi wi −1 gt − aii (x) gt + aii (y) aii (x) i=1

i=1

i=1

Next, by (3) and (2),

d d Y Y −1 a (x) (y) − ≤ c(|x − y|β ∧ 1), a−1 ii ii i=1

i=1

TRANSITION DENSITIES FOR SDES DRIVEN BY CYLINDRICAL STABLE PROCESSES

13

and by Lemma 3.3 together with (3) and (2)

d   d Y  w  Y w i i gt − gt aii (y) aii (x) i=1 i=1     d m X wi |wi ||aii (x) − aii (y)|  Y g ≤ c 1 ∧ t aii (x) t1/α + |wi | i=1 j=1 β

≤ c(|x − y| ∧ 1)

d Y i=1

gt



wi aii (x)



.

The proof is completed.



Corollary 3.6. For any x ∈ Rd , t > 0 we have d    Y  d Y 1 xi − y i xi − yi 1 gt gt − dy2 . . . dyd ≤ c(|x1 − y1 |β ∧ 1), aii (y) aii (˜ y) aii (˜ y) Rd−1 i=2 aii (y) i=2 (33) where y˜ = (x1 , y2 , . . . , yd ). Z

Proof. By the same arguments as in the proof Corollary 3.5 we have

# " d  Y    d d Y Y 1 xi − y i xi − yi 1 gt gt gt (xi − yi ) (|y − y˜|β ∧ 1). − ≤c aii (y) aii (y) aii (˜ y) aii (˜ y) i=2

i=2

i=2

Observing that |y − y˜| = |x1 − y1 | we obtain the conclusion by integration.



Lemma 3.7. For any t ∈ (0, T ], x, x′ , y ∈ Rd satisfying |x − x′ | ≤ t1/α we have d Z X δpy (t, x, zk ek ) − δpy (t, x′ , zk ek ) k=1

R

≤ ctd−1

d Y i=1

!

ρ00 (t, xi )

dzk |zk |1+α

d X (t−1/α |xj − x′j | ∧ 1). j=1

Proof. Fix k ∈ {1, . . . , d} For t > 0, z ∈ R put

hy (t, z) = akk (y)gt



z akk (y)



.

(34)

14

T. KULCZYCKI AND M. RYZNAR

We have Z

δpy (t, x, zk ek ) − δpy (t, x′ , zk ek ) dzk |zk |1+α R     Z Y d xi   =  aii (y)gt  δhy (t, xk , zk ) a (y) ii R i=1 i6=k     d dz ′ Y x   k ′ i −  aii (y)gt  δhy (t, xk , zk ) |zk |1+α aii (y) i=1 i6=k  Y  ′  Z  d d Y xi xi δhy (t, xk , zk ) dzk aii (y)gt − ≤ aii (y)gt aii (y) aii (y) R |zk |1+α i=1 i=1 i6=k i6=k    ′  Z d xi Y  δh (t, xk , zk ) − δh (t, x′ , zk ) dzk . +  aii (y)gt  k y y aii (y) |zk |1+α R i=1 i6=k

By [11, (2.28)] we have Z

R

while, by [11, (2.29)], Z

R

δh (t, xk , zk ) y

δhy (t, xk , zk ) − δhy (t, x′k , zk )

dzk ≤ cρ00 (t, xk ), |zk |1+α

dzk ≤ cρ00 (t, xk )(t−1/α |xk − x′k | ∧ 1). |zk |1+α

Applying the above inequalities and Corollary 3.4 we obtain the desired bound (34).



Lemma 3.8. For any x, x′ , y ∈ Rd , t ∈ (0, T ] and γ ∈ (0, β) we have   |q0 (t, x, y) − q0 (t, x′ , y)| ≤ c |x − x′ |β−γ ∧ 1    d d   X  Y 0  0 β × ρ (t, x − y ) + ρ (t, x − y ) (tρ (t, x − y ))  i i  k k k k γ 0 γ−β i=1 i6=k

k=1

+

d X k=1

  

d Y

i=1 i6=k





    (tρ00 (t, x′i − yi )) ρ0γ (t, x′k − yk ) + ρβγ−β (t, x′k − yk ) 

Proof. Case 1. |x − x′ | > t1/α . By (19) we get, for z = x or x′ , 

′ β−γ

|q0 (t, z, y)| ≤ c |x − x |

Case 2. |x − x′ | ≤ t1/α .

 ∧ 1 td−1

d Y i=1

ρ00 (t, zi

!

− yi )

d X

m=1

β

(|zm − ym | ∧ 1) t

γ−β α

!

.

TRANSITION DENSITIES FOR SDES DRIVEN BY CYLINDRICAL STABLE PROCESSES

15

We have |q0 (t, x, y) − q0 (t, x′ , y)| d Z dzk A X δpy (t, x − y, zk ek )(aαkk (x) − aαkk (y)) = 2 |zk |1+α k=1 R d Z X dz k δpy (t, x′ − y, zk ek )(aαkk (x′ ) − aαkk (y)) − 1+α |zk | R ≤

k=1 d Z X k=1

+

R

δpy (t, x − y, zk ek ) − δpy (t, x′ − y, zk ek ) |aα (x) − aα (y)| kk kk

d Z X δpy (t, x′ − y, zk ek ) aαkk (x) − aαkk (x′ ) k=1

R

dzk |zk |1+α

d Z X δpy (t, x − y, zk ek ) − δpy (t, x′ − y, zk ek ) ≤ c |x − y| ∧ 1



β

k=1

R

d Z  X ′ β δpy (t, x′ − y, zk ek ) +c |x − x | ∧ 1 k=1

dzk |zk |1+α

dzk |zk |1+α

dzk |zk |1+α

R

= I + II.

By Lemma 3.7 we get d    Y β −1/α ′ d−1 ρ00 (t, xi − yi ). I ≤ c |x − y| ∧ 1 t |x − x | ∧ 1 t

(35)

i=1

γ−β

Note that t−1/α |x − x′ | ≤ 1 so t−1/α |x − x′ | ≤ t α |x − x′ |β−γ . Let m ∈ {1, . . . , d} be such that |xm − ym | = maxi∈{1,...,d} |xi − yi |. It follows that (35) is bounded from above by

ct

γ−β α

|x − x′ |β−γ td−1

d Y

!

ρ00 (t, xi − yi )

i=1



≤ c |x − x′ |β−γ



  |xm − ym |β ∧ 1

 d  Y   ∧1  tρ00 (t, xi − yi ) ρβγ−β (t, xm − ym ). i=1 i6=m


We have |x − x′ | ≤ t1/α so 1 ≤ |x − x′ |−γ tγ/α . It follows that |x − x′ |β ∧ 1 ≤
|x − x′ |β−γ ∧ 1 tγ/α . Using this and Lemma 3.1 we get d  Y d−1 ρ00 (t, x′i − yi ) II ≤ c |x − x | ∧ 1 t



′ β

i=1

  ≤ c |x − x′ |β−γ ∧ 1 td−1

d X k=1

  

d Y

i=1 i6=k



 ρ00 (t, x′i − yi ) ρ0γ (t, x′k − yk ). 

16

T. KULCZYCKI AND M. RYZNAR

Theorem 3.9. For any x, x′ , y ∈ Rd , t ∈ (0, T ] and γ ∈ (0, β) we have   |q(t, x, y) − q(t, x′ , y)| ≤ c |x − x′ |β−γ ∧ 1    d d   X Y 0  0 β ×  (tρ0 (t, xi − yi )) ργ (t, xk − yk ) + ργ−β (t, xk − yk ) i=1 i6=k

k=1

+

d X k=1

  

d Y i=1 i6=k

     (tρ00 (t, x′i − yi )) ρ0γ (t, x′k − yk ) + ρβγ−β (t, x′k − yk )  . 

