Stochastic differential equations with Sobolev drifts and driven by <span class="mathjax-formula">$\alpha $</span>-stable processes

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www.imstat.org/aihp 2013, Vol. 49, No. 4, 1057–1079

DOI:10.1214/12-AIHP476

Stochastic differential equations with Sobolev drifts and driven by α-stable processes

Xicheng Zhang

School of Mathematics and Statistics, Wuhan University, Wuhan, Hubei 430072, P. R. China. E-mail:XichengZhang@gmail.com Received 27 March 2011; revised 31 December 2011; accepted 6 January 2012

Abstract. In this article we prove the pathwise uniqueness for stochastic differential equations inR^dwith time-dependent Sobolev drifts, and driven by symmetricα-stable processes provided thatα∈(1,2)and its spectral measure is non-degenerate. In particular, the drift is allowed to have jump discontinuity whenα∈(_d^2d₊₁,2). Our proof is based on some estimates of Krylov’s type for purely discontinuous semimartingales.

Résumé. Dans cet article nous prouvons l’existence et l’unicité d’équations différentielles stochastiques dansR^davec terme de dérive dépendant du temps dans un espace de Sobolev et dirigées par un processus de Lévyα-stable symétrique avecα∈(1,2)et de mesure spectrale non-dégénérée.

En particulier, le terme de dérive peut avoir des discontinuités de saut quandα∈(_d^2d₊₁,2). Notre preuve est basée sur des estimations de type Krylov pour des semimartingales purement discontinues.

MSC:60H10

Keywords:Pathwise uniqueness; Symmetricα-stable process; Krylov’s estimate; Fractional Sobolev space

1. Introduction and main result

Consider the following SDE driven by a symmetricα-stable process inR^d:

dXt=b_t(X_t)dt+dLt, X₀=x, (1.1)

whereb:R+×R^d→R^d is a measurable function,(L_t)_t_≥₀is ad-dimensional symmetricα-stable process defined on some filtered probability space(Ω,F, P;(Ft)_t_≥₀). The aim of this paper is to study the pathwise uniqueness of SDE (1.1) withbin fractional Sobolev spaces.

Let us first briefly recall some well-known results in this direction. WhenLt is a standard d-dimensional Brow- nian motion, Veretennikov [27] firstly proved the existence of a unique strong solution for SDE (1.1) with bounded measurableb. In [17], Krylov and Röckner relaxed the boundedness assumptions onb to the following integrability assumptions:

T 0

R^d

b_t(x)^pdx q/p

dt <+∞, ∀T >0, (1.2)

provided that d p+2

q <1. (1.3)

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Recently, in [28] we extended Krylov and Röckner’s result to the case of non-constant Sobolev diffusion coefficients and meanwhile, obtained the stochastic homeomorphism flow property of solutions and the strong Feller property.

Other generalizations in various directions can be found in [9–12,14], etc.

In the case of symmetricα-stable processes, the pathwise uniqueness for SDE (1.1) with irregular drift is far from being complete. Whend=1,α∈ [1,2)andb is time-independent and bounded continuous, Tanaka, Tsuchiya and Watanabe [25] proved the pathwise uniqueness of solutions to SDE (1.1). Whend >1,α∈ [1,2), the spectral measure ofL_tis non-degenerate, andbis time-independent and bounded Hölder continuous, where the Hölder indexβsatisfies

β >1−α 2,

Priola [21] recently proved the pathwise uniqueness and the stochastic homeomorphism flow property of solutions to SDE (1.1). Whend=1, α∈(1,2)andbis only bounded measurable, Kurenok [18] obtained the existence of weak solutions for SDE (1.1) by proving an estimate of Krylov’s type: for anyT >0,

E _T

0

f_t(X_t)dt≤CfL²([0,T]×R). (1.4)

Whend >1,α∈(1,2)andbis time-independent and belongs to some Kato’s class, Chen, Kim and Song [7], Theo- rem 2.5, proved the existence of martingale solutions (equivalently weak solutions) in terms of Feller semigroup (cf.

