Heat kernel of supercritical SDEs with unbounded drifts
aa r X i v : . [ m a t h . A P ] D ec HEAT KERNEL OF SUPERCRITICAL NONLOCAL OPERATORSWITH UNBOUNDED DRIFTS
STEPHANE MENOZZI AND XICHENG ZHANG
Abstract.
Let α ∈ (0 ,
2) and d ∈ N . Consider the following SDE in R d :d X t = b ( t, X t )d t + a ( t, X t − )d L ( α ) t , X = x, where L ( α ) is a d -dimensional rotationally invariant α -stable process, b : R + × R d → R d and a : R + × R d → R d ⊗ R d are H¨older continuous functions in space,with respective order β, γ ∈ (0 ,
1) such that ( β ∧ γ ) + α >
1, uniformly in t .Here b may be unbounded. When a is bounded and uniformly elliptic, we showthat the unique solution X t ( x ) of the above SDE admits a continuous density,which enjoys sharp two-sided estimates. We also establish sharp upper-boundfor the logarithmic derivative. In particular, we cover the whole supercritical range α ∈ (0 , ad hoc parametrix expansions andprobabilistic techniques. Introduction
Throughout this paper we fix α ∈ (0 , L ( α ) be a d -dimensional rotation-ally invariant α -stable process. We consider the following stochastic differentialequation: d X t = b ( t, X t )d t + a ( t, X t − )d L ( α ) t , (1.1)where b : R + × R d → R d and a : R + × R d → R d ⊗ R d are Borel measurable functionsand satisfy that for some β ∈ ((1 − α ) + ,
1] and κ > | b ( t, | κ , | b ( t, x ) − b ( t, y ) | κ ( | x − y | β ∨ | x − y | ) , ( H βb )and for some γ ∈ ((1 − α ) + ,
1] and κ > κ − I ( aa ∗ )( t, x ) κ I , | a ( t, x ) − a ( t, y ) | κ | x − y | γ , ( H γa )where a ∗ stands for the transpose of a and I is the identity matrix. Under (H βb ) and (H γa ) , it is well known that for each ( s, x ) ∈ R + × R d , there is a unique weaksolution X s,t ( x ) to SDE (1.1) starting from x at time s (see e.g. [12, Theorem 1.1]),and the generator of SDE (1.1) writes as L s f ( x ) := L s f ( x ) + b ( s, x ) · ∇ f ( x ) , (1.2)where L s is given by L s f ( x ) = Z R d δ (2) f ( x ; a ( s, x ) z ) d z | z | d + α = Z R d δ (2) f ( x ; z ) κ ( s, x, z ) | z | d + α d z (1.3) X. Zhang is supported by NNSF of China (No. 11731009) and the DFG through the CRC 1283“Taming uncertainty and profiting from randomness and low regularity in analysis, stochasticsand their applications”.S. Menozzi has been funded by the Russian Science Foundation project (project No. 20-11-20119). with δ (2) f ( x ; z ) := f ( x + z ) + f ( x − z ) − f ( x ) (1.4)and κ ( s, x, z ) := det( a − ( s, x ))( | z | / | a − ( s, x ) z | ) d + α . (1.5)Clearly, by ( H γa ) we have for some ¯ κ > κ − κ ( s, x, z ) ¯ κ , | κ ( s, x, z ) − κ ( s, y, z ) | ¯ κ | x − y | γ . (1.6)The operator L s is called supercritical for α ∈ (0 ,
1) since in this case, the driftterm plays a dominant role. Namely, from the self-similarity properties of thedriving process L ( α ) in (1.1), it holds that for any s > L ( α ) s (law) = s /α L ( α )1 andfor s ∈ (0 , , α ∈ (0 , s /α < s . This precisely means that the fluctuationsinduced by the noise are smaller than the typical order of the drift term in (1.1).For α ∈ (1 , s ∈ (0 , s /α > s ,the fluctuations of the noise prevail in the SDE. From the operator viewpoint, L s plays a dominant role and we say that L s is subcritical . For the remaining case α = 1, the noise and drift both have the same typical order and the operator L s iscalled critical . Note that for α ∈ (0 , z κ ( s, x, z ) is symmetric, we have L s f ( x ) = 2 Z R d δ (1) f ( x ; z ) κ ( s, x, z )d z | z | d + α , where δ (1) f ( x ; z ) := f ( x + z ) − f ( x ) . Let us now indicate that there is a quite large literature concerning stable drivenSDEs. We can first mention the seminal work of Kolokoltsov [20] who obtained, foran SDE driven by a symmetric stable process with smooth non-degenerate spectralmeasure, Lipschitz non-degenerate diffusion coefficient and non trivial Lipschitzbounded drifts when α >
1, two sided estimates for the density of the type: p ( s, x, t, y ) ≍ C ( t − s ) − d/α (cid:18) | x − y | ( t − s ) /α (cid:19) − ( d + α ) , (1.7)where C > T . Here and below, Q ≍ C Q meansthat C − Q Q CQ .Going to weaker regularity of the coefficients in (1.1) then first leads to investigatethe well-posedness of the martingale problem associated with the formal generatorassociated with the dynamics (1.1). In [1], Bass and Chen showed the weak well-posedness for SDE (1.1) when a is only continuous and uniformly elliptic, b isLipschitz and L ( α ) t is cylindrical α -stable process. In the subcritical case we canmention the work by Mikulevicius and Pragarauskas [26] who derived that weakuniqueness holds for equation (1.1) for bounded H¨older coefficients when α > a . The martingale problem was in their framework studiedfrom some related Schauder estimates established on the associated Integro PartialDifferential Equation (IPDE).In the super-critical case, the well-posedness of the martingale problem wasrecently investigated by Kulik et al. [18], [21] (see also [10, 12]). In [21], theauthors consider SDEs of type (1.1) with bounded H¨older drift and non-degenerate EAT KERNEL OF SUPERCRITICAL SDES WITH UNBOUNDED DRIFTS 3 scalar diffusion coefficients under the natural condition α + β > and obtain theexistence of the heat kernel, a corresponding two-sided estimate of the form (1.7)as well as some estimates corresponding to the time derivative through parametrixtype expansions. Let us emphasize that in the super-critical regime, the timederivative of the heat kernel roughly typically behaves as t − at time t whereas thespatial gradient is then more singular, as it is expected to have typical behavior oforder t − /α > t − for t ∈ (0 , α ∈ (0 ,
2) and for unbounded drifts. It is known, and somehowintuitive, that for unbounded drifts the heat kernel bounds must reflect somehowthe transport induced by the drift. This was for instance observed for a Lipschitzdrift in [14] for degenerate Kolmogorov SDEs which can be viewed as ODEs per-turbed on some components by a Brownian noise propagating through the wholechain thanks to a weak type H¨ormander condition on the drift. Before going furtherlet us also mention the work by Huang [17] which establishes two-sided estimatesfor stable driven SDE with unbounded Lipschitz drift and α ∈ (1 , s < t and x, y ∈ R d , p ( s, x, t, y ) ≍ C ( t − s ) − d/α (cid:18) | θ s,t ( x ) − y | ( t − s ) /α (cid:19) − ( d + α ) , (1.8)where θ s,t ( x ) denotes the flow associated to the drift in (1.1). Namely,˙ θ s,t ( x ) = b ( t, θ s,t ( x )) , θ s,s ( x ) = x. In the non-degenerate Brownian case, the type of heat kernel estimate in [14]has recently been extended to drifts satisfying a linear-growth without a priori smoothness assumptions on the drift, see [25]. In the quoted work, under additionalH¨older continuity of the drift for the second derivatives, the estimates also extendto the derivatives up to order two with the corresponding additional parabolicsingularity.We here somehow follow the main line of that work but are faced with manyadditional difficulties. In particular, a common feature to both the Gaussian SDEsconsidered in [25] and the stable driven here is that we first need to establishthe gradient estimates for smooth coefficients, with constants depending on thederivatives of the coefficients. In the current strictly stable framework we cannotrely on the Malliavin calculus arguments of [25] because of integrability issues.We thus establish here some direct bounds on the associated semi-group and itsderivatives when the coefficients are smooth which serve as a starting point to derivethe estimates concerning the logarithmic gradient of the density. This part is crucialand quite intricate (see Theorem 4.1 below). Then, in a second time, exploitingthoroughly the two-sided estimate, which holds independently of the smoothness it is indeed well known that from the seminal work of Tanaka et al. [28] that weak uniquenessmay fail if this condition is not met. STEPHANE MENOZZI AND XICHENG ZHANG of the coefficients, we establish that the gradient estimates hold also independentlyof such a smoothness. We eventually conclude through a compactness argument.To state our main result, we introduce the regularized flow associated with drift b . For ε >
0, let b ε ( t, x ) := b ( t, · ) ∗ ρ ε ( x ), where ρ ε ( x ) = ε − d ρ ( x/ε ) and ρ is asmooth density function with support in the unit ball. Note that under ( H βb ), | b ε ( t, x ) − b ( t, x ) | κ ε β , ε ∈ (0 , , (1.9)and |∇ b ε ( t, x ) | = (cid:12)(cid:12)(cid:12)(cid:12)Z R d ( b ( t, y ) − b ( t, x )) ∇ ρ ε ( x − y )d y (cid:12)(cid:12)(cid:12)(cid:12) κ ( ε β − + 1) k∇ ρ k L . (1.10)In particular, since α + β >
1, for any
T >
0, there is a
C > s < t T , Z ts k∇ b | r − s | /α ( r, · ) k ∞ d r C Z ts (cid:16) | r − s | β − α + 1 (cid:17) d r C ( t − s ) α + β − α . (1.11)Thus, for fixed s >
0, the following ODE admits a unique solution θ s,t ( x ):˙ θ s,t = b | t − s | /α ( t, θ s,t ) , θ s,s = x, t > . (1.12)Note that for t > s , θ s,t ( x ) denotes the forward solution of the above ODE, whilefor t < s , it denotes the backward solution. We carefully mention that our mainresults will be stated w.r.t. to the flow θ in (1.12) which is precisely associated witha mollified drift with parameter corresponding to the typical scale of the drivingprocess of the SDE (1.1) at the current considered time.For notational simplicity, we introduce the following parameter setΘ := ( κ , κ , d, α, β, γ ) . (1.13)We also denote for T ∈ (0 , ∞ ], D T := { ( s, x, t, y ) : 0 s < t < T, x, y ∈ R d } . We will frequently use from now on the notation . . For two quantities Q and Q ,we mean by Q . Q that there exists C := C ( T, Θ) such that Q CQ . Otherpossible dependencies for the constants will be explicitly specified. Moreover, wealso use the following notation |D ( α ) f | ( x ) := Z R d | δ (2) f ( x ; z ) || z | d + α d z. (1.14)The aim of this paper is to show the following result. Theorem 1.1.
Under (H βb ) and (H γa ) , for each s < t < ∞ and x ∈ R d , X s,t ( x ) admits a density p ( s, x, t, y ) (called heat kernel of L s ) that is continuous asa function of x, y , and such that for each t > and x, y ∈ R d and Lebesgue almostall s ∈ [0 , t ) , ∂ s p ( s, x, t, y ) = L s p ( s, · , t, y )( x ) , p ( s, x, t, · ) → δ { x } ( · ) weakly as s ↑ t, where δ { x } (d y ) denotes the Dirac measure concentrated at x . Moreover, we have(i) (Two-sides estimate) For any
T > , there is a constant C = C ( T, Θ) > such that for all ( s, x, t, y ) ∈ D T , p ( s, x, t, y ) ≍ C ( t − s )(( t − s ) /α + | θ s,t ( x ) − y | ) − d − α , (1.15) where θ s,t ( x ) is defined by ODE (1.12) . EAT KERNEL OF SUPERCRITICAL SDES WITH UNBOUNDED DRIFTS 5 (ii) (Fractional derivative estimate)
For any
T > , there is a constant C = C ( T, Θ) > such that for all ( s, x, t, y ) ∈ D T , |D ( α ) p ( s, · , t, y ) | ( x ) C (( t − s ) /α + | θ s,t ( x ) − y | ) − d − α . (1.16) (iii) (Gradient estimate in x ) For any
T > , there is a constant C = C ( T, Θ) > such that for all ( s, x, t, y ) ∈ D T , |∇ x log p ( s, x, t, y ) | C ( t − s ) − /α . (1.17) Remark 1.2. If | b ( t, x ) − b ( t, y ) | κ | x − y | β for any x, y ∈ R d with | x − y | | b (0 , t ) | κ for all t >
0, then (H βb ) holds. In particular, for c ( x ) being abounded β -H¨older continuous function, b ( x ) := x + c ( x ) satisfies (H βb ) . Remark 1.3.