(36)

Proof. By the definition of qn , (28) and Lemma 3.8 we get for n ∈ N

|qn (t, x, y) − qn (t, x′ , y)| Z tZ |q0 (t − s, x, z) − q0 (t − s, x′ , z)||qn−1 (s, z, y)| dz ds ≤ Rd n−1 cC 0

≤

 
|x − x′ |β−γ ∧ 1 A(t, x, y) + A(t, x′ , y) ,

(n!)β/α

(37)

where C is the constant from (28) and A(t, x, y) =

Z tZ

Rd k=1

0

×

d X



  

d Y

i=1 i6=k



   ρ0α (t − s, xi − zi ) ρ0γ (t − s, xk − zk ) + ρβγ−β (t − s, xk − zk ) 

d d  X Y  0 β 0   ρ (s, z − y ) (s, z − y ) + ρ (s, z − y ) dz1 . . . dzd ds. ρ j j  m m m m α β 0 

m=1

j=1 j6=m

We have

A(t, x, y) =

d X k=1

+

  

d Y

i=1 i6=k

d X



 tρ00 (t, xi − yi ) Bk (t, x, y)

d X

k=1 m=1 m6=k

  

d Y

i=1 i6=k, i6=m



 tρ00 (t, xi − yi )

×[Dk,m (t, x, y) + Ek,m (t, x, y) + Fk,m (t, x, y) + Gk,m (t, x, y)], where Z tZ   ρ0γ (t − s, xk − zk ) + ρβγ−β (t − s, xk − zk ) Bk (t, x, y) = 0  R  × ρ0β (s, zk − yk ) + ρβ0 (s, zk − yk ) dzk ds, Dk,m(t, x, y) =

Z tZ 0

×

Z

R

R

ρ0γ (t − s, xk − zk )ρ0α (s, zk − yk ) dzk

ρ0α (t − s, xm − zm )ρ0β (s, zm − ym ) dzm ds,

(38)

TRANSITION DENSITIES FOR SDES DRIVEN BY CYLINDRICAL STABLE PROCESSES

Ek,m(t, x, y) =

Z tZ 0

×

Fk,m (t, x, y) =

R

Z

R

R

ρβγ−β (t − s, xk − zk )ρ0α (s, zk − yk ) dzk

Z

ρ0α (t − s, xm − zm )ρ0β (s, zm − ym ) dzm ds,

Z tZ

ρβγ−β (t − s, xk − zk )ρ0α (s, zk − yk ) dzk

×

R

Gk,m (t, x, y) =

ρ0γ (t − s, xk − zk )ρ0α (s, zk − yk ) dzk

ρ0α (t − s, xm − zm )ρβ0 (s, zm − ym ) dzm ds,

Z tZ 0

0

×

Z

R

R

17

ρ0α (t − s, xm − zm )ρβ0 (s, zm − ym ) dzm ds.

By (12) we get   Bk (t, x, y) ≤ c ρ0γ (t, xk − yk ) + ρβγ−β (t, xk − yk ) .

(39)

Using (11) we obtain

Dk,m (t, x, y) Z t  β  γ−α γ β−α (t − s) α s + (t − s) α ≤c s α + (t − s)s α dsρ00 (t, xk − yk )ρ00 (t, xm − ym ) 0

≤

≤

γ β ct1+ α + α ρ00 (t, xk − yk )ρ00 (t, xm − ym ) ctρ0γ (t, xk − yk )ρ00 (t, xm − ym ),

(40)

Ek,m (t, x, y) Z t   β−α γ−α γ (t − s)β/α + (t − s)s α ds (t − s) α s + (t − s) α ≤c 0 0 ×ρ0 (t, xk Z t

(t − s)

+c

0

≤

≤

− yk )ρ00 (t, xm − ym ) γ−α α

s + (t − s)

γ α



dsρ00 (t, xk − yk )ρβ0 (t, xm − ym )

β γ ct1+ α + α ρ00 (t, xk − yk )ρ00 (t, xm − ym ) ctρ0γ (t, xk − yk )ρ00 (t, xm − ym ),

γ

+ ct1+ α ρ00 (t, xk − yk )ρβ0 (t, xm − ym ) (41)

Fk,m (t, x, y) Z t  β  γ−α γ−β β β−α (t − s) α s + (t − s) α s α ≤c s α + (t − s)s α ds

0 0 ×ρ0 (t, xk Z t

+c

0

≤

≤

− yk )ρ00 (t, xm − ym )  β  γ−β β−α dsρβ0 (t, xk − yk )ρ00 (t, xm − ym ) (t − s) α s α + (t − s)s α

γ β ct1+ α + α ρ00 (t, xk − yk )ρ00 (t, xm − ym ) ctρ0γ (t, xk − yk )ρ00 (t, xm − ym ),

γ

+ ct1+ α ρβ0 (t, xk − yk )ρ00 (t, xm − ym ) (42)

18

T. KULCZYCKI AND M. RYZNAR

Gk,m (t, x, y) Z t   γ−α β γ−β β β−α (t − s) α s + (t − s) α s α (t − s) α + (t − s)s α ds ≤c 0

×ρ00 (t, xk − yk )ρ00 (t, xm − ym ) Z t  γ−β β γ−α (t − s) α s + (t − s) α s α dsρ00 (t, xk − yk )ρβ0 (t, xm − ym ) +c 0 Z t   γ−β β β−α (t − s) α (t − s) α + (t − s)s α +c dsρβ0 (t, xk − yk )ρ00 (t, xm − ym ) 0 Z t γ−β +c (t − s) α dsρβ0 (t, xk − yk )ρβ0 (t, xm − ym ) 0 γ β γ +α 1+ α ρ00 (t, xk − yk )ρ00 (t, xm − ym ) + ct1+ α ρ00 (t, xk − yk )ρβ0 (t, xm − ym ) ≤ ct γ γ β +ct1+ α ρβ0 (t, xk − yk )ρ00 (t, xm − ym ) + ct1+ α − α ρβ0 (t, xk − yk )ρβ0 (t, xm − ym ) ≤ ctρ0γ (t, xk − yk )ρ00 (t, xm − ym ) + ctρβγ−β (t, xk − yk )ρ00 (t, xm − ym ).

(43)

By (38-43) we obtain A(t, x, y) ≤ c

d X k=1

  

d Y i=1 i6=k



   tρ00 (t, xi − yi ) ρ0γ (t, xk − yk ) + ρβγ−β (t, xk − yk ) .

Using this and (37) we obtain that for any n ∈ N, x, x′ , y ∈ Rd , t ∈ (0, T ] and γ ∈ (0, β)  cC n−1  ′ β−γ |x − x | ∧ 1 |qn (t, x, y) − qn (t, x′ , y)| ≤ (n!)β/α    d d   X Y 0  0 β ×  (tρ0 (t, xi − yi )) ργ (t, xk − yk ) + ργ−β (t, xk − yk ) k=1

+

d X k=1

  

i=1 i6=k

d Y i=1 i6=k

     (tρ00 (t, x′i − yi )) ρ0γ (t, x′k − yk ) + ρβγ−β (t, x′k − yk )  . 

This, Lemma 3.8 and the definition of q imply the assertion of the theorem.



Lemma 3.10. For all γ ∈ (0, 1], x, x′ , y ∈ Rd , t > 0, we have #! # " d " d Y Y gt (x′i ) . gt (xi ) + |py (t, x) − py (t, x′ )| ≤ c|x − x′ |γ t−γ/α i=1

i=1

Proof. By Lemma 3.3 we get ′



′ −1/α

|py (t, x) − py (t, x )| ≤ c |x − x |t Since |x −

x′ |t−1/α

 ∧1

"

d Y i=1

#

gt (xi ) +

"

d Y i=1

#!

gt (x′i )


∧ 1 ≤ |x − x′ |γ t−γ/α we obtain the assertion of the lemma.

By Lemma 3.3 and the formula for py (t, x) we obtain Lemma 3.11. For any x, y ∈ Rd and t > 0 we have 1

|∇py (t, ·)(x − y)| ≤ ct− α

d Y i=1

ρ0α (t, xi − yi ).

. 

TRANSITION DENSITIES FOR SDES DRIVEN BY CYLINDRICAL STABLE PROCESSES

19

Lemma 3.12. For any x ∈ Rd and t ∈ (0, T ] we have Z ≤ ct β−1 α . ∇p (t, ·)(x − y) dy y d R

Proof. Let D1 py (t, ·)(x−y) = limh→0 (py (t, x−y+he1 )−py (t, x−y))/h. It is enough to prove R the estimate for I = Rd D1 py (t, ·)(x − y) dy. Let γ ∈ (0, β) and put y˜ = (x1 , y2 , . . . , yd ). We have Z "   "Y #  d 1 x1 − y 1 1 xi − y i ′ g |I| = gt Rd a211 (y) t a11 (y) aii (y) aii (y) i=2 "  Y ##   d 1 x − y 1 x − y i i 1 1 gt dy − 2 g′ y) aii (˜ y) aii (˜ y) a11 (˜ y ) t a11 (˜ i=2 Z    Z    1 1 x1 − y 1 x1 − y 1 ′ ′ ≤ − 2 dy1 gt g 2 a11 (y) y) a11 (˜ y ) t a11 (˜ Rd−1 R a11 (y) " d #  Y 1 xi − y i dy2 , . . . dyd gt × aii (˜ y) aii (˜ y) Z i=2   1 x1 − y 1 + gt′ 2 a11 (y) Rd a11 (y) " d  Y #   d Y 1 1 xi − y i xi − y i gt gt dy × − aii (y) aii (y) aii (˜ y) aii (˜ y) i=2

i=2

= II + III.