[6]). On the other hand, there are many works devoted to the study of weak uniqueness (i.e., the well-posedness of martingale problems) for SDEs with jumps. This is refereed to the survey paper of Bass [5]. Moreover, in the one dimensional case, the pathwise uniqueness for SDEs with Hölder continuous diffusion coefficients has been extensively studied (see the recent works of Fournier [13] and Li and Mytnik [20]). However, to the author’s knowledge, there are few results about the pathwise uniqueness for multidimensional SDE (1.1) with discontinuous drifts.

Before stating our main result, we recall some facts about symmetricα-stable processes. Let (Lt)t≥0 be a d- dimensional symmetricα-stable process. By Lévy–Khinchin’s formula, its characteristic function is given by (cf.

[23])

Eeⁱ^ξ,L^t=e⁻^{tψ (ξ )}, where

ψ (ξ )=

R^d

1−eⁱ^ξ,x+i ξ, x1_|x|≤1

ν(dx),

and the Lévy measureνwithν({0})=0 is given by ν(U )=

S^d⁻¹

_+∞

0

1U(rθ )

r^d⁺^α drμ(dθ ), U∈B R^d

, (1.5)

whereB(R^d)is the Borelσ-field of Euclidean spaceR^d,μis a symmetric finite measure on the unit sphereS^d⁻¹:=

{θ∈R^d: |θ| =1}, calledspectral measureof stable processLt. By an elementary calculation, we have ψ (ξ )=

R^d

1−cos ξ, x

ν(dx)=c_α

S^d−1

ξ, θ^αμ(dθ ).

In particular, ifμis the uniform distribution onS^d⁻¹, thenψ (ξ )=c_α|ξ|^α, herec_αmay be different. Throughout this paper, we make the following assumption:

(H^α): For someα∈(0,2)and constantCα>0,

ψ (ξ )≥C_α|ξ|^α, ξ ∈R^d. (1.6)

We remark that the above condition is equivalent that the support of spectral measureμis not contained in a proper linear subspace ofR^d (cf. [21], p. 4).

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We now introduce the class of local strong solutions for SDE (1.1). Letτ be any (Ft)-stopping time. Forx∈R^d, letS_b^τ(x)be the class of allR^d-valued (Ft)-adapted càdlàg stochastic processX_t on[0, τ )satisfying

P

ω:

_T

0

b_s

X_s(ω)ds <+∞,∀T ∈ 0, τ (ω)

=1, and such that

X_t=x+ t

0

b_s(X_s)ds+L_t, ∀t∈ [0, τ ),a.s.

The main result of the present paper is:

Theorem 1.1. Assume that(H^α)holds withα∈(1,2),andb:R₊×R^d→R^dsatisfies that for someβ∈(1−^α₂,1), p > ^2d_α and anyT , R >0,

sup

t∈[0,T]

BR

|b_t(x)−b_t(y)|^p

|x−y|^d⁺^βp dxdy <+∞ (1.7)

and

sup

(t,x)∈[0,T]×B_R

b_t(x)<+∞, (1.8)

whereB_R= {x∈R^d: |x| ≤R}.Then,for anyx∈R^d,there exists an(Ft)-stopping timeζ (x)(called explosion time) and a unique strong solutionX_t∈S_b^{ζ (x)}(x)to SDE(1.1)with

t↑limζ (x)Xt(x)= +∞ a.s.on

ζ (x) <+∞

. (1.9)

Remark 1.2. LetObe a bounded smooth domain inR^d.It is well-known that for anyβ∈(0,1)andp∈(1,_β¹)(cf.