For α ∈ [1 , θ s,t ( x ) in (1.15) by any regularized flow θ ( ε ) s,t ( x ) defined in (2.1) below. When α ∈ (0 , ε = ( t − s ) /α since we need to use ε to compensate the time singularity inthe supercritical case. For α ∈ (0 , b is continuous in x , we can replace θ s,t ( x )in (1.15) by any measurable Peano flow ϑ s,t ( x ) of the ODE ˙ ϑ s,t ( x ) = b ( t, ϑ s,t ( x )). Remark 1.4.
When b ≡ L ( α ) is a general α -stable-like process, it was provenin [24] that the gradient estimate (1.17) holds for α ∈ ( , α -stable process with α ∈ (0 , ].The paper is organized as follows. We give in Section 2 some preliminary es-timates needed for the main analysis. This concerns the mollified flow, some exitprobabilities, convolution inequalities and the density of the proxy process involvedin the parametrix (which has a dynamic similar to (1.1) with coefficient frozen alonga suitable deterministic flow). Section 3 is then devoted to the derivation of thetwo-sided bound and the fractional derivative estimate under (H βb ) and (H γa ) whenthe coefficients are additionally supposed to be smooth. We specifically addressthe gradient estimate under those same assumptions in Section 4. We eventuallypresent in Section 5 some compactness arguments to derive the results of Theorem1.1 under the sole conditions (H βb ) , (H γa ) .2. Preliminaries
ODE flow.
We first present some basic properties about the solution θ s,t ( x )of the ODE (1.12). Since the drift coefficient therein depends on the initial time s ,the following flow property does no longer hold: θ r,t ◦ θ s,r ( x ) = θ s,t ( x ) , s < r < t. However, the above flow property holds for the following regularized ODE:˙ θ ( ε ) s,t ( x ) = b ε ( t, θ ( ε ) s,t ( x )) , θ ( ε ) s,s ( x ) = x, (2.1)for any fixed regularizing parameter ε >
0. Below we fix α ∈ (0 ,
2) and alwaysassume ( H βb ). The following lemma is easy. Lemma 2.1. (i) For each ε > and s, t > , x θ ( ε ) s,t ( x ) is a C -diffeomorphismand ( θ ( ε ) s,t ) − ( y ) = θ ( ε ) t,s ( y ) . (2.2) Moreover, for all s, r, t > , it holds that θ ( ε ) s,t ( x ) = θ ( ε ) r,t ◦ θ ( ε ) s,r ( x ) . (2.3) STEPHANE MENOZZI AND XICHENG ZHANG (ii) For all ε, ε ′ > and s, t > , x ∈ R d , it holds that | θ ( ε ′ ) s,t ( x ) − θ ( ε ) s,t ( x ) | κ ( ε ∨ ε ′ ) β | t − s | e κ k∇ ρ k L (( ε ∨ ε ′ ) β − +1) | t − s | , (2.4) (iii) For any T > , there is a constant C = C ( T, d, κ ) > such that for all s, t ∈ [0 , T ] , x, y ∈ R d and ε = | t − s | /α , | θ ( ε ) s,t ( x ) − y | ≍ C | x − θ ( ε ) t,s ( y ) | , | θ ( ε ) s,t ( x ) − θ ( ε ) s,t ( y ) | ≍ C | x − y | . (2.5) Proof. (i) Note that by (2.1), for 0 s < t : θ ( ε ) s,t ( x ) = x + Z ts b ( r, θ ( ε ) s,r ( x ))d r, θ ( ε ) t,s ( y ) = y − Z ts b ( r, θ ( ε ) t,r ( y ))d r. Let y = θ ( ε ) s,t ( x ). By the flow property, we have y = (cid:0) θ ( ε ) s,t (cid:1) − ( y ) + Z ts b ( r, θ ( ε ) s,r ◦ (cid:0) θ ( ε ) s,t (cid:1) − ( y ))d r = ( θ ( ε ) s,t ) − ( y ) + Z ts b ( r, (cid:0) θ ( ε ) r,t (cid:1) − ( y ))d r. Since the ODE has a unique solution, we immediately have (cid:0) θ ( ε ) s,t (cid:1) − ( y ) = θ ( ε ) t,s ( y ).As for (2.3), it follows from (2.2) and the flow property.(ii) Without loss of generality, we assume ε ′ < ε . Since by (1.9) and (1.10), | b ε ( t, x ) − b ε ′ ( t, x ) | κ ε β , k∇ b ε k ∞ κ k∇ ρ k L ( ε β − + 1) , (2.6)by definition we have | θ ( ε ′ ) s,t ( x ) − θ ( ε ) s,t ( x ) | Z ts | b ε ′ ( r, θ ( ε ′ ) s,r ( x )) − b ε ( r, θ ( ε ′ ) s,r ( x )) | d r + Z ts | b ε ( r, θ ( ε ′ ) s,r ( x ) − b ε ( r, θ ( ε ) s,r ( x ) | d r κ ε β ( t − s ) + κ k∇ ρ k L ( ε β − + 1) Z ts | θ ( ε ′ ) s,r ( x ) − θ ( ε ) s,r ( x ) | d r. Using Gronwall’s inequality, we obtain (2.4).(iii) Without loss of generality, we assume 0 s < t T . Note that for u ∈ [ s, t ], | θ ( ε ) s,u ( x ) − θ ( ε ) s,u ( y ) | | x − y | + Z us k∇ b ε ( r, · ) k ∞ | θ ( ε ) s,r ( x ) − θ ( ε ) s,r ( y ) | d r. For ε = | t − s | /α , it follows from the Gronwall inequality and (1.10) that | θ ( ε ) s,t ( x ) − θ ( ε ) s,t ( y ) | e κ k∇ ρ k L ( | t − s | ( β − /α +1) | t − s | | x − y | . We thus get (2.5) by (i) and symmetry. (cid:3)
The following result is a consequence of the above lemma, which plays a crucialrole below.
Lemma 2.2. (i) For each s, t > , the map x θ s,t ( x ) , where ( θ s,u ( x )) u ∈ [ s,t ] solves (1.12) , is a C -diffeomorphism and there is a constant C = C ( T, Θ) > such that | det( ∇ θ − s,t ( x )) − | C | t − s | ( α + β − /α . (ii) For any T > , there is a constant C = C ( T, Θ) > such that for all s, t ∈ [0 , T ] and x, y ∈ R d , | t − s | /α + | θ s,t ( x ) − y | ≍ C | t − s | /α + | x − θ t,s ( y ) | . (2.7) EAT KERNEL OF SUPERCRITICAL SDES WITH UNBOUNDED DRIFTS 7 (iii) For any
T > , there is a constant C = C ( T, Θ) > such that for all s, r, t ∈ [0 , T ] and x ∈ R d , | θ s,t ( x ) − θ r,t ◦ θ s,r ( x ) | C | ( r ∨ s ∨ t ) − ( r ∧ s ∧ t ) | /α . (2.8) Proof. (i) It is well known thatdet( ∇ θ s,t ( x )) = 1 + Z ts div b | r − s | /α ( r, θ s,r ( x )) det( ∇ θ s,r ( x ))d r. Thus det( ∇ θ s,t ( x )) = exp (cid:26)Z ts div b | r − s | /α ( r, θ s,r ( x ))d r (cid:27) . The desired estimate follows by (1.11) and ∇ θ − s,t ( x ) = ( ∇ θ s,t ) − ( θ − s,t ( x )).(ii) Fix s < t . For u ∈ [ s, t ], by definition we have θ s,u ( x ) = x + Z us b | r − s | /α ( r, θ s,r ( x ))d r, and for ε = | t − s | /α , θ ( ε ) s,u ( x ) = x + Z us b ε ( r, θ ( ε ) s,r ( x ))d r. By (2.6) with ε ′ = | r − s | /α and ε = | t − s | /α , we have for all u > s , | θ s,u ( x ) − θ ( ε ) s,u ( x ) | Z us (cid:12)(cid:12)(cid:12) b | r − s | /α ( r, θ s,r ( x )) − b ε ( r, θ ( ε ) s,r ( x )) (cid:12)(cid:12)(cid:12) d r . | t − s | β/α +1 + ( t − s ) ( β − /α Z us | θ s,r ( x ) − θ ( ε ) s,r ( x ) | d r, which yields by Gronwall’s inequality that | θ s,t ( x ) − θ ( ε ) s,t ( x ) | . e ( t − s ) ( β − /α +1 | t − s | β/α +1 . | t − s | /α , (2.9)where the second inequality is due to α + β >
1. Thus, by (2.5) and (2.9), we have | θ s,t ( x ) − y | | θ ( ε ) s,t ( x ) − y | + | θ s,t ( x ) − θ ( ε ) s,t ( x ) | . | x − θ ( ε ) s,t ( y ) | + | t − s | /α . | x − θ t,s ( y ) | + | t − s | /α . The right hand side inequality of (2.7) follows. By symmetry, we also have the lefthand side inequality.(iii) Let s, r, t > ε := | r ∨ s ∨ t − r ∧ s ∧ t | /α . By (2.3) we have | θ s,t ( x ) − θ r,t ◦ θ s,r ( x ) | | θ s,t ( x ) − θ ( ε ) s,t ( x ) | + | θ ( ε ) r,t ◦ θ ( ε ) s,r ( x ) − θ ( ε ) r,t ◦ θ s,r ( x ) | + | θ ( ε ) r,t ◦ θ s,r ( x ) − θ r,t ◦ θ s,r ( x ) | . The desired estimate (2.8) again follows by (2.5) and (2.9). (cid:3)
Remark 2.3.
By (2.8), we have | x − θ t,s ◦ θ s,t ( x ) | . C | t − s | /α , (2.10)and by (2.7), | t − s | /α + | x − y | ≍ C | t − s | /α + | θ s,t ( x ) − θ t,s ( y ) | . (2.11) STEPHANE MENOZZI AND XICHENG ZHANG
Put it differently, we have an approximate flow property for the ODE (1.12).Namely, the flow property holds up to an additive time factor which has the samemagnitude as the current typical time (self-similarity index of the driving process).2.2.
Probability estimates.
We need the following master formula.
Lemma 2.4. (L´evy system) Let X t := X ,t be any solution of SDE (1.1) . For anynonnegative measurable function f : R + × R d × R d → R + and finite stopping time τ , E X r ∈ (0 ,τ ] f ( r, X r − , ∆ X r ) = E Z τ Z R d f ( r, X r − , z ) κ ( r, X r − , z ) | z | d + α d z d r, where ∆ X r := X r − X r − and κ ( r, x, z ) is defined in (1.5) .Proof. Let N (d t, d z ) be the counting measure associated with L ( α ) t , i.e., N ((0 , t ] × E ) := X s ∈ [0 ,t ] E (∆ L ( α ) s ) , E ∈ B ( R d ) . Noting that ∆ X t = a ( t, X t − )∆ L ( α ) t , we have for any ε > X r ∈ (0 ,τ ] f ( r, X r − , ∆ X r ) | ∆ X r | > ε = Z τ Z | a ( r,X r − ) z | > ε f ( r, X r − , a ( r, X r − ) z ) N (d r, d z ) . Since the compensated measure of N (d t, d z ) is d z d t | z | d + α , by the change of variable,we have E X r ∈ (0 ,τ ] f ( r, X r − , ∆ X r ) | ∆ X r | > ε = E Z τ Z | a ( r,X r − ) z | > ε f ( r, X r − , a ( r, X r − ) z ) d z d r | z | d + α = E Z τ Z | z | > ε f ( r, X r − , z ) κ ( r, X r − , z ) | z | d + α d z d r, where κ ( r, x, z ) is given in (1.5), which in turn gives the desired formula by themonotone convergence theorem. (cid:3) Fix ( s, x ) ∈ R + × R d . For η >
0, define the stopping time τ ηs,x := inf n t > s : | X s,t ( x ) − θ s,t ( x ) | > η o , (2.12)which corresponds to the exit time of the diffusion from a tube around the de-terministic ODE introduced in (1.12). We now give a tube estimate which roughlysays that, for a given spatial threshold η , the probability that the difference betweenthe process X s, · ( x ) and the deterministic regularized flow θ s, · ( x ) leaves the tube ofradius η before a certain fraction εη α of the corresponding typical time scale η α issomehow small . Lemma 2.5.
Under (H βb ) and (H a ) , for any T > , there is an ε ∈ (0 , onlydepending on T, Θ such that for all ( s, x ) ∈ R + × R d and η ∈ (0 , T /α ] , P ( τ ηs,x < s + εη α ) / . Proof.