By [11, (2.31)] we get     1 1 x1 − y 1 x1 − y1 ′ ′ ≤ c(|x1 −y1 |β ∧1)t−1/α (ρ0α (t, x1 −y1 )+ργα−γ (t, x1 −y1 )). − g g t a2 (y) t a11 (y) 2 (˜ a (˜ y ) a y ) 11 11 11 β−1

Using this and (10) we obtain II ≤ ct α . Note also that   1 x1 − y1 ′ 0 a2 (y) gt a11 (y) ≤ cρα−1 (t, x1 − y1 ). 11

Using this, Corollary 3.6 and (10) we get III ≤ ct

β−1 α

.



Similarly as in [11] we denote φy (t, x, s) = and

Z

Rd

pz (t − s, x − z)q(s, z, y) dz

ϕy (t, x) = Clearly we have

Z

t

φy (t, x, s) ds. 0

pA (t, x, y) = py (t, x − y) + ϕy (t, x).

(44)

By well known estimates of ∇pz (t − s, ·)(x − z) and Theorem 3.2 we easily obtain the following result. Lemma 3.13. For any x, y ∈ Rd , t > 0 and s ∈ (0, t) we have Z ∇pz (t − s, ·)(x − z)q(s, z, y) dz. ∇x φy (t, x, s) = Rd

The next result is the most important step in proving gradient estimates of pA (t, x, y).

20

T. KULCZYCKI AND M. RYZNAR

Lemma 3.14. For any α ∈ (1, 2), x, y ∈ Rd and t ∈ (0, T ] we have Z tZ ∇pz (t − s, ·)(x − z)q(s, z, y) dz ds. ∇x ϕy (t, x) = 0

(45)

Rd

and

|∇x ϕy (t, x)| ≤ ct

1 −α

d Y

ρ0α (t, xi − yi ).

(46)

i=1

Proof. Let x, y ∈ Rd , t ∈ (0, T ] and s ∈ (0, t). The main tool used in this case is Theorem 3.2. Using this theorem, Lemmas 3.11, 3.13 and (11) we obtain Z |∇pz (t − s, ·)(x − z)| |q(s, z, y)| dz |∇x φy (t, x, s)| ≤ Rd Z (t − s)−1/α pz (t − s, x − z) ≤c d R   d d Y X   ρ0α (s, zi − yi ) [ρβ0 (s, zm − ym ) + ρ0β (s, zm − ym )] dz ×  m=1

≤c

i=1 i6=m

d X

m=1

  

d Y

ρ0α (t, xi

i=1 i6=m



 − yi )

Z

R

ρ0α−1 (t − s, xm − zm )

×[ρβ0 (s, zm − ym ) + ρ0β (s, zm − ym )] dz   d d h X Y β−1   ρ0α (t, xi − yi ) (t − s) α ρ00 (t, xm − ym ) ≤c  m=1

α−1 α

β−α α

1

ρ00 (t, xm − ym ) + (t − s)− α ρβ0 (t, xm − ym ) i 1 β +(t − s)− α s α ρ00 (t, xm − ym ) .

+(t − s)

It follows that

i=1 i6=m

∇x

s

Z

t 0

 Z t ∇x φy (t, x, s) ds, φy (t, x, s) ds = 0

which implies (45). We also obtain Z t ∇x φy (t, x, s) ds 0   Z th d d X Y β−1  0 ρ (t, x − y ) ≤c (t − s) α ρ00 (t, xm − ym )  i i  α m=1

0

i=1 i6=m

α−1 α

β−α α

1

ρ00 (t, xm − ym ) + (t − s)− α ρβ0 (t, xm − ym ) i 1 β + (t − s)− α s α ρ00 (t, xm − ym ) ds.   d d X Y  ≤c ρ0α (t, xi − yi ) [ρ0α+β−1 (t, xm − ym ) + ρβα−1 (t, xm − ym )],  +(t − s)

m=1

s

i=1 i6=m

which implies (46).



TRANSITION DENSITIES FOR SDES DRIVEN BY CYLINDRICAL STABLE PROCESSES

21

Proposition 3.15. For any α ∈ (1, 2), t ∈ (0, T ] and x, y ∈ Rd we have 1

|∇x pA (t, x, y)| ≤ ct− α

d Y

gt (xi − yi ).

i=1

Proof. The assertion follows from formula (44) and Lemmas 3.11, 3.14.



4. Feller semigroup For any bounded Borel f : Rd → R, t ∈ (0, ∞) and x ∈ Rd we define Z pA (t, x, y)f (y) dy. PtA f (x) = Rd

The main aim of this section is to show that {PtA } is a Feller semigroup. For any ε ≥ 0 and x ∈ Rd we put d Z Aα X dzi Lε f (x) = δf (x, ei zi ) σi (x) 1+α , 2 |zi | {zi : |zi |>ε} i=1

Lyε f (x) =

d Z dzi Aα X δf (x, ei zi ) σi (y) 1+α . 2 |z i| {zi : |zi |>ε} i=1

Using [11, (3.13)] and the same arguments as in the proof of Lemma 3.12 we obtain Lemma 4.1. For any ε > 0, x, y ∈ Rd and t ∈ (0, T ] we have Z β−α x Lε py (t, ·)(x − y) dy ≤ ct α . Rd

Lemma 4.2. For any x, y ∈ Rd and t > 0 Lx ϕy (t, x) is well defined and we have Z tZ Lx pz (t − s, ·)(x − z)q(s, z, y) dz ds. Lx ϕy (t, x) =

(47)

Rd

0

Fix γ ∈ (0, β). There exists c such that for any ε > 0, t ∈ (0, T ] and x, y ∈ Rd we have |Lε pA (t, ·, y)(x)| ≤ ct

−α+γ−β α

d Y

ρ0α (t, xi − yi ).

(48)

i=1

Moreover, t → Lx ϕy (t, x) is continuous on (0, T ) for any x, y ∈ Rd . Proof. Let ε > 0. We have Lε ϕy (t, x) =

Z tZ 0

Rd

Lxε pz (t − s, ·)(x − z)q(s, z, y) dz ds.

By Lemma 3.1 and Theorem 3.2 one easily gets Z Z x Lε pz (t − s, ·)(x − z)q(s, z, y) dz = lim ε→0+

Rd

Rd

Lx pz (t − s, ·)(x − z)q(s, z, y) dz.

The most difficult part of the proof is to justify Z tZ lim Lxε pz (t − s, ·)(x − z)q(s, z, y) dz ds + d ε→0 0 R Z t Z Lxε pz (t − s, ·)(x − z)q(s, z, y) dz ds. lim = + 0 ε→0

Rd

(49)

(50)

22

T. KULCZYCKI AND M. RYZNAR

We have Z

Lε ϕy (t, x) =

0

t/2 t

Z

+

Lxε pz (t − s, ·)(x − z)q(s, z, y) dz ds

Rd

t

Z

+

t/2 Z

t/2

Z

Rd

Z

Rd

Lxε pz (t − s, ·)(x − z) dzq(s, x, y) ds Lxε pz (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 |Lxε pz (t − s, ·)(x − z)q(s, z, y)| dz d R   d d X Y  ρ0α (t, xi − yi ) ≤c  m=1

i=1 i6=m

Z

ρ00 (t − s, xm − zm )[ρβ0 (s, zm − ym ) + ρ0β (s, zm − ym )] dz R   d d X Y  ρ0α (t, xi − yi ) ≤c  ×

m=1

h

i=1 i6=m

× (t − s)

β−α α

ρ00 (t, xm − ym ) + s

It follows that lim

ε→0+

=

Z

Z

0 t/2

t/2 Z

lim

i ρ00 (t, xm − ym ) + (t − s)−1 ρβ0 (t, xm − ym ) .

Lxε pz (t − s, ·)(x − z)q(s, z, y) dz ds

Rd

ε→0+

0

β−α α

Z

Rd

Lxε pz (t − s, ·)(x − z)q(s, z, y) dz ds.

(51)

and D(t, x, y) ≤ ct−1

d Y

ρ0α (t, xi − yi ).

(52)

i=1

For s ∈ (t/2, t) by Theorem 3.2 and Lemma 4.1 we obtain Z d Y β−α −1 x α t ρ0α (t, xi − yi ). L p (t − s, ·)(x − z) dzq(s, x, y) ≤ c(t − s) ε z Rd

i=1

It follows that

lim

ε→0+

=

Z

Z

t

t/2

t

Z

lim

Rd

+ t/2 ε→0

Z

Lxε pz (t − s, ·)(x − z) dzq(s, x, y) ds

Rd

Lxε pz (t − s, ·)(x − z) dzq(s, x, y) ds.