[1]),

1_O∈W^βp, (1.10)

whereW^βpis the fractional Sobolev space defined by(2.3)below.Hence,ifα∈(_d^2d₊₁,2),then one can choose β∈

1−α

2, α 2d

, p∈

2d α ,1

β

so that Theorem1.1can be used to uniquely solve the following discontinuous SDE:

dXt= b⁽¹⁾1_O+b⁽²⁾1_O^c

(Xt)dt+dLt, X0=x,

whereb⁽ⁱ⁾, i=1,2,are two bounded and locally Hölder continuous functions with Hölder index greater thanβ.In one dimensional case,ifα∈(1,2),it is well-known that regularity condition(1.7)can be dropped(cf. [25],p. 82, Remark1).The key point in this case is that the weak uniqueness is equivalent to the pathwise uniqueness.However, in the case ofd ≥2,it is still open that whether SDE(1.1)has a unique strong solution whenb is only bounded measurable.

For proving this theorem, as in [17,28,30], we mainly study the following partial integro-differential equation (abbreviated as PIDE) by using some interpolation techniques:

∂tu=L0u+bⁱ∂iu+f, u0(x)=0,

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whereL0is the generator of Lévy process(L_t)_t_≥₀given by L0u(x)=

R^d

u(x+z)−u(x)−1_|z|≤1zⁱ∂_iu(x)

ν(dz)=lim

ε↓0

|z|>ε

u(x+z)−u(x)

ν(dz), (1.11)

where the second equality is due to the symmetry ofν. Here and below, we use the convention that the repeated index will be summed automatically. However, we need to firstly extend Krylov’s estimate (1.4) to the multidimensional case. As in [18], we shall investigate the following semi-linear PIDE:

∂_tu=L0u+κ|∇u| +f, u₀(x)=0,

whereκ >0 and∇is the gradient operator with respect to the spatial variablex. We want to emphasize that Fourier’s transform used in [18,19] seems only work for one-dimensional case.

Our method for studying the above two PIDEs is based on the semigroup argument. For this aim, we shall derive some smoothing and asymptotic properties about the Markov semigroup associated withL0. In particular, the interpolation techniques will be used frequently. This will be done in Section2. In Section3, partly following Kurenok’s idea, we shall prove two Krylov’s estimates for multidimensional purely discontinuous semimartingales. In Section4, we prove our main result by using a transformation to remove the drift as adopted in [12,21,30].

In the remainder of this paper, the letterCwith or without subscripts will denote a positive constant whose value may change in different occasions.

2. Preliminaries

Forp≥1, the norm inL^p-spaceL^p(R^d)is denoted by · p. Forβ≥0 andp≥1, letH^βp:=(I−Δ)⁻^β/2(L^p(R^d)) be the Bessel potential space with the norm

fβ,p:=(I−Δ)^β/2f

p.

Notice that forβ=m∈N, an equivalent norm ofH^βpis given by (cf. [26], p. 177) fm,p= fp+∇^mf

p.

By Sobolev’s embedding theorem, ifβ−^d_p>0 is not an integer, then (cf. [26], p. 206, (16)) H^βp→C^β⁻^d/p

R^d

, (2.1)

where forγ >0,C^γ(R^d)is the usual Hölder space with the norm:

fC^γ:=

[γ]

k=0

sup

x∈R^d

∇^kf (x)+sup

x=y

|∇^[^γ^]f (x)− ∇^[^γ^]f (y)|

|x−y|^γ^−[^γ^] , where[γ] :=max{m∈N∪ {0}: m≤γ}is the integer part ofγ.

LetA andB be an interpolation pair of Banach spaces. For θ∈ [0,1], we use[A, B]θ to denote the complex interpolation space betweenAandB (cf. [26]). We have the following relation (cf. [26], p. 185, (11)): for p >1, β1=β2andθ∈(0,1),

H^βp¹,H^βp²

θ=H^βp¹⁺^{θ (β}²⁻^β¹⁾. (2.2)

On the other hand, for 0< β=integer, the fractional Sobolev spaceW^βpis defined by (cf. [26], p. 190, (15)) f^∼β,p:= fp+

[β]

k=0

R^d

|∇^kf (x)− ∇^kf (y)|^p

|x−y|^d⁺^(β^−[^β^]^)p dxdy 1/p

<+∞. (2.3)

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Forβ=0,1,2, . . . ,we setW^βp:=H^βp. The relation betweenH^βpandW^βp is given as follows (cf. [26], p. 180, (9)):

for anyβ >0,ε∈(0, β)andp >1,

H^βp⁺^ε→W^βp→H^βp⁻^ε. (2.4)

We recall the following complex interpolation theorem (cf. [26], p. 59, Theorem (a)).