Without loss of generality, we assume ( s, x ) = (0 , X t := X ,t (0) , θ t := θ ,t (0) , τ := τ η , . EAT KERNEL OF SUPERCRITICAL SDES WITH UNBOUNDED DRIFTS 9
Let f ∈ C b ( R d ) with f (0) = 0 and f ( x ) = 1 for | x | >
1. For η >
0, set f η ( x ) := f ( x/η ) , u := εη α . Note that Y t := X t − θ t = Z t ( b ( r, X r ) − b r /α ( r, θ r ))d r + Z t a ( r, X r − )d L ( α ) r . By Itˆo’s formula, we have E f η ( Y u ∧ τ ) = E Z u ∧ τ h ( b ( r, X r ) − b r /α ( r, θ r )) · ∇ f η ( Y r ) + L r f η ( Y r ) i d r. Note that | b ( r, X r ) − b r /α ( r, θ r ) | | b ( r, X r ) − b ( r, θ r ) | + | b ( r, θ r ) − b r /α ( r, θ r ) | ( H βb ) , (2.6) κ ( | X r − θ r | β + | X r − θ r | ) + κ r β/α . Also, with the notation of (1.4), | δ (2) f η ( x ; z ) | ( | z | k∇ f η k ∞ ) ∧ (4 k f η k ∞ ) . ( | z | /η ) ∧ . Hence, E f η ( Y u ∧ τ ) . E Z u (cid:20) ( | Y r | β + | Y r | + r β/α ) · | Y r | η η + Z R d ( | z | /η ) ∧ | z | d + α d z (cid:21) d r . Z u ( η β − + 1 + r β/α η − + η − α )d r . u = εη α ε ( η α + β − + η α + ε β/α η α + β − + 1) . ε ( T β − /α + T + 1) , where the last inequality is due to α + β > η T /α . Importantly, theimplicit constant is independent of ε . Note that P ( τ < u ) = E τ K ( t − s ) /α , after a time ε ( t − s ) with ε as in Lemma 2.5,the stochastic forward transport of x by the SDE, i.e. X s,s + ε ( t − s ) ( x ) and thebackward deterministic transport of y by the regularized flow θ t,s + ε ( t − s ) ( y ) belongto a diagonal tube, with radius K ( t − s ) /α , where again ( t − s ) /α corresponds tothe current typical scale between times s and t . Lemma 2.6.
Suppose that (H βb ) and (H a ) hold. Let ε ∈ (0 , be as in Lemma2.5. For any T > , there are constants c ∈ (0 , , K > depending only on T, Θ such that for all s < t T and | x − θ t,s ( y ) | > K ( t − s ) /α , P (cid:16) | X s,s + ε ( t − s ) ( x ) − θ t,s + ε ( t − s ) ( y ) | K ( t − s ) /α (cid:17) > c ( t − s ) d/α | x − θ t,s ( y ) | d + α . This Lemma will be crucial for the lower bound estimate of the heat kernel,since it precisely gives the control needed for a chaining argument, see Theorem3.5 below. As opposed to the continuous case, for SDEs driven by pure jumpprocesses, a single intermediate time, associated with a large jump , is needed forthe chaining. Roughly speaking between times s + ε ( t − s ) and t we will use the global diagonal bound of order ( t − s ) − d/α , since ε is meant to be small enough , and theabove Lemma controls the probability that the process enters a good neigborhoodof the backward flow to do so. The lower bound is the sought one in the sensethat when multiplying it by ( t − s ) − d/α exactly makes the expression in (1.15),( t − s )(( t − s ) /α + | θ s,t ( x ) − y | ) − ( d + α ) ≍ C ( t − s ) | θ s,t ( x ) − y | − ( d + α ) appear since | x − θ s,t ( y ) | > K ( t − s ) /α ( off-diagonal regime ). Proof of Lemma 2.6.
Without loss of generality, we assume s = 0 and for simplicity,we write η := t /α , u := εη α = εt, X r ( x ) := X ,r ( x ) . Define a stopping time σ := inf n r > | X r ( x ) − θ t,r ( y ) | η o . By the right continuity of r X r ( x ) − θ t,r ( y ), one sees that | X σ ( x ) − θ t,σ ( y ) | η, a.s. In particular, for σ u , by (2.11) and (2.8), there is a constant C = C (Θ) > | X u ( x ) − θ t,u ( y ) | | X σ,u ( X σ ( x )) − θ σ,u ( X σ ( x )) | + | θ σ,u ( X σ ( x )) − θ σ,u ( θ t,σ ( y )) | + | θ σ,u ( θ t,σ ( y )) − θ t,u ( y ) | | X σ,u ( X σ ( x )) − θ σ,u ( X σ ( x )) | + C η. Let K > C + 1. Then (cid:8) | X σ,u ( X σ ( x )) − θ σ,u ( X σ ( x )) | < η (cid:9) ⊂ (cid:8) | X u ( x ) − θ t,u ( y ) | Kη (cid:9) . Thus, by the strong Markov property, we have P ( | X u ( x ) − θ t,u ( y ) | Kη ) > P ( σ u ; | X u ( x ) − θ t,u ( y ) | Kη ) > P ( σ u ; | X σ,u ( X σ ( x )) − θ σ,u ( X σ ( x )) | < η ) > P ( σ u ) inf ( s,z ) ∈ [0 ,u ] × R d P ( | X s,u ( z ) − θ s,u ( z ) | < η ) . Let τ ηs,z be defined by (2.12). By Lemma 2.5 we have P ( | X s,u ( z ) − θ s,u ( z ) | > η ) P ( τ ηs,z u ) P ( τ ηs,z s + εη α ) / , (2.13)which implies that inf ( s,z ) ∈ [0 ,u ] × R d P ( | X s,u ( z ) − θ s,u ( z ) | < η ) > / P ( | X u ( x ) − θ t,u ( y ) | Kη ) > P ( σ u ) / . (2.14)Next we need to obtain a lower bound estimate for P ( σ u ). Let τ := τ η ,x . For r < u ∧ τ , by (2.7), there are constants c , C > | X r ( x ) − θ t,r ( y ) | > | θ ,r ( x ) − θ t,r ( y ) | − | X r ( x ) − θ ,r ( x ) | > c | x − θ t, ( y ) | − C t /α − η. In particular, if we choose K > ( C + 1) ∨ (( C + 2) /c ), then since by assumption | x − θ t, ( y ) | > Kη,
EAT KERNEL OF SUPERCRITICAL SDES WITH UNBOUNDED DRIFTS 11 it holds that for r < u ∧ τ , | X r ( x ) − θ t,r ( y ) | > c Kη − ( C + 1) η > η. (2.15)Thus we have {| X u ∧ τ ( x ) − θ t,u ∧ τ ( y ) | η } = X r ∈ (0 ,u ∧ τ ] {| X r ( x ) − θ t,r ( y ) | η } , i.e. we have at most one term in the above summand. We then derive from Lemma2.4, P {| X u ∧ τ ( x ) − θ t,u ∧ τ ( y ) | η } = E X r ∈ (0 ,u ∧ τ ] {| X r ( x ) − θ t,r ( y ) | η } = E Z u ∧ τ Z | z − θ t,r ( y ) | η κ ( r, X r − ( x ) , z − X r − ( x )) | z − X r − ( x ) | d + α d z d r. On the other hand, noting that for r < u ∧ τ and | z − θ t,r ( y ) | η , we get | z − X r − ( x ) | | z − θ t,r ( y ) | + | θ t,r ( y ) − θ ,r ( x ) | + | θ ,r ( x ) − X r − ( x ) | η + C | x − θ t, ( y ) | + C t /α + η C | x − θ t, ( y ) | + C η, using as well (2.7) for the last but one inequality. Since {| X u ∧ τ ( x ) − θ t,u ∧ τ ( y ) | η } ⊂ { σ u } , we further have P { σ u } > P {| X u ∧ τ ( x ) − θ t,u ∧ τ ( y ) | η } > E Z u ∧ τ Z | z − θ t,r ( y ) | η κ − ( C | x − θ t, ( y ) | + C η ) d + α d z d r = E ( u ∧ τ ) κ − η d · Vol( B )( C | x − θ t, ( y ) | + C η ) d + α > c t d/α | x − θ t, ( y ) | d + α , (2.16)where the last step is due to | x − θ t, ( y ) | > Kη and E ( u ∧ τ ) > u P ( τ > u ) (2.13) > u/ εt/ . Combining (2.14) and (2.16), we obtain the desired estimate. (cid:3)
Convolution inequalities.
This Section is dedicated to some useful convo-lution controls associated with functions that are known to be upper-bounds of theisotropic stable density and its gradient, see e.g. [20], [4]. Though a bit technical,these results will turn out to be crucial in order to control the parametrix seriesrepresentation of the density and its gradient (see e.g. equation (3.10), Lemma 3.3and Theorem 3.5 below).For η ∈ (0 ,
2) and ( t, x ) ∈ R + × R d , let ̺ ( η ) ( x ) := (1 + | x | ) − d − η , ̺ ( η ) ( t, x ) := t − d/α ̺ ( η ) ( t − /α x ) . For β > γ ∈ R , we introduce the following functions for later use ̺ ( η ) β,γ ( t, x ) := (1 ∧ ( t /α + | x | )) β t ( γ − η ) /α ̺ ( η ) ( t, x ) (2.17)and φ ( η ) β,γ ( s, x, t, y ) := ̺ ( η ) β,γ ( t − s, x − θ t,s ( y )) . (2.18)Note that ̺ ( η ) β,γ ( t, x ) = (1 ∧ ( t /α + | x | )) β t γ/α ( t /α + | x | ) d + η . For
T >
0, by (2.7) we have for ( s, x, t, y ) ∈ D T , φ ( η ) β,γ ( s, x, t, y ) ≍ ̺ ( η ) β,γ ( t − s, θ s,t ( x ) − y ) , (2.19)and for β ∈ [0 , η ], Z R d φ ( η ) β,γ ( s, x, t, y )d y . Z R d ̺ ( η ) β,γ ( t − s, y )d y . ( t − s ) β + γ − ηα . (2.20)For two functions f, g on D ∞ , we write( f ⊙ g ) r ( s, x, t, y ) := Z R d f ( s, x, r, z ) g ( r, z, t, y )d z and ( f ⊗ g )( s, x, t, y ) := Z ts ( f ⊙ g ) r ( s, x, t, y )d r. The following lemma is the same as in [8, Lemma 2.1].
Lemma 2.7.
Fix α ∈ (0 , . For any β , β ∈ [0 , α ] and T > , there is a C = C ( T, Θ , β , β ) > such that for all γ > − β and γ > − β , r ∈ [ s, t ] and x, y ∈ R d , (cid:0) φ ( α ) β , ⊙ φ ( α ) β , (cid:1) r ( s, x, t, y ) . C (cid:0) ( r − s ) β − αα + ( t − r ) β − αα (cid:1) φ ( α ) β ∧ β , ( s, x, t, y )(2.21) and φ ( α ) β ,γ ⊗ φ ( α ) β ,γ ( s, x, t, y ) . C B ( β + γ α , β + γ α ) φ ( α ) β ∧ β ,β + β + γ + γ ( s, x, t, y ) , (2.22) where B ( γ, β ) is the usual Beta function defined by B ( γ, β ) := Z (1 − s ) γ − s β − d s, γ, β > . Proof.
We follow the proof in [11]. Let ℓ ( u ) := u d + α ∧ u β . It is easy to see that, as soonas d + α > β , ℓ is increasing on R + and for any λ > ℓ ( λu ) λ d + α ℓ ( u ) . (2.23)Hence, ℓ ( u + w ) ℓ (2( u ∨ w )) d + α ℓ ( u ∨ w ) d + α ( ℓ ( u ) + ℓ ( w )) . (2.24)Now for r ∈ [ s, t ] and x, y ∈ R d , since | t + s | /α + | x + y | /α (cid:0) | s | /α + | x | + | t | /α + | y | (cid:1) , by (2.23) and (2.24), we have ℓ ( | t + s | /α + | x + y | ) . C ℓ ( | s | /α + | x | ) + ℓ ( | t | /α + | y | ) . In particular,(( t + s ) /α + | x + y | ) d + α ∧ (( t + s ) /α + | x + y | ) β ∧ β . ( s /α + | x | ) d + α ∧ ( s /α + | x | ) β + ( t /α + | y | ) d + α ∧ ( t /α + | y | ) β . Hence, 1 ∧ ( s /α + | x | ) β ( s /α + | x | ) d + α × ∧ ( t /α + | y | ) β ( t /α + | y | ) d + α . C (cid:20) ∧ ( s /α + | x | ) β ( s /α + | x | ) d + α + 1 ∧ ( t /α + | y | ) β ( t /α + | y | ) d + α (cid:21) EAT KERNEL OF SUPERCRITICAL SDES WITH UNBOUNDED DRIFTS 13 × ∧ (( t + s ) /α + | x + y | ) β ∧ β (( t + s ) /α + | x + y | ) d + α . By this, the desired estimates follow by (2.19), (2.20) and (2.8). (cid:3)
Density estimate.