(53)

and E(t, x, y) ≤ ct−1

d Y i=1

ρ0α (t, xi − yi ).

(54)

TRANSITION DENSITIES FOR SDES DRIVEN BY CYLINDRICAL STABLE PROCESSES

23

Now, we need to obtain some estimates which will be crucial in studying the most difficult 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 ∈ Rd we have "Z # Z X d −1−α |δpz (t − s, x − z, wi ei )| |wi | dwi |q(s, z, y) − q(s, x, y)| dz Rd i=1

|wi |>ε

≤ c(t − s)−1

Z

(|x − z|β−γ ∧ 1)

Rd

"

d Y

#

ρ0α (t − s, xi − zi ) dz

i=1

d X

m=1

  

d Y

i=1 i6=m



 ρ0α (s, xi − yi )

× [ρβγ−β (s, xm − ym ) + ρ0γ (s, xm − ym )]   # d " d Z d X Y Y  ρ0α (t − s, xi − zi ) ρ0α (s, zi − yi ) + c(t − s)−1 (|x − z|β−γ ∧ 1)  Rd

m=1

i=1

i=1 i6=m

× [ρβγ−β (s, zm − ym ) + ρ0γ (s, zm − ym )] dz

= B1 (s, t, x, y) + B2 (s, t, x, y). P Clearly (|x − z|β−γ ∧ 1) ≤ dk=1 (|xk − zk |β−γ ∧ 1). It follows that " d # Z Y (|x − z|β−γ ∧ 1) ρ0α (t − s, xi − zi ) dz (t − s)−1 Rd

≤ c

d Z X k=1

R

≤ c(t − s)

i=1

ρ0β−γ (t − s, xk − zk ) dzk

β−γ−α α

.

Hence B1 (s, t, x, y) ≤ c(t − s)

β−γ−α α

d X

m=1

  

d Y

i=1 i6=m



 ρ0α (s, xi − yi )

× [ρβγ−β (s, xm − ym ) + ρ0γ (s, xm − ym )]. We also have B2 (s, t, x, y) ≤ c

d Z X

m=1

Rd



 d Y   ρ0α (t − s, xi − zi )ρ0α (s, zi − yi )  i=1 i6=m

× ρ0β−γ (t − s, xm − zm )[ρβγ−β (s, zm − ym ) + ρ0γ (s, zm − ym )] dz   Z d d d X X  Y 0  + c ρα (t − s, xi − zi )ρ0α (s, zi − yi )  d m=1 k=1 R k6=m

i=1 i6=m,k

× ρ0α (t − s, xm − zm )[ρβγ−β (s, zm − ym ) + ρ0γ (s, zm − ym )]

× ρ0β−γ (t − s, xk − zk )ρ0α (s, zk − yk ) dz = B3 (s, t, x, y) + B4 (s, t, x, y).

(55)

24

T. KULCZYCKI AND M. RYZNAR

By (11), we have B3 (s, t, x, y)   d d h X Y β−α 2β−γ−α γ−β  ρ0α (t, xi − yi ) (t − s) α s α ρ00 (t, xm − ym ) + s α ρ00 (t, xm − ym ) ≤ c  m=1

i=1 i6=m

β−γ−α α

γ−β α

ρβ0 (t, xm − ym ) + s i β−γ−α γ +(t − s) α s α ρ00 (t, xm − ym ) .

+(t − s)

s

γ−α α

ρ0β−γ (t, xm − ym )

(56)

By (11), we also have

B4 (s, t, x, y)   d d d X X Y  ≤ c ρ0α (t, xi − yi )  m=1 k=1 k6=m

hh

i=1 i6=m,k

i i γ−β γ α γ−α + (t − s) α s α + s α ρ00 (t, xm − ym ) + s α ρβ0 (t, xm − ym ) i i hh β−γ β−γ−α α (57) × (t − s) α s α + s α ρ00 (t, xk − yk ) + ρ0β−γ (t, xk − yk )

×

β

(t − s) α s

γ−β α

By (55-57) and (10), we get Z t Z lim Lxε pz (t − s, ·)(x − z)(q(s, z, y) − q(s, x, y)) dz ds ε→0+

=

Z

Rd

t/2

t

lim

+ t/2 ε→0

Z

Rd

Lxε pz (t − s, ·)(x − z)(q(s, z, y) − q(s, x, y)) dz ds.

(58)

and F (t, x, y) ≤ ct

−α+γ−β α

d Y

ρ0α (t, xi − yi ).

(59)

i=1

By (51), (53), (58) we get (50). We also get continuity t → Lx ϕy (t, x). By (49) and (50) we obtain (47). Using (52), (54), (59), Lemma 3.1 and formula (44) we get (48).  The next result is an analogue of [11, Theorem 4.1]. Its proof is almost the same as the proof of [11, Theorem 4.1] and is omitted. Proposition 4.3. Let u(t, x) ∈ Cb ([0, T ] × Rd ) with lim sup |u(t, x) − u(0, x)| = 0.

(60)

t→0+ x∈Rd

Assume that t → Lu(t, x)

is continuous on

(0, T ]

for each

x ∈ Rd

(61)

and for any ε ∈ (0, 1) and some γ ∈ ((α − 1) ∨ 0, 1) sup |u(t, x) − u(t, x′ )| ≤ Kε |x − x′ |γ ,

x, x′ ∈ Rd .

(62)

t∈(ε,T )

If u satisfies ∂ u(t, x) = Lu(t, x), ∂t

t ∈ (0, T ], x ∈ Rd ,

(63)

then sup sup u(t, x) ≤ sup u(0, x). t∈(0,T ) x∈Rd

x∈Rd

(64)

TRANSITION DENSITIES FOR SDES DRIVEN BY CYLINDRICAL STABLE PROCESSES

25

Lemma 4.4. Let x, y ∈ Rd . Then the mapping t → ϕy (t, x) is absolutely continuous on (0, T ]. For any t ∈ (0, T ) we have Z tZ ∂ϕy Lz pz (t − s, ·)(x − z)q(s, z, y) dz ds. (65) (t, x) = q(t, x, y) + ∂t d 0 R Proof. Let h > 0 be such that t + h < T . We have ϕy (t + h, x) − ϕy (t, x) h Z Z 1 t+h = pz (t + h − s, x − z)q(s, z, y) dz ds h 0 Rd Z Z 1 t pz (t − s, x − z)q(s, z, y) dz ds − h 0 Rd Z Z 1 t+h = pz (t + h − s, x − z)q(s, z, y) dz ds h 0 Rd Z Z 1 t − pz (t + h − s, x − z)q(s, z, y) dz ds h 0 Rd Z Z 1 t + pz (t + h − s, x − z)q(s, z, y) dz ds h 0 Rd Z Z 1 t pz (t − s, x − z)q(s, z, y) dz ds − h 0 Rd Z Z 1 t+h = pz (t + h − s, x − z)q(s, z, y) dz ds h t Rd Z tZ pz (t + h − s, x − z) − pz (t − s, x − z) + q(s, z, y) dz ds h 0 Rd = I + II. After change of variables t + h − s = u we have I = =

1 h 1 h

Z Z

0

0

hZ

Rd

hZ

Rd

pz (u, x − z)q(t + h − u, z, y) dz du (pz (u, x − z) − px (u, x − z))q(t + h − u, z, y) dz du

Z Z 1 h px (u, x − z)q(t + h − u, z, y) dz du h 0 Rd = I1 + I2 .

+

By Theorem 3.2 , supu≤h,z∈Rd q(t + h − u, z, y) ≤ M < ∞. Moreover, from Corollary 3.5, |pz (u, x − z) − px (u, x − z)| ≤ cpx (u, x − z)(|x − z|β ∧ 1). Hence, lim sup |I1 | ≤ cM lim sup h→0+

h→0+

1 h

Z

0

hZ

Rd

px (u, x − z)(|x − z|β ∧ 1) dz du = 0.