Theorem 2.1. LetAi ⊂Bi, i=0,1,be Banach spaces.LetT :Ai→Bi, i=0,1,be bounded linear operators.For θ∈ [0,1],we have

TAθ→Bθ≤ T¹_A⁻₀_→^θ _B₀T^θ_A₁_→_B₁,

whereA_θ:= [A₀, A₁]θ,B_θ:= [B₀, B₁]θ andTA_θ→B_θ denotes the operator norm ofT mappingA_θ toB_θ. Letf be a locally integrable function onR^d. The Hardy–Littlewood maximal function is defined by

Mf (x):= sup

0<r<∞

1

|B_r|

B_r

f (x+y)dy,

whereB_r := {x∈R^d:|x|< r}. The following well-known results can be found in [8,22,29] and [24], p. 5, Theorem 1.

Lemma 2.2. (i)Forf∈W¹₁,there exists a constantC_d>0and a Lebesgue zero setEsuch that for allx, y /∈E, f (x)−f (y)≤Cd|x−y|

M|∇f|(x)+M|∇f|(y)

. (2.5)

(ii)Forp >1,there exists a constantC_d,p>0such that for allf∈L^p(R^d),

Mfp≤C_d,pfp. (2.6)

For fixedz∈R^d, define the shift operator Tzf (x):=f (x+z)−f (x).

We have the following useful estimate.

Lemma 2.3. Forp >1andγ∈ [1,2],there exists a constantC=C(p, γ , d) >0such that for anyf∈H^γp,

Tzf1,p≤C|z|^γ⁻¹fγ ,p. (2.7)

Proof. By (2.5), we have for Lebesgue almost allx∈R^d, Tzf (x)≤C|z|

M|∇f|(x+z)+M|∇f|(x) , and so, by (2.6),

Tzfp≤C|z| ·M|∇f|

p≤C|z| · ∇fp≤C|z| · f1,p. On the other hand, it is clear that for anyβ >0,

Tzfβ,p≤2fβ,p.

By Theorem2.1and (2.2), forθ∈(0,1), we immediately have Tzfθβ,p≤C|z|¹⁻^θf1+θ (β−1),p,

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which gives the desired result by lettingθ=2−γ andβ=₂₋¹_γ. We now recall the following well-known properties about the symmetricα-stable process(L_t)_t_≥0(cf. [23], Theo- rem 25.3, and [21], Lemma 3.1).

Proposition 2.4. Letμ_t be the law ofα-stable processL_t.

(i) Scaling property:For anyλ >0,(L_t)_t_≥₀and(λ⁻^1/αL_λt)_t_≥₀have the same finite dimensional law.In particular, for anyt >0andA∈B(R^d),μ_t(A)=μ₁(t⁻^1/αA).

(ii) Existence of smooth density:For anyt >0,μ_t has a smooth densityp_t with respect to the Lebesgue measure, which is given by

p_t(x)= 1 (2π)^d

R^de⁻ⁱ^x,ξe⁻^{t ψ(ξ )}dξ.

Moreover,p_t(x)=p_t(−x)and for anyk∈N,∇^kp_t∈L¹(R^d).

(iii) Moments:For anyt >0,ifβ < α,thenE|L_t|^β<+∞;ifβ≥α,thenE|L_t|^β= ∞.