Let a : R + → R d ⊗ R d be a measurable d × d -matrix-valuedfunction satisfying the non-degeneracy condition κ − | ξ | | a ( s ) ξ | κ | ξ | . (2.25)Fix α ∈ (0 ,
2) and consider the following jump process X as,t := Z ts a ( r )d W S r , (2.26)where W is a d -dimensional Brownian motion and S is an α/ W , both defined on some probability space (Ω , F , P ). Note that X as,t ( d ) = ( t − s ) /α X ˜ a , , where ˜ a ( r ) := a ( s + r ( t − s )) . We have the following lemma that can be derived from the approach initiallyused in [2] (see also [4]). We provide below a proof for completeness.
Lemma 2.8.
For any s < t < ∞ , X as,t has a smooth density p as,t ( x ) with thescaling property p as,t ( x ) = ( t − s ) − d/α p ˜ a , (( t − s ) − /α x ) , (2.27) which enjoys the following estimates: p as,t ( x ) ≍ C ̺ ( α )0 ,α ( t − s, x ) , (2.28) and for any j ∈ N , |∇ j p as,t ( x ) | . C j ̺ ( α + j )0 ,α ( t − s, x ) , (2.29) where the constants only depend on κ , d, α . Moreover, suppose that the integrandin (2.26) writes as a ξ ( r ) and smoothly depends on the parameter ξ ∈ R d so that (2.25) holds uniformly and sup r,ξ |∇ kξ a ξ ( r ) | < ∞ for any k ∈ N . Let p a ξ s,t be thedensity of the integral in (2.26) associated with a ξ . Then we have for k ∈ N and j ∈ N , |∇ kξ ∇ jx p a ξ s,t ( x ) | . C j,k ̺ ( α + j )0 ,α ( t − s, x ) . (2.30) Importantly, this last bound means that, the differentiation w.r.t. the parameter ξ appearing in the diffusion coefficient a ξ does not yield an additional time singularity.Proof. The two sided estimate (2.28) is well known (see e.g. [9]). We show (2.29).Without loss of generality, we assume s = 0 and write X t := Z t a ( r )d W S r . Fix a c`adl`ag path ℓ s . Consider the following Gaussian random variable: X ℓt := Z t a ( r )d W ℓ r . It has a density g a,ℓt ( x ) = (2 π ) − d/ q det (cid:0) ( C a,ℓt ) − (cid:1) exp {−h (cid:0) C a,ℓt (cid:1) − x, x i / } , (2.31) where C a,ℓt := Z t ( aa ∗ )( r )d ℓ r . From the non-degeneracy assumption (2.25), we have h (cid:0) C a,ℓt (cid:1) − x, x i ≍ | x | /ℓ t , det (cid:0) ( C a,ℓt ) − (cid:1) ≍ ℓ − dt , and |∇ g a,ℓt ( x ) | . | x | /ℓ t exp {− λ | x | /ℓ t } . The density p a ,t ( x ) =: p at ( x ) of X t is given by p at ( x ) = E g a,St ( x ) . (2.32)The bound of (2.29) is direct from the Fourier representation of the density when | x | t /α . On the other hand, for | x | > t /α , from the global bound on the law ofthe subordinator µ S t (d r ) := P ◦ S − t (d r ) . t r − α/ − d r, it readily follows that |∇ p at ( x ) | E |∇ g a,St ( x ) | . | x | E ( S − d/ − t exp {− λ | x | /S t } ) < + ∞ . Hence, from the bounded convergence theorem it holds that |∇ p at ( x ) | . | x | Z ∞ r − ( d +2) / e − λ | x | /r µ S t (d r ) , and the integral expression in the r.h.s. precisely corresponds to the stable heatkernel in dimension d + 2 at time t and point ˜ x ∈ R d +2 s.t. | ˜ x | = √ λ | x | . Thus,from (2.28), |∇ p at ( x ) | . | x | t − ( d +2) /α t − /α | ˜ x | ) d +2+ α . t ( t /α + | x | ) − d − α − = ̺ ( α +1)0 ,α ( t, x ) . The approach is similar for higher order derivatives. This is also the case for (2.30)recalling that differentiating a Gaussian density w.r.t. the variance does not induceadditional singularities. The proof is complete. (cid:3)
Remark 2.9.
We would like to emphasize that the gradient estimate (2.29) playsa crucial role for two-sided estimates due to the fact that for any β ∈ [0 , | x | β ̺ ( α +1)0 ,α ( t, x ) = t | x | β ( t /α + | x | ) d + α +1 t ( α + β − /α ( t /α + | x | ) d + α = ̺ ( α )0 ,β + α − ( t, x ) . In particular, for any β ∈ [0 , | x − θ t,s ( x ) | β φ ( α +1)0 ,α ( s, x, t, y ) φ ( α )0 ,β + α − ( s, x, t, y ) . (2.33)We carefully point out that the Gradient estimate (2.29), which remarkably em-phasizes a concentration gain, does not hold for a general α -stable like process [15].This is also why, for the driving process in (1.1), we limit ourselves to the rota-tionally invariant, and thus symmetric, α -stable process and do not handle general α -stable like processes.The following lemma is taken from [9, Lemmas 3.2 and 3.3 ]. EAT KERNEL OF SUPERCRITICAL SDES WITH UNBOUNDED DRIFTS 15
Lemma 2.10.
Under (2.25) , there is a constant C = C ( d, α, κ ) > such that |∇ p as,t − ∇ p ¯ as,t | ( x ) . C k a − ¯ a k ∞ ̺ ( α +1)0 ,α ( t − s, x ) . (2.34) Also, |D ( α ) p as,t | ( x ) . C ̺ ( α )0 , ( t − s, x ) , (2.35) and |D ( α ) ( p as,t − p ¯ as,t ) | ( x ) . C k a − ¯ a k ∞ ̺ ( α )0 , ( t − s, x ) . (2.36) Moreover, we also have Z R d | δ (2) p as,t ( x ; z ) − δ (2) p as,t ( x ; z ) | d z | z | d + α . C (cid:18) | x − x | ( t − s ) /α ∧ (cid:19) X i =1 , ̺ ( α )0 , ( t − s, x i ) ! . (2.37) Proof.
From the scaling property (2.27), it suffices to consider s = 0 and t = 1.Note that | δ (2) p a ( x ; z ) | = | p a ( x + z ) + p a ( x − z ) − p a ( x ) | . ( | z | ∧ ̺ ( α ) ( x + z ) + ̺ ( α ) ( x − z ) + ̺ ( α ) ( x )) . By elementary calculations, one sees that Z R d ̺ ( α ) ( x + z ) ( | z | ∧ z | z | d + α . C ̺ ( α ) ( x ) . (2.38)Thus (2.35) follows. As for (2.36) and (2.37), they can be derived similarly to [11,Lemma 2.7 and Lemma 2.8]. The statement (2.34) can also be derived from thearguments developed therein. We omit the details. (cid:3) Heat kernel of nonlocal operators with smooth coefficients
In this section we assume that (H βb ) and (H γa ) hold and additionally that forany j ∈ N , k∇ j b k ∞ + k∇ j a k ∞ < ∞ . (3.1)We shall denote C := (cid:8) ( b, a ) : satisfying (H βb ) , (H γa ) with common bounds κ , κ and (3.1). (cid:9) Under (H γa ) and (3.1), for each ( s, x ) ∈ R + × R d , it is well known that there isa unique solution ( X s,t ( x )) t > s to SDE (1.1), and X s,t ( x ) has for t > s a density p ( s, x, t, y ) so that (cf. [13, 7]) P s,t f ( x ) := E f ( X s,t ( x )) = Z R d f ( y ) p ( s, x, t, y )d y, f ∈ L ∞ ( R d ) . The density is also a mild solution of the Kolmogorov equation in the sense thatfor all ϕ ∈ C ( R d ) P s,t ϕ ( x ) = ϕ ( x ) + Z ts P s,r L r ϕ ( x )d r. (3.2)Fix ( τ, ξ ) ∈ [ s, t ] × R d . Consider the following freezing process X ( τ,ξ ) s,t := x + Z ts b | r − τ | /α ( r, θ τ,r ( ξ ))d r + Z ts a ( r, θ τ,r ( ξ ))d L ( α ) r . By Lemma 2.8, the density of X ( τ,ξ ) s,t is given by˜ p ( τ,ξ ) ( s, x, t, y ) = p a ( τ,ξ ) s,t (cid:18) x − y + Z ts b | r − τ | /α ( r, θ τ,r ( ξ ))d r (cid:19) , (3.3)where a ( τ,ξ ) ( r ) := a ( r, θ τ,r ( ξ )) and p a ( τ,ξ ) s,t is the density of R ts a ( τ,ξ ) ( r )d L ( α ) r given inLemma 2.8. In particular, ∂ s ˜ p ( τ,ξ ) ( s, x, t, y ) + ˜ L ( τ,ξ ) s ˜ p ( τ,ξ ) ( s, · , t, y )( x ) = 0 , (3.4)where ˜ L ( τ,ξ ) s f ( x ) := ˜ L ( τ,ξ ) s f ( x ) + b | s − τ | /α ( s, θ τ,s ( ξ )) · ∇ f ( x )and ˜ L ( τ,ξ ) s f ( x ) = Z R d δ (2) f ( x ; z ) κ ( s, θ τ,s ( ξ ) , z ) | z | d + α d z with κ ( s, θ s,τ ( ξ ) , z ) := det( a − ( s, θ τ,s ( ξ )) | z | d + α | a − ( s, θ τ,s ( ξ )) z | d + α . For simplicity, we shall write A ( τ,ξ ) s f ( x ) := ( L s − ˜ L ( τ,ξ ) s ) f ( x ) = K ( τ,ξ ) s f ( x ) + B ( τ,ξ ) s f ( x ) , (3.5)where K ( τ,ξ ) s f ( x ) := ( L s − ˜ L ( τ,ξ ) s ) f ( x ) , and B ( τ,ξ ) s f ( x ) := (cid:0) b ( s, x ) − b | s − τ | /α ( s, θ τ,s ( ξ )) (cid:1) · ∇ f ( x ) . Let us introduce the corresponding frozen semi-group: e P ( τ,ξ ) s,t f ( x ) := E f ( X ( τ,ξ ) s,t ( x )) . (3.6)We have the following Duhamel type representation formula: Lemma 3.1.
For any f ∈ C ∞ b ( R d ) and ( τ, ξ ) ∈ [ s, t ] × R d , it holds that P s,t f = e P ( τ,ξ ) s,t f + Z ts P s,r A ( τ,ξ ) r e P ( τ,ξ ) r,t f d r = e P ( τ,ξ ) s,t f + Z ts e P ( τ,ξ ) s,r A ( τ,ξ ) r P r,t f d r. Proof.
We drop for the proof the superscript ( τ, ξ ) for notational simplicity. Sincefrom (3.2) and (3.4), ∂ t P s,t f = P s,t L t f, ∂ s e P s,t f = − f L s e P s,t f, by the chain rule, we have ∂ r ( P s,r e P r,t f ) = P s,r L r e P r,t f − P s,r f L r e P r,t f = P s,r A r e P r,t f. Integrating both sides from s to t with respect to r yields P s,t f = e P s,t f + Z ts P s,r A r e P r,t f d r. Similarly, one can show that e P s,t f = P s,t f − Z ts e P s,r A r P r,t f d r. The proof is complete. (cid:3)
EAT KERNEL OF SUPERCRITICAL SDES WITH UNBOUNDED DRIFTS 17
By Lemma 3.1, we have for each ( τ, ξ ) ∈ [ s, t ] × R d and x, y ∈ R d , p ( s, x, t, y ) = e p ( τ,ξ ) ( s, x, t, y ) + Z ts Z R d p ( s, x, r, z ) A ( τ,ξ ) r e p ( τ,ξ ) ( r, · , t, y )( z )d z d r. In particular, if we take ( τ, ξ ) = ( t, y ) and define p ( s, x, t, y ) := ˜ p ( t,y ) ( s, x, t, y ) = p a ( t,y ) s,t ( x − θ t,s ( y )) , (3.7)then we obtain the forward representation, p ( s, x, t, y ) = p ( s, x, t, y ) + Z ts Z R d p ( s, x, r, z ) A ( t,y ) r p ( r, · , t, y )( z )d z d r. (3.8)Let q ( s, x, t, y ) := A ( t,y ) s p ( s, · , t, y )( x ) , and define recursively for n > q n := q ⊗ q n − , q = ∞ X n =0 q n . (3.9)By iteration, we formally obtain from (3.8) and (3.9), p = p + p ⊗ q = p + ∞ X n =0 p ⊗ q n = p + p ⊗ q. (3.10)The following lemma is a direct consequence of (3.7), (2.35) and (2.29). Lemma 3.2.