(66)

Next, by Theorem 3.2, the function q(s, z, y) is continuous and bounded on [t, T ] × Rd , as a function of s and z. Since the measures µu (dz) = px (u, x − z)dz converge weakly to δx as u → 0+ , we obtain lim I2 = q(t, x, y). (67) h→0+

26

T. KULCZYCKI AND M. RYZNAR

We have t/2 Z

pz (t + h − s, x − z) − pz (t − s, x − z) q(s, z, y) dz ds h 0 Rd Z t Z pz (t + h − s, x − z) − pz (t − s, x − z) + q(s, z, y) dz ds h t/2 Rd = III + IV. Z

II =

∂ ∂t pz (t

− s, x − z) following from (17) and Theorem 3.2 we get Z t/2 Z ∂ pz (t − s, x − z)q(s, z, y) dz ds. lim III = + h→0 0 Rd ∂t

By estimates of

We have IV =

t

Z

t/2 t

+

Z

t/2

Z

Rd

Z

Rd

(68)

pz (t + h − s, x − z) − pz (t − s, x − z) (q(s, z, y) − q(s, x, y)) dz ds h pz (t + h − s, x − z) − pz (t − s, x − z) dzq(s, x, y) ds h

= V + VI. Note that for h > 0, s ∈ (t/2, t), γ ∈ (0, β), x, y, z ∈ Rd , by Theorem 3.9 and the ∂ estimates of ∂t pz (t − s, x − z), we obtain pz (t + h − s, x − z) − pz (t − s, x − z) |(q(s, z, y) − q(s, x, y))| h ∂ β−γ ≤ B(|x − z| ∧ 1) pz (t + θh − s, x − z) ∂t ≤ cB(|x − z|β−γ ∧ 1)(t − s)−1

d Y

ρ0α (t − s, xi − zi ),

i=1

where B = B(t, α, d, b1 , b2 , b3 , β, γ) ∈ (0, ∞) and θ = θ(s, t, h, α, d, b1 , b2 , b3 , β, γ, x, z) ∈ (0, 1). We also have Z d Y β−γ −1 ρ0α (t − s, xi − zi ) dz (|x − z| ∧ 1)(t − s) Rd

≤ c

i=1

d Z X i=1

R

≤ c(t − s)

ρ0β−γ (t − s, xi − zi ) dzi

β−γ−α α

.

It follows that lim V =

h→0+

Z

t t/2

Z

Rd

∂ pz (t − s, x − z)(q(s, z, y) − q(s, x, y)) dz ds. ∂t

Note that for h > 0, s ∈ (t/2, t), x, z ∈ Rd we have  Z Z 1 pz (t − s, x − z) dz pz (t + h − s, x − z) dz − h Rd Rd Z ∂ = pz (t + θh − s, x − z) dz ∂t Rd Z ∂ pz (t + θh − s, x − z) dz, = Rd ∂t

where θ = θ(s, t, h, α, d, b1 , b2 , b3 , β, γ, x) ∈ (0, 1).

(69)

TRANSITION DENSITIES FOR SDES DRIVEN BY CYLINDRICAL STABLE PROCESSES

27

Using this, (13) and the definition of q0 (t, x, y) we get Z pz (t + h − s, x − z) − pz (t − s, x − z) dz h d R Z Lz pz (t + θh − s, ·)(x − z) dz = Rd Z Z Lx pz (t + θh − s, ·)(x − z) dz. q0 (t + θh − s, x, z) dz + = − Rd

Rd

By (19) and (10), we have Z

R

By Lemma 4.1 we get Z

Rd

It follows that

β−α q0 (t + θh − s, x, z) dz ≤ c(t − s) α . d

β−α Lx pz (t + θh − s, ·)(x − z) dz ≤ c(t − s) α .

lim VI =

h→0+

Z

t

t/2

Z

Rd

∂ pz (t − s, x − z) dzq(s, x, y) ds. ∂t

(70)

By (66-70) we obtain lim

h→0+

ϕy (t + h, x) − ϕy (t, x) = q(t, x, y) + h

Z tZ 0

Rd

Lz pz (t − s, ·)(x − z)q(s, z, y) dz ds.

The proof of the analogous result for limh→0− is very similar and it is omitted.



Proposition 4.5. For all t ∈ (0, ∞) and x, y ∈ Rd we have ∂ A p (t, x, y) = LpA (t, ·, y)(x). ∂t Proof. By the definition of q(t, x, y) we obtain Z tZ q0 (t − s, x, z)q(s, z, y) dz ds. q(t, x, y) = q0 (t, x, y) + 0

(71)

Rd

Using (44), (13), Lemma 4.4 and the definition of q0 (t, x, y) we obtain

∂py ∂ϕy ∂pA (t, x, y) = (t, x − y) + (t, x) ∂t ∂t ∂t Z tZ Lz pz (t − s, ·)(x − z)q(s, z, y) dz ds = Ly py (t, x − y) + q(t, x, y) + d 0 R Z tZ x Lz pz (t − s, ·)(x − z)q(s, z, y) dz ds. = L py (t, x − y) − q0 (t, x, y) + q(t, x, y) + 0

Rd

By (71) this is equal to x

L py (t, x − y) +

Z tZ 0

= Lx py (t, x − y) +

Z

0

Rd tZ

(q0 (t − s, x, z) + Lz pz (t − s, ·)(x − z)) q(s, z, y) dz ds

Rd

Lx pz (t − s, ·)(x − z)q(s, z, y) dz ds.

By (47) and (44) this is equal to Lx pA (t, ·, y)(x), which completes the proof. Lemma 4.6. For any bounded Borel f : Rd → R, t ∈ (0, ∞) and x ∈ Rd we have L(PtA f )(x) =

∂ A P f (x). ∂t t



28

T. KULCZYCKI AND M. RYZNAR

Proof. We have L(PtA f )(x) = lim Lε (PtA f )(x) = lim ε→0+

ε→0+

By Lemma 4.2 this is equal to Z Z A lim Lε p (t, ·, y)(x)f (y) dy = + Rd ε→0

Z

Rd

Lε pA (t, ·, y)(x)f (y) dy.

LpA (t, ·, y)(x)f (y) dy.

(72)

Rd

By Proposition 4.5 this is equal to Z Z ∂ A ∂ pA (t, x, y)f (y) dy. p (t, x, y)f (y) dy = ∂t ∂t d d R R



Proposition 4.7. For t ∈ (0, T ] and x, y ∈ Rd we have pA (t, x, y) ≤ c

d Y

gt (xi − yi ).

i=1

Proof. By Theorem 3.2, estimates of pz and (12) we obtain Z t Z |ϕy (t, x)| = pz (t − s, x − z)q(s, z, y) dz ds ≤c

Z tZ 0

×

d X

m=1

≤c

  

d X

m=1

Rd

d Y

0 d Y

ρ0α (t − s, xi − zi )

i=1

i=1 i6=m

  

Rd



 ρ0α (s, zi − yi ) [ρβ0 (s, zm − ym ) + ρ0β (s, zm − ym )] dz ds

d Y



ρ0α (t, xi

 − yi )

i=1 i6=m

Z tZ 0

R

ρ0α (t − s, xm − zm )

×[ρβ0 (s, zm − ym ) + ρ0β (s, zm − ym )] dz ds   d d X Y  ρ0α (t, xi − yi ) [ρ0α+β (t, xm − ym ) + ρβα (t, xm − ym )] ≤c  m=1

≤c

"

d Y i=1

i=1 i6=m

ρ0α (t, xi

#"

− yi )

t

β/α

# d   X β |xm − ym | ∧ 1 . +

(73)

m=1

Now the conlusion follows from (16) and estimates of py .



The following result shows that {PtA } is a Feller semigroup. Theorem 4.8. We have: (i) PtA : C0 (Rd ) → C0 (Rd ) for any t ∈ (0, ∞), (ii) limt→0+ ||PtA f − f ||∞ = 0 for any f ∈ C0 (Rd ). (iii) RpA (t, x, y) ≥ 0 for any (t, x, y) ∈ (0, ∞) × Rd × Rd , (iv) Rd pA (t, x, y) dy = 1 for any (t, x) ∈ (0, ∞) × Rd , R (v) Rd pA (t, x, z)pA (s, z, y) dz = pA (s + t, x, y) for any (s, t, x, y) ∈ (0, ∞) × (0, ∞) × Rd × Rd .