The Markov semigroup associated with the Lévy process(L_t)_t_≥₀is given by Ttf (x)=E

f (L_t+x)

=

R^dp_t(z−x)f (z)dz=

R^dp_t(x−z)f (z)dz. (2.8)

We have:

Lemma 2.5. (i)For anyα∈(0,2),p >1andβ, γ≥0,we have for allf ∈H^βp,

Ttfβ+γ ,p≤Ct⁻^{γ /α}fβ,p. (2.9)

(ii)For anyα∈(1,2),θ∈ [0,1]andp >1,there exists a constantC=C(d, p, θ ) >0such that for allf∈H^θ_p,

Ttf −fp≤Ct^θ/αfθ,p. (2.10)

Proof. (i) Letf ∈C₀^∞(R^d). For anyk, m∈N, by the scaling property, we have

∇^k⁺^mTtf (x)=t⁻^(d⁺^k)/α

R^d

∇^kp1

t⁻^1/α(x−z)

∇^mf (z)dz.

Hence,

∇^k⁺^mTtf

p≤t⁻^k/α∇^mf

p

R^d

∇^kp1(x)dx.

SinceC₀^∞(R^d)is dense inH^mp, we further have for anyf∈H^mp, ∇^k⁺^mTtf

p≤t⁻^k/αfm,p

R^d

∇^kp₁(x)dx.

On the other hand, it is clear that Ttfp≤ fp.

By Theorem2.1, we obtain (2.9).

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(ii) First, we assume thatf ∈H¹p. By (2.5), we have for Lebesgue almost allx∈R^d, Ttf (x)−f (x)≤

R^d

f (x+y)−f (x)·pt(y)dy

≤C

R^d

M|∇f|(x+y)+M|∇f|(x)

· |y| ·p_t(y)dy, and so, by (2.6) and the scaling property,

Ttf−fp≤CM|∇f|

p

R^d|y| ·p_t(y)dy≤C∇fpE|L_t| =Ct^1/α∇fpE|L₁|.

Estimate (2.10) follows by (iii) of Proposition2.4and Theorem2.1again.

We also need the following simple result for proving the uniqueness.

Lemma 2.6. Let(Zt)t≥0be a locally bounded and(Ft)-adapted process and(At)t≥0a continuous real valued non- decreasing(F_t)-adapted process withA0=0.Assume that for any stopping timeηandt≥0,

E|Z_t_∧_η| ≤E _t_∧_η

0

|Z_s|dAs. ThenZ_t=0a.s.for allt≥0.

Proof. By replacingA_t byA_t+t, one may assume thatt→A_t is strictly increasing. Fort≥0, define the stopping time

τ_t:=inf{s≥0:A_s≥t}.

It is clear thatτt is the inverse oft→At. FixT >0. By the assumption and the change of variables, we have E|Z_T_∧_τ_t| ≤E

_T_∧_τ_t

0 |Z_s|dAs≤E _τ_t

0 |Z_T_∧_s|dAs= _t

0 E|Z_T_∧_τ_s|ds.

By Gronwall’s lemma, we obtainZ_T_∧_τ_t =0. Lettingt→ ∞gives the conclusion.

3. Krylov’s estimates for purely discontinuous semimartingales

Let(L_t)_t_≥0be a symmetricα-stable process. The associated Poisson random measure is defined by N

(0, t] ×U :=

s∈(0,t]

1U(L_s−L_s₋), U∈B

R^d\ {0}

, t >0.

The compensated Poisson random measure is given byN ((0, t˜ ] ×U )=N ((0, t] ×U )−t ν(U ). By Lévy–Itô’s de- composition, one may write (cf. [23])

L_t= _t

0

|x|≤1

xN (ds,˜ dx)+ _t

0

|x|>1

xN (ds,dx). (3.1)

LetX_ta purely discontinuous semimartingale with the form X_t=X₀+

_t

0

ξ_sds+L_t, (3.2)

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whereX₀∈F0and(ξ_t)_t_≥₀is a measurable and(Ft)-adaptedR^d-valued process. Letube a bounded smooth function onR₊×R^d. By Itô’s formula (cf. [3]), we have

u_t(X_t)=u₀(X₀)+ _t

0

[∂_su_s+L0u_s](X_s)+ξ_sⁱ∂_iu_s(X_s) ds +

_t

0

R^d\{0} u_s(X_s₋+y)−u_s(X_s₋)N (ds,˜ dy).