For any α ∈ (0 , and j = 0 , , · · · , we have |∇ j p ( s, · , t, y ) | ( x ) . φ ( α + j )0 ,α ( s, x, t, y ) (3.11) and |D ( α ) p ( s, · , t, y ) | ( x ) . φ ( α )0 , ( s, x, t, y ) . (3.12)The following lemma corresponds to [8, Theorem 3.1]. Lemma 3.3.
The series q = P ∞ n =0 q n is absolutely convergent, and for each s < t , ( x, y ) q ( s, x, t, y ) is equi-continuous in ( b, a ) ∈ C . Moreover, for any T > ,there is a constant C = C ( T, Θ) > such that for all ( s, x, t, y ) ∈ D T , | q ( s, x, t, y ) | . C (cid:0) φ ( α ) γ , + φ ( α )0 ,γ (cid:1) ( s, x, t, y ) , (3.13) where γ := ( α + β − ∧ γ , and for any γ ∈ (0 , γ ) , | q ( s, x, t, y ) − q ( s, x ′ , t, y ) | . C ( | x − x ′ | γ ∧ × (cid:16)(cid:0) φ ( α ) γ , − γ + φ ( α )0 ,γ − γ (cid:1) ( s, x, t, y ) + (cid:0) φ ( α ) γ , − γ + φ ( α )0 ,γ − γ (cid:1) ( s, x ′ , t, y ) (cid:17) . (3.14) Proof. (i) First of all, note that by (1.6), | κ ( s, x, z ) − κ ( s, θ t,s ( y ) , z ) | . ( | x − θ t,s ( y ) | γ ∧ H βb ), | b ( s, x ) − b | s − t | /α ( s, θ t,s ( y )) | . | x − θ t,s ( y ) | β + | x − θ t,s ( y ) | + | t − s | β/α . Thus, we have by (3.12), | K ( t,y ) s p ( s, · , t, y )( x ) | . φ ( α ) γ, ( s, x, t, y ) , and by (3.11) and (2.33), | B ( t,y ) s p ( s, · , t, y )( x ) | . φ ( α )0 ,α + β − ( s, x, t, y ) . So, for γ = γ ∧ ( α + β − | q ( s, x, t, y ) | . (cid:16) φ ( α ) γ, + φ ( α )0 ,α + β − (cid:17) ( s, x, t, y ) . (cid:16) φ ( α ) γ , + φ ( α )0 ,γ (cid:17) ( s, x, t, y ) . Suppose now that for some k ∈ N , | q k − ( s, x, t, y ) | C k (cid:16) φ ( α ) γ , ( k − γ + φ ( α )0 ,kγ (cid:17) ( s, x, t, y ) . By Lemma 2.7, we have | q k ( s, x, t, y ) | CC k (cid:0) φ ( α ) γ , + φ ( α )0 ,γ (cid:1) ⊗ (cid:0) φ ( α ) γ , ( k − γ + φ ( α )0 ,kγ (cid:1) ( s, x, t, y ) C C k B ( γ α , kγ α ) (cid:0) φ ( α ) γ ,kγ + φ ( α )0 , ( k +1) γ (cid:1) ( s, x, t, y ) . (3.15)Hence, C k +1 = C C k B ( γ α , kγ α ) . From the relation B ( γ, β ) = Γ( γ )Γ( β )Γ( γ + β ) , where Γ is the usual Gamma function, weobtain C k = C k k − Y i =1 B ( γ α , ( k − γ α ) = ( C Γ( γ /α )) k Γ( kγ /α ) , with the usual convention that Q i =1 = 1. Thus ∞ X k =0 | q k ( s, x, t, y ) | ∞ X k =0 ( C Γ( γ /α )) k Γ( kγ /α ) (cid:16) φ ( α ) γ ,kγ + φ ( α )0 , ( k +1) γ (cid:17) ( s, x, t, y ) ∞ X k =0 ( C Γ( γ /α )) k Γ( kγ /α ) (cid:16) φ ( α ) γ , + φ ( α )0 ,γ (cid:17) ( s, x, t, y ) . This gives (3.13).(ii) For fixed s < t , by Lemma 2.8 and the definition of q , one sees that ( x, y ) q ( s, x, t, y ) is equi-continuous in ( b, a ) ∈ C . Furthermore, it follows by inductionthat, for each k ∈ N , ( x, y ) q k ( s, x, t, y ) is also equi-continuous in ( b, a ) ∈ C .Hence, ( x, y ) q ( s, x, t, y ) is equi-continuous in ( b, a ) ∈ C .(iii) If | x − x ′ | > ( t − s ) /α , then we have | q ( s, x, t, y ) | . ( | x − x ′ | γ ∧ t − s ) − γ /α (cid:0) φ ( α ) γ , + φ ( α )0 ,γ (cid:1) ( s, x, t, y )= ( | x − x ′ | γ ∧ (cid:0) φ ( α ) γ , − γ + φ ( α )0 ,γ − γ (cid:1) ( s, x, t, y ) . Next we assume | x − x ′ | ( t − s ) /α . In this case, it is easy to see from (2.17)-(2.18),that φ ( η )0 , ( s, x, t, y ) ≍ φ ( η )0 , ( s, x ′ , t, y ) , η > . (3.16)By (2.35), (2.37) and (3.16), we have | K ( t,y ) s p ( s, · , t, y )( x ) − K ( t,y ) s p ( s, · , t, y )( x ′ ) | k κ ( · , x, · ) − κ ( · , x ′ , · ) k ∞ Z R d | δ (2) p ( s, · ,t,y ) ( x ; z ) | d z | z | d + α + k κ ( · , x, · ) − κ ( · , θ t,s ( y ) , · ) k ∞ × Z R d | δ (2) p ( s, · ,t,y ) ( x ; z ) − δ (2) p ( s, · ,t,y ) ( x ′ ; z ) | d z | z | d + α ( | x − x ′ | γ ∧ φ ( α )0 , ( s, x, t, y ) + ( | x − θ t,s ( y ) | γ ∧ × (cid:18) | x − x ′ | ( t − s ) /α ∧ (cid:19) (cid:16) φ ( α )0 , ( s, x, t, y ) + φ ( α )0 , ( s, x ′ , t, y ) (cid:17) EAT KERNEL OF SUPERCRITICAL SDES WITH UNBOUNDED DRIFTS 19 . ( | x − x ′ | γ ∧ (cid:16) φ ( α )0 ,γ − γ ( s, x, t, y ) + φ ( α ) γ, − γ ( s, x, t, y ) (cid:17) . Moreover, by (3.11), (3.16) and (2.33), we also have | B ( t,y ) s p ( s, · , t, y )( x ) − B ( t,y ) s p ( s, · , t, y )( x ′ ) | | b ( s, x ) − b ( s, x ′ ) | · |∇ p ( s, · , t, y ) | ( x ′ )+ (cid:12)(cid:12) b ( s, x ) − b | s − t | /α ( s, θ t,s ( y )) (cid:12)(cid:12) · |∇ p ( s, · , t, y )( x ′ ) − ∇ p ( s, · , t, y )( x ) | . | x − x ′ | β φ ( α +1)0 ,α ( s, x, t, y ) + ( | x − θ t,s ( y ) | β + | t − s | β/α ) | x − x ′ | φ ( α +2)0 ,α ( s, x, t, y ) . ( | x − x ′ | γ ∧ φ ( α )0 ,α + β − − γ ( s, x, t, y ) . Combining the above calculations and recalling γ = γ ∧ ( α + β − | q ( s, x, t, y ) − q ( s, x ′ , t, y ) | . C ( | x − x ′ | γ ∧ × (cid:16)(cid:0) φ ( α ) γ , − γ + φ ( α )0 ,γ − γ (cid:1) ( s, x, t, y ) + (cid:0) φ ( α ) γ , − γ + φ ( α )0 ,γ − γ (cid:1) ( s, x ′ , t, y ) (cid:17) . Using this last estimate, equation (3.14) follows from the same iterative argumentas in (i). (cid:3)
Remark 3.4.
This lemma allows to iterate the representation (3.8) which leads tothe representation (3.10) of the density.We now aim at proving the following a priori estimate about p ( s, x, t, y ). Theorem 3.5.
Under (H γa ) , (H βb ) and (3.1) , for each s < t < ∞ , X s,t ( x ) admits a density p ( s, x, t, y ) that is equi-continuous in ( b, a ) ∈ C as a function of x, y ∈ R d , and there is a constant C = C ( T, Θ) > so that for all ( s, x, t, y ) ∈ D T , p ( s, x, t, y ) ≍ C φ ( α )0 ,α ( s, x, t, y ) . (3.17) Proof.
Note that by (3.7), (2.28) and (2.7), p ( s, x, t, y ) ≍ C φ ( α )0 ,α ( s, x, t, y ) . By Lemma 2.7, we have | p ⊗ q | ( s, x, t, y ) . C ( φ ( α )0 ,α + γ + φ ( α ) γ ,α )( s, x, t, y ) . The upper bound follows from (3.10).Next we use Lemma 2.6 to show the lower bound estimate. Let K be as inLemma 2.6. Suppose that | x − θ t,s ( y ) | K ( t − s ) /α . Then we have p ( s, x, t, y ) > p ( s, x, t, y ) − | p ⊗ q ( s, x, t, y ) | > c φ ( α )0 ,α ( s, x, t, y ) − ( φ ( α )0 ,α + γ + φ ( α ) γ ,α )( s, x, t, y ) > ( c − C ( t − s ) γ α ) φ ( α )0 ,α ( s, x, t, y ) . In particular, if t − s ℓ with ℓ small enough and | x − θ t,s ( y ) | K ( t − s ) /α , then p ( s, x, t, y ) > c φ ( α )0 ,α ( s, x, t, y ) > c ( t − s ) − d/α . (3.18)Next we prove the above estimate still holds for | x − θ t,s ( y ) | > K ( t − s ) /α . Let ε ∈ (0 , /
2) be as in Lemma 2.6 and small enough so that 2(1 − ε ) /α >
1. Let r := s + ε ( t − s ) , B := { z : | z − θ t,r ( y ) | K ( t − r ) /α } . Since 2(1 − ε ) /α >
1, we clearly have B ⊃ { z : | z − θ t,r ( y ) | K ( t − s ) /α } =: B ′ . Now from the Chapman-Kolmogorov equation, we have for t − s ℓ , p ( s, x, t, y ) = Z R d p ( s, x, r, z ) p ( r, z, t, y )d z > Z B p ( s, x, r, z ) p ( r, z, t, y )d z > inf z ∈ B p ( r, z, t, y ) Z B p ( s, x, r, z )d z (3.18) > c ( t − r ) − d/α P ( X s,r ( x ) ∈ B ) > c ( t − s ) − d/α P ( X s,r ( x ) ∈ B ′ ) > c ( t − s ) | x − θ t,s ( y ) | − d − α , where the last step is due to Lemma 2.6. Thus we obtain that for some c > s, t ∈ [0 , T ], p ( s, x, t, y ) > c φ ( α )0 ,α ( s, x, t, y ) , t − s ℓ, x, y ∈ R d . For t − s > ℓ , the bound follows iteratively from the Chapman-Kolmogorov equation.The proof is complete. (cid:3) For the fractional derivative estimates, we need the following lemma.
Lemma 3.6.
For s < t , let h s,t ( x ) := R R d p ( s, x, t, y )d y . We have for some C > , |D ( α ) h s,t | ( x ) . C ( t − s ) γ /α − , γ := γ ∧ ( α + β − . Proof.