TRANSITION DENSITIES FOR SDES DRIVEN BY CYLINDRICAL STABLE PROCESSES

29

Proof. (i) follows by the fact that x → pA (t, x, y) is continuous and by Proposition 4.7. It is shown in the proof of Proposition 4.7 that   d d X Y   |ϕy (t, x)| ≤ c ρ0α (t, xi − yi ) [ρ0α+β (t, xm − ym ) + ρβα (t, xm − ym )].  m=1

i=1 i6=m

R Let f ∈ C0 (Rd ). It follows that limt→0+ supx∈Rd Rd ϕy (t, x)f (y) dy = 0 for any f ∈ R C0 (Rd ). It is clear that limt→0+ supx∈Rd Rd py (t, x − y)f (y) dy − f (x) = 0 for any f ∈ C0 (Rd ). Hence we obtain (ii). For any (t, x) ∈ (0, T ] × Rd put u(t, x) = PtA 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 we get (61). By Theorem 1.1 (iv) and Proposition 3.15 we obtain (62). Lemma 4.6 gives ˜(t, x) = (63). Applying Proposition 4.3 to f ∈ Cc∞ , f ≤ 0 we obtain (iii). Note that u A −1 + Pt 1(x), u(0, x) = 0 also satisfies the assumptions of Proposition 4.3. Using this proposition we get that PtA 1 ≡ 1 which implies (iv). Fix s ∈ (0, T ], f ∈ Cc∞ , f ≥ 0 A f (x), u (t, x) = P A P A f (x), u (0, x) = u (0, x) = P A f (x), and denote u1 (t, x) = Pt+s 2 1 2 t s s u(t, x) = u1 (t, x) − u2 (t, x). By Proposition 4.3 applied to u(t, x) we get u1 ≡ u2 which implies (v).  Using similar ideas as in the proof of (47) one can easily obtain the following result. Lemma 4.9. For any t ∈ (0, ∞), x ∈ Rd and any bounded, H¨ older continuous function f we have  Z t Z t A LPsA f (x) ds. (74) Ps f (·) ds (x) = L 0

0

Proposition 4.10. For any t ∈ (0, ∞), x ∈

Rd

PtA f (x) = f (x) + Rt

Z

and f ∈ Cb2 (Rd ) we have

t 0

PsA Lf (x) ds.

(75)

PsA Lf (x) ds. By (74) we get Z t
Lu(t, x) = Lf (x) + L PsA Lf (x) ds.

Proof. Put u(t, x) = f (x) +

0

0

By Lemma 4.6 this is equal to Z t
∂ ∂ PsA Lf (x) ds = PtA Lf (x) = u(t, x). Lf (x) + ∂t 0 ∂s

It is easy to check that u(t, x) satisfies (60-62). Put u ˜(t, x) = PtA 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 ≡ 0 which implies the assertion of the lemma.  The next theorem gives that L is a generator of the semigroup {PtA }. Theorem 4.11. For any f ∈ Cb2 (Rd ) we have PtA f (x) − f (x) = Lf (x), t t→0+ and the convergence is uniform. lim

x ∈ Rd

Proof. By Proposition 4.10 we have Z PtA f (x) − f (x) 1 t A lim = lim Ps Lf (x) ds. t t→0+ t→0+ t 0 By Theorem 4.8 (ii) this is equal to Lf (x) and the convergence is uniform.



30

T. KULCZYCKI AND M. RYZNAR

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 and ˜ t with the transition kernel Theorem 4.11 we conclude that there is a Feller process X pA (t, x, y) and the generator L. Let Px , Ex be the distribution and expectation for the process starting from x ∈ Rd . First, note that for any function f ∈ Cb2 (Rd ), the process Z t ˜ X,f ˜ s )ds ˜ ˜ Lf (X Mt = f (Xt ) − f (X0 ) − 0

(Px , F

x is a t ) martingale, where Ft is a natural filtration. That is P solves the martingale problem for (L, Cb2 (Rd )). On the other hand, according to [1, Theorem 6.3], the unique weak solution X to the stochastic equation (1) has the law which is the unique solution ˜ and X have the same to the martingale problem for (L, Cb2 (Rd )). It follows that that X A law and p (t, x, y) is the transition kernel of X. The continuity of pA (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 next section. Finally, (ii) follows from Proposition 4.5. The upper bound estimate in (iii) follows from Proposition 4.7. 

proof of Theorem 1.1 (iv). The main tool used in this proof is Theorem 3.2. By Lemma 3.10 and Theorem 3.2 we get ϕy (t, x) − ϕy (t, x′ ) Z tZ pz (t − s, x − z) − pz (t − s, x′ − z) |q(s, z, y)| dz ds ≤ 0 Rd
≤ c A(t, x, y) + A(t, x′ , y) , (76) where

′ γ

A(t, x, y) = |x − x |

Z tZ 0

×s

d−1

" d Y

−γ/α

(t − s) Rd

#

gt−s (xi − zi )

i=1

ρ00 (s, zi

#"

− yi )

i=1

We have

" d Y

s

β/α

+

d  X

m=1

#  |zm − ym | ∧ 1 dz ds. β

  d d X Y   A(t, x, y) ≤ c|x − x′ |γ gt (xi − yi )  m=1

i=1 i6=m

Z tZ

ρ0α−γ (t − s, xm − zm )ρβ0 (s, zm − ym ) dzm ds #Z Z " d t Y ′ γ ρ0α−γ (t − s, x1 − z1 )ρ0β (s, z1 − y1 ) dz1 ds. gt (xi − yi ) +c|x − x | ×

0

R

i=2

0

R

By (12) we have Z tZ ρ0α−γ (t − s, xm − zm )ρβ0 (s, zm − ym ) dzm ds ≤ ρ0α−γ+β (t, xm − ym ) + ρβα−γ (t, xm − ym ), 0

and

R

Z tZ 0

R

ρ0α−γ (t − s, x1 − z1 )ρ0β (s, z1 − y1 ) dz1 ds ≤ ρ0α−γ+β (t, x1 − y1 ).

TRANSITION DENSITIES FOR SDES DRIVEN BY CYLINDRICAL STABLE PROCESSES

It follows that A(t, x, y) ≤ c|x − x′ |γ t−γ/α

"

d Y

31

#

gt (xi − yi ) .

i=1

Using this, (76), Lemma 3.10 and (44) we get the assertion of Theorem 1.1 (iv).



5. Lower bound estimates 5.1. L´ evy system. Let Px , Ex be the distribution and expectation for the process Xt starting from x ∈ Rd . By Ft we denote a natural filtration. For x ∈ Rd and Borel A ⊂ Rd we define the jumping measure d Z X dyi ⊗k6=i δxk (dyk )aαii (x) J(x, A) = Aα , |y − xi |1+α i A i=1

where δxk is a Dirac measure on R concentrated at xk . The purpose of this subsection is to provide arguments for the L´evy system formula. Namely, we will show that for any x ∈ Rd and any non-negative measurable function f on R+ × Rd × Rd vanishing on {(s, x, y) ∈ R+ × Rd × Rd ; x = y} and Ft stopping time T , we have Z TZ X x x f (s, Xs− , Xs ) = E E f (s, Xs , y)J(Xs , dy)ds. (77) Rd

0

s≤T

Since we exactly follow the approach of [11] we only briefly sketch the arguments. It is well known that for f ∈ Cb2 (Rd ), d Z dwi Aα X [f (x + aii (x)wi ei ) + f (x − aii (x)wi ei ) − 2f (x)] . Lf (x) = 1+α 2 |w i| R i=1

Rd

For y ∈ we denote |y|∞ = maxi {|yi |} the sup-norm in Rd . For x ∈ Rd and r > 0 we denote B(x, r) = {y ∈ Rd , |y − x|∞ < r}. Then for f ∈ Cb2 (Rd ), we can rewrite the formula of the generator as Z (f (y) − f (x))J(x, dy). Lf (x) = lim rց0 B c (x,r)

As it has been already observed in the last section, for any function f ∈ Cb2 (Rd ), the process Z t Lf (Xs )ds Mtf = f (Xt ) − f (X0 ) − 0

is a (Px , Ft ) martingale. Suppose that A and B are two bounded closed subsets of Rd having a positive distance from each other. Let f ∈ Cb2 (Rd ) be such that f (x) = 0, x ∈ A and f (x) = 1, x ∈ B. We consider a martingale transform of Mtf , Z t f 1A (Xs− )dMsf . Nt = 0

By the Ito formula, if Xs− ∈ A, we have

dMsf = f (Xs ) − f (Xs− ) − Lf (Xs )ds = f (Xs ) − Lf (Xs )ds. This implies that Ntf

=

X

1A (Xs− )f (Xs ) −

X

1A (Xs− )f (Xs ) −

s≤t

=

s≤t

Z Z

t

1A (Xs )Lf (Xs )ds 0 t

1A (Xs ) 0

Z

f (y)J(Xs , dy)ds

32

T. KULCZYCKI AND M. RYZNAR

Approximating 1B by a decreasing sequence of smooth functions we show that Z Z t X J(Xs , dy)ds 1A (Xs ) 1A (Xs− )1B (Xs ) − B

0

s≤t

is a martingale, hence E

x

X

1A (Xs− )1B (Xs ) = E

x

s≤t

Z

t

1A (Xs ) 0

Z

J(Xs , dy)ds. B

Using this and a routine measure theoretic argument, we get Z tZ X x x f (Xs− , Xs ) = E E f (Xs , y)J(Xs , dy)ds 0

s≤t

Rd

for any x ∈ Rd and any non-negative measurable function f on Rd × Rd vanishing on the diagonal. Finally, following the same arguments as in [10, Appendix A], we obtain (77).