In this section, we prove two estimates of Krylov’s type for the aboveX_t. Let us first prove the following simple Krylov’s estimate, which will be used in Section4to prove the existence of weak solutions for SDE (1.1) with singular driftb.

Theorem 3.1. Suppose thatα∈(1,2),p >_α₋^d₁andq >_p(α₋^pα₁₎₋_d.Then,for anyT0>0,there exist a constantC= C(T0, d, α, p, q) >0such that for any(Ft)-stopping timeτ,and0≤S≤T ≤T0,and allf∈L^q([S, T];L^p(R^d)),

E _T_∧_τ

S∧τ

f_s(X_s)dsFS

≤C

1+E _T_∧_τ

S∧τ

|ξ_s|dsFS

fL^q([S,T];L^p(R^d)). (3.3)

Proof. Let us first assume thatf ∈C₀^∞(R₊×R^d)and define u_t(x)=

_t

0 Tt−sf_s(x)ds,

whereTt is defined by (2.8). By Lemma2.5, it is easy to see thatu_t(x)∈C^∞(R₊×R^d)and solves the following PIDE:

∂_tu_t(x)=L0u_t(x)+f_t(x).

Choosingγ∈(1+^d_p, α−^α_q), by (2.9) and Hölder’s inequality, we have u_tγ ,p≤

t 0

Tt−sf_sγ ,pds≤C t

0

(t−s)⁻^{γ /α}f_spds

≤C t

0

(t−s)⁻^q^∗^{γ /α}ds 1/q^∗

fL^q(R+;L^p)≤CfL^q(R+;L^p), (3.4) whereq^∗=q/(q−1).

FixT₀>0 and an (Ft)-stopping timeτ. Using Itô’s formula foru_T₀₋_t(X_t)and by Doob’s optimal theorem, we have

E

u_T₀₋_T_∧_τ(X_T_∧_τ)|F_S

−u_T₀₋_S_∧_τ(X_S_∧_τ)

=E _T_∧_τ

S∧τ

[∂_su_T₀₋_s+L0u_T₀₋_s](X_s)+ξ_sⁱ∂_iu_T₀₋_s(X_s) dsFS

≤E _T_∧_τ

S∧τ

−f_s(X_s)+ |ξ_s| · |∇u_T₀₋_s|(X_s) dsFS

, which yields by (3.4) and (2.1) that

E _T_∧_τ

S∧τ

f_s(X_s)dsFS

≤2 sup

s,x

u_s(x)+sup

s,x |∇u_s|(x)·E _T_∧_τ

S∧τ |ξ_s|dsFS

≤CfL^q(R+;L^p)

1+E

T∧τ S∧τ

|ξ_s|dsF_S

,

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where we have used p(γ −1) > d. By a standard density argument, we obtain (3.3) for general f ∈L^q([0, T];

L^p(R^d)).

In one-dimensional case, as in [19], we even have:

Theorem 3.2. LetX_t take the following form:

X_t=X₀+ _t

0

ξ_sds+ _t

0

h_sdLs,

whereh_s is a bounded predictable process.Suppose thatα∈(1,2)andp >_α₋¹₁.Then,for anyT₀>0,there exist a constantC=C(T0, α, p, q) >0such that for any(Ft)-stopping timeτ,and0≤S≤T ≤T0,and allf ∈L^p(R^d),

E T∧τ

S∧τ |h_s|^αf (X_s)dsFS

≤C

1+E T∧τ

S∧τ

|ξ_s| + |h_s|^α dsFS

fp. (3.5)

Proof. FixT₀>0. Forf ∈C₀^∞(R^d), let us define uT₀(x):=

T₀ 0

TT₀−sf (x)ds.