By definition we have |D ( α ) h s,t | ( x ) = Z R d (cid:12)(cid:12)(cid:12)(cid:12)Z R d δ (2)˜ p ( t,y ) ( s, · ,t,y ) ( x ; z )d y (cid:12)(cid:12)(cid:12)(cid:12) d z | z | d + α = Z R d (cid:12)(cid:12)(cid:12)(cid:12)Z R d δ (2) p a ( t,y ) s,t ( x − θ t,s ( y ); z )d y (cid:12)(cid:12)(cid:12)(cid:12) d z | z | d + α Z R d (cid:12)(cid:12)(cid:12)(cid:12)Z R d δ (2) p a ( t,y ) s,t − p a ( s,x ) s,t ( x − θ t,s ( y ); z )d y (cid:12)(cid:12)(cid:12)(cid:12) d z | z | d + α + Z R d (cid:12)(cid:12)(cid:12)(cid:12)Z R d δ (2) p a ( s,x ) s,t ( x − θ t,s ( y ); z )d y (cid:12)(cid:12)(cid:12)(cid:12) d z | z | d + α =: I + I . For I , noting that by ( H γa ) and Lemma 2.2, | a ( r, θ s,r ( x )) − a ( r, θ t,r ( y )) | . ∧ | x − θ t,s ( y ) | γ + | t − s | γ/α , we have I Z R d |D ( α ) ( p a ( t,y ) s,t − p a ( s,x ) s,t ) | ( x − θ t,s ( y ))d y (2.36) . Z R d (cid:16) φ ( α ) γ, + φ ( α )0 ,γ (cid:17) ( s, x, t, y )d y (2.20) . ( t − s ) γ/α − . For I , by the change of variable we have I = Z R d (cid:12)(cid:12)(cid:12)(cid:12)Z R d δ (2) p a ( s,x ) s,t ( x − y ; z ) det( ∇ θ − s,t ( y ))d y (cid:12)(cid:12)(cid:12)(cid:12) d z | z | d + α = Z R d (cid:12)(cid:12)(cid:12)(cid:12)Z R d δ (2) p a ( s,x ) s,t ( x − y ; z ) (cid:16) det( ∇ θ − s,t ( y )) − (cid:17) d y (cid:12)(cid:12)(cid:12)(cid:12) d z | z | d + α , EAT KERNEL OF SUPERCRITICAL SDES WITH UNBOUNDED DRIFTS 21 where we have used that Z R d p a ( s,x ) s,t ( x − y )d y = 1 ⇒ Z R d δ (2) p a ( s,x ) s,t ( x − y )d y = 0 . Thus by (i) of Lemma 2.2 and (2.35), we have I . ( t − s ) β + α − α Z R d |D ( α ) p a ( s,x ) s,t | ( x − y )d y . ( t − s ) β + α − α Z R d ̺ ( α )0 , ( t − s, x − y )d y . ( t − s ) β − α = ( t − s ) − α + β − α . The proof is complete. (cid:3)
Lemma 3.7. (Fractional derivative estimate) For any
T > , we have for some C = C ( T, Θ) > , |D ( α ) p ( s, · , t, y ) | ( x ) . C φ ( α )0 , ( s, x, t, y ) . Proof.
Let u = ( s + t ) /
2. By (3.10) and the definition of δ (2) , we have δ (2) p ( s, · ,t,y ) ( x ; ¯ z ) = δ (2) p ( s, · ,t,y ) ( x ; ¯ z ) + Z ts Z R d δ (2) p ( s, · ,r,z ) ( x ; ¯ z ) q ( r, z, t, y )d z d r = δ (2) p ( s, · ,t,y ) ( x ; ¯ z ) + Z us (cid:18)Z R d δ (2) p ( s, · ,r,z ) ( x ; ¯ z )d z (cid:19) q ( r, θ s,r ( x ) , t, y )d r + Z us Z R d δ (2) p ( s, · ,r,z ) ( x ; ¯ z )( q ( r, z, t, y ) − q ( r, θ s,r ( x ) , t, y ))d z d r + Z tu Z R d δ (2) p ( s, · ,r,z ) ( x ; ¯ z ) q ( r, z, t, y )d z d r. With the notations of Lemma 3.6, set h s,r ( x ) = R R d p ( s, x, r, z )d z . By (1.14) andthe Fubini theorem, we have |D ( α ) p ( s, · , t, y ) | ( x ) |D ( α ) p ( s, · , t, y ) | ( x ) + Z us |D ( α ) h s,r | ( x ) | q ( r, θ s,r ( x ) , t, y ) | d r + Z us Z R d |D ( α ) p ( s, · , r, z ) | ( x ) | q ( r, z, t, y ) − q ( r, θ s,r ( x ) , t, y ) | d z d r + Z tu Z R d |D ( α ) p ( s, · , r, z ) | ( x ) | q ( r, z, t, y ) | d z d r =: I ( x ) + I ( x ) + I ( x ) + I ( x ) . For I , by (3.12) we have I ( x ) . φ ( α )0 , ( s, x, t, y ) . Recall γ =( α + β − ∧ γ, γ ∈ (0 , γ ) . For I , by Lemma 3.6, (3.13), (2.19) and (2.8), we have I ( x ) . Z us ( r − s ) γ α − (cid:0) φ ( α ) γ , + φ ( α )0 ,γ (cid:1) ( r, θ s,r ( x ) , t, y )d r . (cid:18)Z us ( r − s ) γ α − d r (cid:19) (cid:0) φ ( α ) γ , + φ ( α )0 ,γ (cid:1) ( s, x, t, y ) . (cid:0) φ ( α ) γ ,γ + φ ( α )0 , γ (cid:1) ( s, x, t, y ) . For I , by (3.12), (3.14) and (2.22), we have I ( x ) . Z us Z R d φ ( α ) γ , ( s, x, r, z ) (cid:0) φ ( α ) γ , − γ + φ ( α )0 ,γ − γ (cid:1) ( r, z, t, y )d z d r + Z us Z R d φ ( α ) γ , ( s, x, r, z )d z (cid:0) φ ( α ) γ , − γ + φ ( α )0 ,γ − γ (cid:1) ( s, x, t, y )d r . (cid:0) φ ( α ) γ , + φ ( α )0 ,γ (cid:1) ( s, x, t, y ) . For I , by (3.12), (3.13) and (2.21), we have I ( x ) . Z tu (cid:0) φ ( α )0 , ⊙ ( φ ( α ) γ , + φ ( α )0 ,γ ) (cid:1) r ( s, x, t, y )d r . φ ( α )0 , ( s, x, t, y ) . Combining the above estimates, we complete the proof. (cid:3) A priori gradient estimates
The aim of this section is to show the following a priori gradient estimate.
Theorem 4.1.
Under (H βb ) , (H γa ) and (3.1) , for any T > , there is a constant C = C ( T, Θ) > such that for all f ∈ B b ( R d ) , s < t T and x ∈ R d , |∇ P s,t f ( x ) | . C ( t − s ) − /α P s,t | f | ( x ) . (4.1) Moreover, x
7→ ∇ P s,t f ( x ) is equi-continuous in ( b, a ) ∈ C . We shall prove this theorem for α ∈ [1 ,
2) and α ∈ (0 ,
1) separately by differentmethods.4.1.
Critical and Subcritical cases: α ∈ [1 , . In this subsection we start fromthe series expansion (3.10) for the density to derive the estimate |∇ x p ( s, x, t, y ) | . C φ ( α )0 ,α − ( s, x, t, y ) , (4.2)when (H βb ) , (H γa ) and (3.1) are in force and α ∈ [1 , p ( s, x, t, y ) = p ( s, x, t, y ) + ( p ⊗ q )( s, x, t, y ) . Therefore, for u = s + t and ξ = θ s,r ( x ), ∇ x p ( s, x, t, y ) = ∇ x p ( s, x, t, y ) + Z tu Z R d ∇ x p ( s, x, r, z ) q ( r, z, t, y )d z d r + Z us Z R d ( ∇ x p − ∇ x ˜ p ( r,ξ ) )( s, x, r, z ) q ( r, z, t, y )d z d r + Z us Z R d ∇ x ˜ p ( r,ξ ) ( s, x, r, z )( q ( r, z, t, y ) − q ( r, ξ, t, y ))d z d r =: G ( s, x, t, y ) + G ( s, x, t, y ) + G ( s, x, t, y ) + G ( s, x, t, y ) , where for the last term, we have used precisely the cancellation property Z R d ∇ x ˜ p ( r,ξ ) ( s, x, r, z )d z = 0 . For G , by (3.11) we clearly have | G ( s, x, t, y ) | . φ ( α +1)0 ,α ( s, x, t, y ) φ ( α )0 ,α − ( s, x, t, y ) , EAT KERNEL OF SUPERCRITICAL SDES WITH UNBOUNDED DRIFTS 23 using Remark 2.9, equation (2.33), for the last inequality. For G , by (3.11), (3.13)and (2.22), we have | G ( s, x, t, y ) | Z tu Z R d φ ( α +1)0 ,α ( s, x, r, z ) | q ( r, z, t, y ) | d z d r . ( t − s ) − α φ ( α )0 ,α ⊗ (cid:0) φ ( α ) γ , + φ ( α )0 ,γ (cid:1) ( s, x, t, y ) . ( t − s ) − α φ ( α )0 ,α + γ ( s, x, t, y ) = φ ( α )0 ,α + γ − ( s, x, t, y ) . For G , noting that by (3.3), ∇ x p ( s, x, r, z ) = ∇ x p a ( r,z ) s,r (cid:18) x − z + Z rs b | r ′ − r | /α ( r ′ , θ r,r ′ ( z ))d r ′ (cid:19) , ∇ x ˜ p ( r,ξ ) ( s, x, r, z ) = ∇ x p a ( r,ξ ) s,r (cid:18) x − z + Z rs b | r ′ − r | /α ( r ′ , θ r,r ′ ( ξ ))d r ′ (cid:19) , by (2.34), (2.29), (1.11) and (2.5), one finds that |∇ x p − ∇ x ˜ p ( r,ξ ) | ( s, x, r, z ) . φ ( α +1)0 ,α ( s, x, r, z )(1 ∧ | z − θ s,r ( x ) | γ )+ φ ( α +2)0 ,α ( s, x, r, z ) (cid:0) | z − θ s,r ( x ) | β + ( r − s ) βα (cid:1) ( r − s ) . (2.33) (cid:0) φ ( α )0 ,α + γ − + φ ( α )0 , α + β − (cid:1) ( s, x, r, z ) . φ ( α )0 ,α − γ ( s, x, r, z ) , (4.3)where γ = γ ∧ ( α + β − α ∈ [1 , | G ( s, x, t, y ) | . φ ( α )0 ,α − γ ⊗ (cid:0) φ ( α ) γ , + φ ( α )0 ,γ (cid:1) ( s, x, t, y ) . φ ( α )0 ,α − γ ( s, x, t, y ) φ ( α )0 ,α − ( s, x, t, y ) . (4.4)For G , by (3.3), (2.29) and (3.14) we have for γ ∈ (0 , γ ), | G ( s, x, t, y ) | Z us Z R d |∇ x ˜ p ( r,ξ ) ( s, x, r, z ) | | ( q ( r, z, t, y ) − q ( r, ξ, t, y )) | d z d r . Z us d r Z R d d zφ ( α )0 ,α − ( s, x, r, z )(1 ∧ | z − ξ | γ ) × h(cid:0) φ ( α ) γ , − γ + φ ( α )0 ,γ − γ (cid:1) ( r, z, t, y ) + (cid:0) φ ( α ) γ , − γ + φ ( α )0 ,γ − γ (cid:1) ( r, ξ, t, y ) i . Since t − r ≍ t − s for r ∈ [ s, u ] and ξ = θ s,r ( x ), from (2.7) in Lemma 2.2, it holds (cid:0) φ ( α ) γ , − γ + φ ( α )0 ,γ − γ (cid:1) ( r, ξ, t, y ) . (cid:0) φ ( α ) γ , − γ + φ ( α )0 ,γ − γ (cid:1) ( s, x, t, y ) . Thus by (2.22), we eventually have | G ( s, x, t, y ) | . Z us d r Z R d d zφ ( α )0 ,α + γ − ( s, x, r, z ) × h(cid:0) φ ( α ) γ , − γ + φ ( α )0 ,γ − γ (cid:1) ( r, z, t, y ) + (cid:0) φ ( α ) γ , − γ + φ ( α )0 ,γ − γ (cid:1) ( s, x, t, y ) i . (cid:0) φ ( α )0 ,γ − γ + φ ( α ) γ ,α − (cid:1) ( s, x, t, y ) . Combining the above calculations, we obtain (4.2). Moreover, by the dominatedconvergence theorem, from the above calculations, it is easy to see thatlim x → x sup ( b,a ) ∈ C | G b,ai ( s, x, t, y ) − G b,ai ( s, x , t, y ) | = 0 , i = 1 , , , , where G b,ai are defined as above through the coefficients b, a . For instance,lim x → x sup ( b,a ) ∈ C | G b,a ( s, x, t, y ) − G b,a ( s, x , t, y ) | Z tu lim x → x sup ( b,a ) ∈ C Z R d |∇ x p ( s, x, r, z ) − ∇ x p ( s, x , r, z ) | | q ( r, z, t, y ) | d z d r, and for each r ∈ ( u, t ), by (3.13) and (3.11),lim x → x sup ( b,a ) ∈ C Z R d |∇ x p ( s, x, r, z ) − ∇ x p ( s, x , r, z ) | | q ( r, z, t, y ) | d z . Z R d lim x → x sup ( b,a ) ∈ C |∇ x p ( s, x, r, z ) − ∇ x p ( s, x , r, z ) | φ ( α )0 , ( r, z, t, y )d z = 0 . In particular, Theorem 4.1 holds for α ∈ [1 , Remark 4.2.
We remark that for α ∈ (0 , α + β >
1, the second inequalityin (4.4) may not hold since α + γ − α + β >
1. Eventually, we also point out that the previous argumentscan be simplified if α ∈ (1 ,
2) for which the full parametrix expansion (3.10) of thedensity can actually be directly differentiated since the induced singularity in timeremains integrable.4.2.
Supercritical case α ∈ (0 , . The following gradient estimate comes in [29].