5.2. Lower bound of pA . We essentially follow the approach from [11], where an argument relied on certain exit and hitting times estimates was applied, but the singularity of the jumping measure forces us to use an induction argument. We start with the near diagonal estimate of the transition kernel. Lemma 5.1. For any a > 0 there is c = c(a, d, α, b1 , b2 , b3 , β) and 0 < t0 ≤ 1, t0 = t0 (a, d, α, b1 , b2 , b3 , β) such that for t ≤ t0 and x, y ∈ Rd with |y − x|∞ ≤ at1/α , pA (t, x, y) ≥ ct−d/α .

(78)

Proof. By (73), if |y − x|∞ ≤ at1/α , we have # " d  d  X Y |xm − ym |β ∧ 1 ≤ c1 t−d/α tβ/α . ρ0α (t, xi − yi ) tβ/α + |ϕy (t, x)| ≤ c m=1

i=1

Hence, we can find t0 ≤ 1 such that for t ≤ t0 and |y − x|∞ ≤ at1/α we have pA (t, x, y)

= py (t, x − y) + ϕy (t, x) ≥ py (t, x − y) − |ϕy (t, x)| ≥ c2 t−d/α − c1 t−d/α tβ/α ≥ ct−d/α . 

Let for a Borel D ⊂ Rd , τD = inf{t > 0; Xt ∈ / D} and TD = inf{t > 0; Xt ∈ D} be the first exit and hitting time of D, respectively. Lemma 5.2. There is c such that, for t ≤ 1, R > 0, x ∈ Rd , Px (τB(x,R) ≤ t) ≤ c

t . Rα

TRANSITION DENSITIES FOR SDES DRIVEN BY CYLINDRICAL STABLE PROCESSES

33

Proof. Applying the strong Markow property, we obtain Px (τB(x,R) ≤ t) ≤ Px (τB(x,R) ≤ t; |X(t) − x|∞ ≤ R/8) + Px (|X(t) − x|∞ ≥ R/8) ≤ Px (τB(x,R) ≤ t; |X(t) − X(τB(x,R) )|∞ ≥ R/8) + Px (|X(t) − x|∞ ≥ R/8) = Ex (τB(x,R) ≤ t; P X(τB(x,R) ) (|X(t − τB(x,R) ) − X(τB(x,R) )|∞ ≥ R/8)) + Px (|X(t) − x|∞ ≥ R/8) ≤ 2 sup sup Pz (|X(s) − z|∞ ≥ R/8) z

s≤t

t ≤ c α. R The last step follows from the upper estimate (5) of the heat kernel pA (t, x, y).



Lemma 5.3. Let r > 0 and x, y ∈ Rd . Assume that |x1 −y1 | ≥ 6r and max2≤i≤d |xi −yi | ≤ r. Then for t > 0, Px (X(t) ∈ B(y, 4r)) ≥ c

rt Px (τB(x,r) ≥ t) inf Pz (τB(z,2r) > t) z |y1 − x1 |1+α

Proof. Let σ = TB(y,2r) be the first hitting time. By the strong Markow property Px (X(t) ∈ B(y, 4r)) ≥ Px (σ ≤ t; sup |X(s) − X(σ)|∞ ≤ 2r) σ≤s≤σ+t    x X(σ) = E σ ≤ t; P sup |X(s) − X(σ)|∞ ≤ 2r s≤t

≥ Px (σ ≤ t) inf Pz (sup |X(s) − z|∞ ≤ 2r) z

s≤t

z

≥ inf P (τB(z,2r) > t)Px (X(t ∧ τB(x,r) ) ∈ B(y, 2r)). z

By the L´evy system formula (77), we have x

P ((X(t ∧ τB(x,r) ) ∈ B(y, 2r)) = E

x

Z

0

t∧τB(x,r)

Z

J(Xs , du)ds.

B(y,2r)

We may assume x1 < y1 . Since |x1 − y1 | ≥ 6r and max2≤i≤d |xi − yi | ≤ r, for z ∈ B(x, r), we have Z y1 +2r α Z a11 (z)dw1 r J(z, du) = ≥c . 1+α |y1 − x1 |1+α y1 −2r |w1 − z1 | B(y,2r) Hence, r Ex [t ∧ τB(x,r) ] |y1 − x1 |1+α rt ≥ c Px (τB(x,r) ≥ t). |y1 − x1 |1+α

Px ((X(t ∧ τB(x,r) ) ∈ B(y, 2r)) ≥ c

 Lemma 5.4. There is t0 > 0, t0 = t0 (d, α, b1 , b2 , b3 , β), such that for 0 < t ≤ t0 , x, y ∈ Rd satisfying max2≤i≤d |xi − yi | ≤ 2t1/α we have pA (t, x, y) ≥ c

d Y i=1

gt (xi − yi ).

34

T. KULCZYCKI AND M. RYZNAR

Proof. We pick t0 > 0 corresponding to a = 12 in Lemma 5.1. Due to the near diagonal estimate (78), it is enough to consider |x1 − y1 | ≥ 12t1/α and max2≤i≤d |xi − yi | ≤ 2t1/α . Applying Lemma 5.3 with r = 2t1/α , we obtain Z A pA (t1 , x, z)pA (t2 , z, y)dz p (t, x, y) ≥ B(y,4r)

≥

inf

z∈B(y,4r)

≥ c ×

inf

pA (t2 , z, y)Px ((X(t1 ) ∈ B(y, 4r)) pA (t2 , z, y)

z∈B(y,4r) inf Pz (τB(z,2r) z

rt1 Px (τB(x,r) ≥ t1 ) |y1 − x1 |1+α

> t1 ),

(79)

where ti > 0 with t1 + t2 = t. Now, due to Lemma 5.2, we can pick 0 < λ < 1, independently of t, such that inf Pz (τB(z,r) ≥ λt) ≥ 1/2. z

Moreover, we can select λ so small that 8 ≤ 12(1 − λ)1/α . Then for |z − y|∞ ≤ 4r = 8t1/α ≤ 12((1 − λ)t)1/α , by Lemma 5.1, we have pA ((1 − λ)t, z, y) ≥ ct−d/α . Taking t1 = λt, t2 = (1 − λ)t and applying (79) we arrive at d

pA (t, x, y) ≥ ct−(d−1)/α

Y t gt (xi − yi ). ≥c 1+α |y1 − x1 | i=1

 Proof of the lower bound estimates in Theorem 1.1 (iii). For a natural k ≤ d−1 we define Vk (t) = {(x, y) ∈ R2d ; min |xi − yi | ≥ t1/α and max |xi − yi | ≤ t1/α }. 1≤i≤k

k+1≤i≤d

We set V0 (t) = {(x, y) ∈ R2d ; max |xi − yi | ≤ t1/α } 1≤i≤d

and Vd (t) = {(x, y) ∈ R2d ; min |xi − yi | ≥ t1/α }. 1≤i≤d

By a renumeration argument it is enough to prove the corresponding lower bound on Vk (t), k = 0, . . . , d. At first, we assume that t ≤ t0 , where t0 was found in Lemma 5.4. We have already proved the lower bound on V0 (t) and V1 (t). We show how to extend it to V2 (t). Thus , we consider the case |x1 −y1 | ≥ t1/α , |x2 −y2 | ≥ t1/α and max3≤i≤d |xi −yi | ≤ t1/α . Let x′ = (y1 , x2 , . . . , xd ). If z ∈ B(x′ , t1/α /4) then |x1 − z1 | ≥ (3/4)t1/α , max2≤i≤d |xi − zi | ≤ 2t1/α and |y2 − z2 | ≥ (3/4)t1/α , maxi6=2 |yi − zi | ≤ 2t1/α . Hence, by Lemma 5.4, pA (t, x, z) ≥ ct−(d−1)/α

t t ≥ ct−(d−1)/α , |z1 − x1 |1+α |y1 − x1 |1+α

pA (t, z, y) ≥ ct−(d−1)/α

t t ≥ ct−(d−1)/α . 1+α |z2 − y2 | |y2 − x2 |1+α

TRANSITION DENSITIES FOR SDES DRIVEN BY CYLINDRICAL STABLE PROCESSES

35

Finally, pA (2t, x, y) ≥

Z

pA (t, x, z)pA (t, z, y)dz

B(x′ ,t1/α /4)

t t t−(d−1)/α ≥ ct 1+α |y1 − x1 | |y2 − x2 |1+α t t ≥ c t−(d−2)/α 1+α |y1 − x1 | |y2 − x2 |1+α −(d−1)/α

≥ c

d Y

Z

dz B(x′ ,t1/α /4)

g2t (xi − yi ).

i=1

This concludes the proof of the lower bound on V2 (t). In a similar fashion, by induction argument, we show that, if (x, y) ∈ Vk (t) and t ≤ t0 , then pA (t, x, y) ≥ c ≥ c

t t × ··· × t−(d−k)/α 1+α |y1 − x1 | |yk − xk |1+α d Y

gt (xi − yi ) ≥ cp0 (t, x − y),

i=1

which ends the proof for the case t ≤ t0 . If t > t0 then we can write t = nt0 + s, with s < t0 and n ∈ N. Then by already proved lower bound Z Z pA (t0 , x, z1 ) . . . pA (t0 , zn , zn+1 )pA (s, zn+1 , y)dz1 . . . dzn+1 ··· pA (t, x, y) = d d R Z R Z p0 (t0 , x − z1 ) . . . p0 (t0 , zn − zn+1 )p0 (s, zn+1 − y)dz1 . . . dzn+1 ≥ cn+1 ··· Rd

n+1

= c

Rd

p0 (t, x − y).

The proof is completed.  proof of Theorem 1.1 (v). The assertion follows from Proposition 3.15 and the lower bound estimate in Theorem 1.1 (iii).  References [1] R. Bass, Z.-Q. Chen, Systems of equations driven by stable processes, Probab. Theory Related Fields 134 (2006), no. 2, 175-214. [2] K. Bogdan, T. Grzywny, M. Ryznar, Density and tails of unimodal convolution semigroups, J. Funct. Anal. 266 (2014), 3543-3571. [3] K. Bogdan, T. Jakubowski, Estimates of heat kernel of fractional Laplacian perturbed by gradient operators, Comm. Math. Phys. 271 (1) (2007), 179-198. [4] K. Bogdan, P. Sztonyk, Estimates of the potential kernel and Harnacks inequality for the anisotropic fractional Laplacian, Studia Math., 181(2) (2007), 101-123. [5] K. Bogdan, P. Sztonyk, V. Knopova, Heat kernel of anisotropic nonlocal operators, arXiv:1704.03705 (2017). [6] Z.-Q. Chen, E. Hu, Heat kernel estimates for ∆ + ∆α/2 under gradient perturbation, Stochastic Processes Appl. 125 (2015), 2603-2642. [7] Z.-Q. Chen, P. Kim, T Kumagai, Global Heat Kernel Estimates for Symmetric Jump Processes, Trans. Amer. Math. Soc. 363, no. 9 (2011), 5021-5055. [8] Z.-Q. Chen, P. Kim, R. Song, Dirichlet heat kernel estimates for fractional Laplacian with gradient perturbation, Ann. Probab. 40 (2012), 2483-2538. [9] Z.-Q. Chen, P. Kim, R. Song, Stability of Dirichlet heat kernel estimates for non-local operators under Feynmman-Kac perturbation, Trans. Amer. Math. Soc. 376 (2015), 5237-5270. [10] Z.-Q. Chen, T. Kumagai, Heat kernel estimates for jump processes of mixed types on metric measure spaces, Probab. Theory Relat. Fields 140, no. 1-2 (2008), 277-317.

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[11] Z.-Q. Chen, X. Zhang, Heat kernels and analyticity of non-symmetric jump diffusion semigroups, Probab. Theory Relat. Fields 165 (2016), no. 1-2, 267-312. [12] A. Friedman, Partial Differential Equations of Parabolic Type, Prentice-Hall, Englewood Cliffs, N.J., (1975). [13] Y. Ishikawa, Density estimate in small time for jump processes with singular L´evy measures, Tohoku Math. J. 53(2) (2001), 183-202. [14] T. Jakubowski, K. Szczypkowski, Estimates of gradient perturbation series, J. Math. Anal. Appl. 389 (2012), 452-460. [15] T. Jakubowski, K. Szczypkowski, Time-dependent gradient perturbations of fractional Laplacian, J. Evol.Equ. 10 (2010), 319-339. [16] K. Kaleta, P. Sztonyk, Estimates of transition densities and their derivatives for jump L´evy processes, J. Math. Anal. Appl. 431(1) (2015), 260-282. [17] K. Kaleta, P. Sztonyk, Small time sharp bounds for kernels of convolution semigroups, J. Anal. Math. 132 (2017), 355-394. [18] K. Kaleta, P. Sztonyk, Upper estimates of transition densities for stable-dominated semigroups, J. Evol. Equ. 13(3) (2013), 633-650. [19] P.Kim, R.Song, Z.Vondraˇcek, Heat kernels of non-symmetric jump processes: beyond the stable case, Potential Analysis, to appear. [20] V. Knopova, Compound kernel estimates for the transition probability density of a L´evy process in Rn , Theor. Probability and Math. Statist. 89 (2014), 57-70. [21] V. Knopova, A. Kulik, Exact asymptotic for distribution densities of L´evy functionals, Electronic Journal of Probability 16 (2011), 1394-1433. [22] V. Knopova, R. Schilling, Transition density estimates for a class of L´evy and L´evy-type processes, J. Theoret. Probab. 25(1) (2012), 144-170. [23] A. N. Kochubei, Parabolic pseudodifferential equations, hypersingular integrals and Markov processes, Math. USSR Izv. 33 (1989), 233-259; translation from Izv. Akad. Nauk SSSR, Ser. Mat. 52 (1988), 909-934. [24] V. Kolokoltsov, Symmetric stable laws and stable-like jump diffusions, Proc. Lond. Math. Soc. 83 (2000), 725-768. [25] T. Kulczycki, M. Ryznar, Gradient estimates of harmonic functions and transition densities for L´evy processes, Trans. Amer. Math. Soc. 368, no. 1 (2016), 281-318. [26] O. A. Ladyzenskaja, V. A. Solonnikov, N. N. Uralceva, Linear and Quasi-linear Equations of Parabolic Type (translated from the Russian by S.Smith), American Mathematical Society, Providence, (1968). [27] E. E. Levi, Sulle equazioni lineari totalmente ellittiche alle derivate parziali, Rend. Circ. Mat. Palermo 24 (1907), 275317. [28] A. Mimica, Heat kernel upper estimates for symmetric jump processes with small jumps of high intensity, Potential Anal. 36, no. 2 (2012), 203-222. [29] J. Picard, Density in small time at accessible points for jump processes, Stochastic Process. Appl. 67, no.2 (1997), 251-279. [30] J. Picard, Density in small time for L´evy processes, ESAIM Probab. Statist. 1 (1997), 358-389. [31] J. Picard, On the existence of smooth densities for jump processes, Probab. Theory Related Fields 105 (1996), 481-511. [32] E. Priola, A. Shirikyan, L. Xu, J. Zabczyk, Exponential ergodicity and regularity for equations with L´evy noise, Stochastic Process. Appl. 122 (2012), no. 1, 106133. [33] E. Priola, J. Zabczyk, Structural properties of semilinear SPDEs driven by cylindrical stable processes, Probab. Theory Related Fields 149 (2011), no. 1-2, 97-137. [34] R. Schilling, P. Sztonyk and J. Wang, Coupling property and gradient estimates for L´evy processes via the symbol, Bernoulli 18 (2012), 1128-1149. [35] X. Sun, Y. Xie, Ergodicity of stochastic dissipative equations driven by α-stable process Stoch. Anal. Appl. 32 (2014), no. 1, 61-76. [36] P. Sztonyk, Approximation of stable-dominated semigroups, Potential Anal. 33(3) (2010), 211-226. [37] P. Sztonyk, Estimates of densities for L´evy processes with lower intensity of large jumps, Math. Nachr., 290(1) (2017), 120-141. [38] P. Sztonyk, Regularity of harmonic functions for anisotropic fractional Laplacians, Math. Nachr. 283 no. 2 (2010) 289-311. [39] L. Xie, X. Zhang, Heat kernel estimates for critical fractional diffusion, Studia Math. 224(3) (2014), 221-263. [40] L. Wang, X. Zhang, Harnack Inequalities for SDEs Driven by Cylindrical α-Stable Processes, Potential Anal. 42(3) (2015), 657669. [41] X. Zhang, Derivative formulas and gradient estimates for SDEs driven by α-stable processes, Stochastic Process. Appl. 123 (2013) 12131228.

TRANSITION DENSITIES FOR SDES DRIVEN BY CYLINDRICAL STABLE PROCESSES

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Faculty of Pure and Applied Mathematics, Wroclaw University of Science and Technol´ skiego 27, 50-370 Wroclaw, Poland. ogy, Wyb. Wyspian E-mail address: [email protected] E-mail address: [email protected]