Theorem 4.3. (Gradient estimate) Under (H βb ) , (H γa ) and (3.1) , for any T > ,there is a constant C > such that for all f ∈ B b ( R d ) and s < t T , |∇ P s,t f ( x ) | . C ( t − s ) − /α k f k ∞ , where the constant C may depend on k∇ b k ∞ and k∇ a k ∞ . Below we fix s < t and x ∈ R d and divide the proof into six steps. (Step 1). For notational simplicity, we shall write for r ∈ [ s, t ], f A r := A ( s,x ) r = K ( s,x ) r + B ( s,x ) r =: f K r + e B r , and h ( s, x, t, y ) := (cid:0) ∇ ˜ p ( τ,ξ ) ( s, · , t, y )( x ) (cid:1) ( τ,ξ )=( s,x ) (3.3) = −∇ y g ( s,x ) s,t ( θ s,t ( x ) − y ) , (4.5)and for a function f , H s,t f ( x ) := Z R d h ( s, x, t, y ) f ( y )d y. By Lemma 3.3 we have ∇ P s,t f ( x ) = ∇ e P ( τ,ξ ) s,t f ( x ) + Z ts ∇ e P ( τ,ξ ) s,t A ( τ,ξ ) r P r,t f ( x )d r. Taking ( τ, ξ ) = ( s, x ) and using the above notations, we can write ∇ P s,t f ( x ) = H s,t f ( x ) + Z ts H s,r f A r P r,t f ( x )d r = H s,t f ( x ) + X i =1 I ( i ) s,t f ( x ) , (4.6)where for u := s + t , I (1) s,t f ( x ) := Z us H s,r f K r P r,t f ( x )d r, EAT KERNEL OF SUPERCRITICAL SDES WITH UNBOUNDED DRIFTS 25 I (2) s,t f ( x ) := Z us H s,r e B r P r,t f ( x )d r,I (3) s,t f ( x ) := Z tu H s,r f K r P r,t f ( x )d r,I (4) s,t f ( x ) := Z tu H s,r e B r P r,t f ( x )d r. (Step 2). Note that for j ∈ N , |∇ jy h ( s, x, t, y ) | (4.5) = |∇ j +1 y g ( s,x ) s,t ( θ s,t ( x ) − y ) | (2.29) . φ ( α + j +1)0 ,α ( s, x, t, y ) . (4.7)Thus we have | H s,t f ( x ) | . Z R d φ ( α +1)0 ,α ( s, x, t, y ) | f ( y ) | d y ( t − s ) − α Z R d φ ( α )0 ,α ( s, x, t, y ) | f ( y ) | d y. For I (1) s,t f ( x ), noting that by Lemma 3.7, | f K r P r,t f ( z ) | . (1 ∧ | z − θ s,r ( x ) | γ ) |D ( α ) P r,t f | ( z ) . (1 ∧ | z − θ s,r ( x ) | γ ) Z R d φ ( α )0 , ( r, z, t, y ) | f ( y ) | d y, and using (2.33) and Lemma 2.7, we have | I (1) s,t f ( x ) | (4.7) . Z us Z R d φ ( α +1) γ,α ( s, x, r, z ) Z R d φ ( α )0 , ( r, z, t, y ) | f ( y ) | d y d z d r (2.33) . Z us Z R d (cid:0) φ ( α )0 ,α + γ − ⊙ φ ( α )0 , (cid:1) r ( s, x, t, y ) | f ( y ) | d y d r (2.21) . Z R d φ ( α )0 ,α + γ − ( s, x, t, y ) | f ( y ) | d y. For I (2) s,t f ( x ), noting that | e B r P r,t f ( z ) | . ( | θ s,r ( x ) − z | β + | θ s,r ( x ) − z | + ( r − s ) β/α ) |∇ P r,t f ( z ) | , using (4.7) and (2.33), we have | I (2) s,t f ( x ) | . Z us Z R d φ ( α )0 ,α + β − ( s, x, r, z ) |∇ P r,t f ( z ) | d z d r . ( t − s ) − α Z ts Z R d φ ( α )0 ,α + β − ( s, x, r, z )( t − r ) α |∇ P r,t f ( z ) | d z d r. (Step 3). In this step we treat the hard term I (3) s,t f ( x ). Let ε := ( t − r ) /α and κ ε ( r, z, z ′ ) := κ ( r, · , z ′ ) ∗ ρ ε ( z ) , ¯ κ ε ( r, z, z ′ ) := κ ε ( r, z, z ′ ) − κ ε ( r, θ r,s ( x ) , z ′ )and f K ( ε ) r f ( z ) = 2 Z R d δ (1) f ( z ; z ′ ) ¯ κ ε ( r, z, z ′ ) | z ′ | d + α d z ′ . Let us write I (3) s,t f ( x ) = Z tu (cid:16) H s,r ( f K r − f K ( ε ) r ) P r,t f ( x ) + H s,r f K ( ε ) r P r,t f ( x ) (cid:17) d r =: Z tu (cid:16) J ( ε )1 ,r ( s, x, t ) + J ( ε )2 ,r ( s, x, t ) (cid:17) d r. Let γ ∈ (0 , γ ). Noting that | ( κ − κ ε )( r, z, z ′ ) − ( κ − κ ε )( r, θ r,s ( x ) , z ′ ) | . C ( | z − θ r,s ( x ) | γ ∧ ε γ − γ , by definition and Lemma 3.7, we have | ( f K r − f K ( ε ) r ) P r,t f ( z ) | . ( | z − θ r,s ( x ) | γ ∧ ε γ − γ |D ( α ) P r,t f | ( z ) . ( | z − θ r,s ( x ) | γ ∧ ε γ − γ Z R d φ ( α )0 , ( r, z, t, y ) | f ( y ) | d y. For J ( ε )1 ,r , recalling ε = ( t − r ) /α , we have Z tu | J ( ε )1 ,r ( s, x, t ) | d r (4.7) . Z tu Z R d (cid:0) φ ( α )0 ,α + γ − ⊙ φ ( α )0 ,γ − γ (cid:1) r ( s, x, t, y ) | f ( y ) | d y d r (2.21) . Z R d φ ( α )0 ,α + γ − ( s, x, t, y ) | f ( y ) | d y. For J ( ε )2 ,r , by the change of variables and Fubini’s theorem, we have J ( ε )2 ,r ( s, x, t ) = Z R d h ( s, x, r, z ) Z R d δ (1) P r,t f ( z ; z ′ ) ¯ κ ε ( r, z, z ′ ) | z ′ | d + α d z ′ d z = Z R d Z R d δ (1) h ( s,x,r, · )¯ κ ε ( r, · ,z ′ ) ( z ; z ′ ) d z ′ | z ′ | d + α P r,t f ( z )d z = Z R d h ( s, x, r, z ) Z R d δ (1)¯ κ ε ( r, · ,z ′ ) ( z ; z ′ ) d z ′ | z ′ | d + α P r,t f ( z )d z + Z R d Z R d δ (1) h ( s,x,r, · ) ( z ; z ′ )¯ κ ε ( r, z + z ′ , z ′ ) d z ′ | z ′ | d + α P r,t f ( z )d z. Noting that by (H γa ) , | δ (1)¯ κ ε ( r, · ,z ′ ) ( z ; z ′ ) | . ( ε γ − | z ′ | ) ∧ | z ′ | γ ∧ , we have Z R d | δ (1)¯ κ ε ( r, · ,z ′ ) ( z ; z ′ ) | d z ′ | z ′ | d + α . Z R d (( ε γ − | z ′ | ) ∧ | z ′ | γ ∧
1) d z ′ | z ′ | d + α . ε ( γ − α ) ∧ . On the other hand, by (4.5) and (2.29), | δ (1) h ( s,x,r, · ) ( z ; z ′ ) | . (cid:0) (( r − s ) − α | z ′ | ) ∧ (cid:1)(cid:0) φ ( α )0 ,α − ( s, x, r, z + z ′ ) + φ ( α )0 ,α − ( s, x, r, z ) (cid:1) . Thus, as in (2.38) we have Z R d | δ (1) h ( s,x,r, · ) ( z ; z ′ ) | d z ′ | z ′ | d + α . Z R d (cid:0) (( r − s ) − α | z ′ | ) ∧ (cid:1) φ ( α )0 ,α − ( s, x, r, z + z ′ ) d z ′ | z ′ | d + α + φ ( α )0 ,α − ( s, x, r, z ) Z R d (cid:0) (( r − s ) − α | z ′ | ) ∧ (cid:1) d z ′ | z ′ | d + α . φ ( α )0 ,α − ( s, x, r, z )( r − s ) − = φ ( α )0 , − ( s, x, r, z ) . Therefore, | J ( ε )2 ,r ( s, x, t ) | . Z R d h ε ( γ − α ) ∧ φ ( α )0 ,α − ( s, x, r, z ) + φ ( α )0 , − ( s, x, r, z ) i P r,t | f | ( z )d z. Recall ε = ( t − r ) α . By (2.33), we obtain Z tu | J ( ε )2 ,r ( s, x, t ) | d r . ( t − s ) − α Z R d φ ( α )0 ,α ( s, x, t, y ) | f ( y ) | d y. EAT KERNEL OF SUPERCRITICAL SDES WITH UNBOUNDED DRIFTS 27 (Step 4).
For ε = ( t − r ) /α , we define¯ b ε ( r, z ) := ( b ∗ ρ ε )( r, z ) − ( b ∗ ρ ε ∗ ρ | r − s | /α )( r, θ r,s ( x ))and e B ( ε ) r f ( z ) := ¯ b ε ( r, z ) · ∇ f ( z ) . For I (4) s,t , we similarly write I (4) s,t f ( x ) = Z tu (cid:16) H s,r ( e B r − e B ( ε ) r ) P r,t f ( x ) + H s,r e B ( ε ) r P r,t f ( x ) (cid:17) d r =: Z tu (cid:16) J ( ε )3 ,r ( s, x, t ) + J ( ε )4 ,r ( s, x, t ) (cid:17) d r. For J ( ε )3 ,r , since | ¯ b − ¯ b ε | ( r, z ) κ ε β = κ ( t − s ) β/α , by (4.7) we have | J ( ε )3 ,r ( s, x, t ) | = (cid:12)(cid:12)(cid:12)(cid:12)Z R d h ( s, x, r, z )(¯ b ( r, z ) − ¯ b ε ( r, z )) · ∇ P r,t f ( z )d z (cid:12)(cid:12)(cid:12)(cid:12) . Z R d φ ( α +1)0 ,α ( s, x, r, z )( t − r ) βα |∇ P r,t f ( z ) | d z, and Z tu | J ( ε )3 ,r ( s, x, t ) | d r . ( t − s ) − α Z ts Z R d φ ( α )0 ,α ( s, x, r, z )( t − r ) βα |∇ P r,t f ( z ) | d z d r. For J ( ε )4 ,r , we derive integrating by parts that | J ( ε )4 ,r ( s, x, t ) | = (cid:12)(cid:12)(cid:12)(cid:12)Z R d h ( s, x, r, z ) ¯ b ε ( r, z ) · ∇ z P r,t f ( z )d z (cid:12)(cid:12)(cid:12)(cid:12) (cid:12)(cid:12)(cid:12)(cid:12)Z R d h ( s, x, r, z ) div¯ b ε ( r, z ) P r,t f ( z )d z (cid:12)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)(cid:12)Z R d ¯ b ε ( r, z ) · ∇ z h ( s, x, r, z ) P r,t f ( z )d z (cid:12)(cid:12)(cid:12)(cid:12) . Since | div¯ b ε ( r, z ) | = | div b ε ( r, z ) | κ ε β − = κ ( t − r ) ( β − /α and | ¯ b ε | ( r, z ) . | z − θ r,s ( x ) | β + ( r − s ) β/α , by (4.7) and (2.33) we have | J ( ε )4 ,r ( s, x, t ) | . Z R d φ ( α )0 ,α − ( s, x, r, z )( t − r ) ( β − /α | P r,t f ( z ) | d z + Z R d φ ( α )0 ,α + β − ( s, x, r, z ) | P r,t f ( z ) | d z. Thus, Z tu | J ( ε )4 ,r ( s, x, t ) | d r . Z tu Z R d (cid:0) φ ( α )0 ,α − ⊙ φ ( α )0 ,α + β − (cid:1) r ( s, x, t, y ) | f ( y ) | d y d r + Z tu Z R d (cid:0) φ ( α )0 ,α + β − ⊙ φ ( α )0 ,α (cid:1) r ( s, x, t, y ) | f ( y ) | d y d r (2.21) . Z tu (cid:2) ( r − s ) α − α ( t − r ) α + β − α + ( r − s ) α + β − α ( t − r ) (cid:3) × (cid:2) ( r − s ) − + ( t − r ) − (cid:3) d r Z R d φ ( α )0 , ( s, x, t, y ) | f ( y ) | d y . Z R d φ ( α )0 , α + β − ( s, x, t, y ) | f ( y ) | d y . ( t − s ) − α Z R d φ ( α )0 ,α ( s, x, t, y ) | f ( y ) | d y, recalling that α + β > | I (4) s,t f ( x ) | . ( t − s ) − α (cid:16) Z ts Z R d φ ( α )0 ,α ( s, x, r, z )( t − r ) βα |∇ P r,t f ( z ) | d z d r + Z R d φ ( α )0 ,α ( s, x, t, y ) | f ( y ) | d y (cid:17) . (Step 5). Combining the above calculations, we obtain |∇ P s,t f ( x ) | . ( t − s ) − α Z R d φ ( α )0 ,α ( s, x, t, y ) | f ( y ) | d y + ( t − s ) − α Z ts Z R d φ ( α )0 ,α ( s, x, r, z )( t − r ) βα |∇ P r,t f ( z ) | d z d r + ( t − s ) − α Z ts Z R d φ ( α )0 ,α + β − ( s, x, r, z )( t − r ) α |∇ P r,t f ( z ) | d z d r. By the lower bound estimate, we further have( t − s ) α |∇ P s,t f ( x ) | . P s,t | f | ( x ) + Z ts ( t − r ) βα P s,r |∇ P r,t f | ( x )d r + Z ts ( r − s ) β − α ( t − r ) α P s,r |∇ P r,t f | ( x )d r. (4.8)For fixed 0 u < t T and s ∈ ( u, t ), we letΓ tu ( s, x ) := ( t − s ) α P u,s |∇ P s,t f | ( x ) . Using P u,s act on both sides of (4.8) and by P u,s P s,r = P u,r , we derive thatΓ tu ( s, x ) . P u,t | f | ( x ) + Z ts h ( r − s ) β − α + ( t − r ) β − α i Γ tu ( r, x )d r. Note that by Theorem 4.3, sup s ∈ [ u,t ] k Γ tu ( s, · ) k ∞ < ∞ . Since α + β >
1, from the Volterra-Gronwall inequality, we obtain that for all s ∈ ( u, t ), Γ tu ( s, x ) . P u,t | f | ( x ) . Taking limit u ↑ s , we obtain( t − s ) α |∇ P s,t f | ( x ) . P s,t | f | ( x ) , which eventually yields the desired gradient estimate. (Step 6). Finally, by (4.6) and the dominated convergence theorem, one canshow that lim x → x sup ( b,a ) ∈ C |∇ P b,as,t f ( x ) − ∇ P b,as,t f ( x ) | = 0 . Indeed, from the above proof, it suffices to show thatlim x → x sup ( b,a ) ∈ C | H b,as,t f ( x ) − H b,as,t f ( x ) | = 0 . EAT KERNEL OF SUPERCRITICAL SDES WITH UNBOUNDED DRIFTS 29
This follows by Lemma 2.8.5.
Proof of Theorem 1.1
The point is here to prove Theorem 1.1. Namely, we want to extend the boundsof Theorem 3.5 and Lemmas 3.7 and (4.2) under the sole assumptions (H γa ) , (H βb ) .Let a ε and b ε be the smooth approximations of a and b , respectively. Hence,assumptions (H γa ) , (H βb ) and (3.1) are met by a ε , b ε for the SDEd X εt = b ε ( t, X εt )d t + a ε ( t, X εt − )d L ( α ) t . (5.1)The following convergence in law result was established in [7], see Theorem 1.1therein. Theorem 5.1.
Let X εs,t ( x ) be the unique solution of SDE (5.1) . Then X εs,t ( x ) weakly converges to X s,t ( x ) .Proof. For fixed ( s, x ) ∈ R + × R d , since the coefficients b, a are linear growth, it isby now standard to show that the law of X εs, · ( x ) is tight in the space of all c´adl´agfunctions. By a standard way, one can show that any weak accumulation point ofthe law of X εs, · ( x ) is a weak solution of SDE (1.1). Finally, by the weak uniqueness,one sees that X εs,t ( x ) weakly converges to X s,t ( x ). (cid:3) Denoting by p ε the associated density, it therefore holds from Theorem 3.5,Lemma 3.7 and Theorem 4.1 that(i) (Two-sides estimate) For any
T >
0, there is a constant C = C ( T, Θ) > s < t T and x, y ∈ R d , p ε ( s, x, t, y ) ≍ C φ ( α )0 ,α ( s, x, t, y ) . (5.2)(ii) (Fractional derivative estimate) For any
T >
0, there is a constant C = C ( T, Θ) > s < t T and x, y ∈ R d , |D ( α ) p ε ( s, · , t, y ) | ( x ) . C φ ( α )0 , ( s, x, t, y ) . (5.3)(iii) (Gradient estimate in x ) For any
T >
0, there is a constant C = C ( T, Θ) > s < t T and x, y ∈ R d , |∇ P εs,t f ( x ) | . C ( t − s ) − /α P εs,t | f | ( x ) . (5.4)where the constants in the above controls only depend on (H γa ) , (H βb ) through Θ(see precisely (1.13)).By Theorem 5.1, we have for any f ∈ C b ( R d ),lim ε → P εs,t f ( x ) := lim ε → E f ( X εs,t ( x )) = E f ( X s,t ( x )) =: P s,t f ( x ) . (5.5)(i) (Two-sided estimates) For nonnegative measurable functions f , we get from(5.2) C − Z R d φ ( α )0 ,α ( s, x, t, y ) f ( y )d y E f ( X s,t ( x )) C Z R d φ ( α )0 ,α ( s, x, t, y ) f ( y )d y, which implies that X s,t ( x ) has a density p ( s, x, t, y ) having lower and upper boundas in (1.15). On the other hand, for fixed s < t , by Theorem 3.5 we have( x, y ) p ε ( s, x, t, y ) is equi-continuous in ε ∈ (0 , . From the Ascoli-Arzel`a theorem, there are a subsequence ε k and a continuous func-tion ¯ p ( s, x, t, y ) as a function of x, y ∈ R d such that p ε k ( s, x, t, y ) → ¯ p ( s, x, t, y ) locally uniformly in x, y ∈ R d , (5.6) which together with (5.5) yields that p ( s, x, t, y ) = ¯ p ( s, x, t, y ) is continuous as a function of x, y ∈ R d . (5.7)(ii) (Fractional derivative estimates) It follows by (5.3), (5.6), (5.7) and Fatou’slemma that |D ( α ) p ( s, · , t, y ) | ( x ) = Z R d lim k →∞ | δ (2) p εk ( s, · ,t,y ) ( x ; z ) | d z | z | d + α lim k →∞ Z R d | δ (2) p εk ( s, · ,t,y ) ( x ; z ) | d z | z | d + α = lim k →∞ |D ( α ) p ε k ( s, · , t, y ) | ( x ) . C φ ( α )0 , ( s, x, t, y ) . (iii) (Gradient estimates) For fixed f ∈ C b ( R d ), by (5.4), x
7→ ∇ P εs,t f ( x ) is equi-continuous in ε, which together with (5.5) implies that x P s,t ( x ) is continuous differentiable. Bytaking limits along a subsequence ε k for (5.4), we obtain |∇ P s,t f ( x ) | . C ( t − s ) − /α P s,t | f | ( x ) . Finally, for fixed t ′ > t and y ∈ R d , we let f ( x ) := p ( t, x, t ′ , y ), then by theChapman-Kolmogorov equation, we obtain |∇ p ( s, · , t ′ , y )( x ) | . C ( t − s ) − /α p ( s, x, t ′ , y ) . This then readily gives estimate (4.2) (logarithmic derivative) of Theorem 1.1.
References [1] R. F. Bass and Z.-Q. Chen, Systems of equations driven by stable processes.
Probab. TheoryRelat. Fields , 134 (2006), 175-214.[2] Bendikov A., Asymptotic formulas for symmetric stable semigroups.
Expositiones Mathemat-icae , 13(1994), 381-384.[3] R. M. Blumenthal and R. K. Getoor, Some theorems on stable processes.
Trans. Amer. Math.Soc. (1960), 263-273.[4] K. Bogdan and T. Jakubowski. Estimates of heat kernel of fractional Laplacian perturbed bygradient operators. Comm. Math. Phys. , 271(1):179–198, 2007.[5] K. Bogdan, P. Sztonyk and V. Knopova, Heat kernel of anisotropic nonlocal operators.arXiv:1704.03705.[6] Z.-Q. Chen E. Hu, L. Xie and X. Zhang, Heat kernels for non-symmetric diffusions operatorswith jumps.
J. Differential Equations (2017), 6576-6634.[7] Z.-Q. Chen Z. Hao and X. Zhang. H¨older regularity and gradient estimates for SDEs drivenby cylindrical α -stable processes. Electron. J. Probab.
25 (2020), article no. 137, 1-23.[8] Z.-Q. Chen and X. Zhang. Heat kernels and analyticity of non-symmetric jump diffusionsemigroups.
Probab. Theory Related Fields , 165(1-2):267–312, 2016.[9] Z.-Q. Chen and X. Zhang. Heat kernels for time-dependent non-symmetric stable-like oper-ators
J. Math. Anal. Appl. , 465 (2018) 1-21(2019).[10] Z.-Q. Chen and X. Zhang. Uniqueness of stable-like processes. arXiv preprintarXiv:1604.02681(2016).[11] Z.-Q. Chen and X. Zhang. Heat kernels for time-dependent non-symmetric mixed L´evy-typeoperators. arXiv preprint arXiv:1604.02681(2020).[12] Z.-Q. Chen, X. Zhang and G. Zhao, Supercritical SDEs driven by multiplicative stable-likeL´evy processes. To appear in
Trans. Amer. Math. Soc. , (2021+).[13] A. Debussche and N. Fournier. Existence of densities for stable-like driven SDE’s with H¨oldercontinuous coefficients,
J. Funct. Anal. , , No. 4, 1757–1778(2013).[14] F. Delarue and S. Menozzi. Density estimates for a random noise propagating through a chainof differential equations. Journal of Functional Analysis , 259–6:1577–1630, 2010.
EAT KERNEL OF SUPERCRITICAL SDES WITH UNBOUNDED DRIFTS 31 [15] K. Du and X. Zhang. Optimal gradient estimates of heat kernels of stabe-like operators.
Proc.of Amer. Math. Soci. . Vol. 147, No 8, 3559-3565(2019).[16] A. Friedman. Partial differential equations of parabolic type. Prentice-Hall Inc., EnglewoodCliffs, N.J., 1964.[17] L. Huang. Density estimates for SDEs driven by tempered stable processes. arXiv:1504.04183 , 2015.[18] V. Knopova and A.M. Kulik. Parametrix construction of the transition probability densityof the solution to an SDE driven by α -stable noise, A nn. Inst. Henri Poincar´e Probab. Stat.,54(1), 100–140 (2018).[19] V. Knopova, A. Kulik, R. Schilling. Construction and heat kernel estimates of general stable-like Markov processes. arXiv:2005.08491 , (2020).[20] V. N. Kolokoltsov. Symmetric Stable Laws and Stable-Like Jump-Diffusions. Proceedings ofthe London Mathematical Society , 80(3):725–768, May 2000.[21] A. M. Kulik. On weak uniqueness and distributional properties of a solution to an SDE with α -stable noise. Stochastic Process. Appl. , 129(2):473–506, 2019.[22] A. Kulik, S. Peszat, E. Priola.Gradient formula for transition semigroup corresponding to sto-chastic equation driven by a system of independent L´evy processes. a rXiv:2006.09133 (2020).[23] R. L´eandre. R´egularit´e de processus de sauts d´eg´en´er´es. Ann. Inst. H. Poincar´e Probab.Statist. , 21(2), 125–146, 1985.[24] W. Liu, R. Song and L. Xie, Gradient estimates for the fundamental solution of L´evy typeoperator.
Adv. Nonlinear Anal.
J. Differential Equations , , 330–369(2021)[26] R. Mikulevicius and H. Pragarauskas. On the Cauchy problem for integro-differential op-erators in H¨older classes and the uniqueness of the martingale problem. Potential Anal. ,40(4):539–563, 2014.[27] K. Sato: L´evy processes and infinitely divisible distributions. Cambridge University Press,London 1999.[28] H. Tanaka, M. Tsuchiya, and S. Watanabe. Perturbation of drift-type for L´evy processes.
J.Math. Kyoto Univ. , 14:73–92, 1974.[29] F.Y. Wang, L. Xu and X. Zhang: Gradient estimates for SDEs driven by multiplicative L´evynoise.
Journal of Functional Analysis , 269 (2015) 3195-3219.[30] F.Y. Wang, X. Zhang. Heat kernel for fractional diffusion operators with perturbations.
Fo-rum Math.
27 (2015) 973-994.27 (2015) 973-994.