It is easy to see that

L0u_T₀(x)=TT₀f (x)−f (x). (3.6)

Using Itô’s formula foru_T₀(X_t)(cf. [4], Proposition 2.1), one finds that E

u_T₀(X_T_∧_τ)|FS

=u_T₀(X_S_∧_τ)+E _T_∧_τ

S∧τ

|h_s|^αL0u_T₀(X_s)+ξ_su_T

0(X_s) dsFS

, which together with (3.6) yields that

T∧τ S∧τ

|hs|^αf (Xs)dsFS

≤2uT₀_∞+u_T

0

∞E T∧τ

S∧τ

ξsdsFS

+ TT₀f_∞E T∧τ

S∧τ

|hs|^αdsFS

≤C

1+E _T_∧_τ

S∧τ

|ξ_s| + |h_s|^α dsFS

fp,

where we have used p(α−1) > d, (2.9) and (2.1). By a standard density argument, we obtain (3.5) for general

f ∈L^p(R^d).

In the above two theorems, the requirement ofp > _α^d₋₁ is too strong to prove Theorem1.1. It is clear that this is caused by directly controlling the∞-norm of∇u_s(x)by Sobolev embedding theorem. In what follows, we shall relax it top >^d_α. The price to pay is that we need to assume thatξ_t is a bounded (Ft)-adapted process. Nevertheless, Theorem3.1can be used to prove the existence of weak solutions for SDE (1.1) with globally integrable drift.

We now start by solving the following semi-linear PIDE:

∂_tu=L0u+κ|∇u| +f, u₀≡0, t≥0, (3.7)

whereκ >0,L0is the generator of Lévy process(L_t)_t_≥₀given by (1.11), andf is a locally integrable function on R+×R^d.

We first give the following definition of generalized solutions to PIDE (3.7).

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Definition 3.3. Forp≥1,a functionu∈C([0,∞);H¹p)is called a generalized solution of(3.7),if for all test function ϕ∈C₀^∞([0,∞)×R^d),it holds that

− _∞

0

R^du ∂tϕ= _∞

0

R^duL^∗0ϕ+ _∞

0

R^d

κ|∇u| +f ϕ, whereL^∗₀is the adjoint operator ofL0given by

L^∗₀ϕ(x)=

R^d

ϕ(x−z)−ϕ(x)+1_|z|≤1zⁱ∂_iϕ(x) ν(dz).

Remark 3.4. If we extenduandf toRby settingu_t=f_t≡0fort≤0,then for allϕ∈C₀^∞(R^d⁺¹),

−

R^d⁺¹u ∂_tϕ=

R^d⁺¹uL^∗0ϕ+

R^d⁺¹

κ|∇u| +f ϕ.

Since the Lévy measureνis symmetric,L^∗₀is in fact the same asL0.

The following proposition is now standard. For the reader’s convenience, a proof is provided here.

Proposition 3.5. Forp≥1,letu∈C([0,∞);H¹p)and f ∈L¹_loc([0,∞);L^p(R^d)).The following three statements are equivalent:

(i) uis a generalized solution of(3.7).

(ii) For anyφ∈C_b^∞(R^d),it holds that for allt≥0,

R^du_tφ= _t

0

R^du_sL^∗₀φ+ _t

0

R^d

κ|∇u_s| +f_s φ.

(iii) usatisfies the following integral equation:

u_t(x)= _t

0

Tt−s

κ|∇u_s| +f_s

(x)ds, ∀t≥0.

Proof. For simplicity of notation, we writeF_s(x):=κ|∇u_s| +f_s. By the assumptions, we have F∈L¹_loc

[0,∞);L^p R^d

.

(i)⇒(ii). Letφ∈C^∞_b (R^d). Letρ∈C₀^∞(R)with support in[0,1]and₁

0ρ(r)dr=1. Fixt >0 and define for ε >0,

χε(s):=

_∞

(s−t )/ε

ρ(r)dr.

It is easy to see thatχ_ε∈C₀^∞([0,∞))converges to 1_[0,t]for anys≥0. Letζ_R(x)∈C₀^∞(R^d)converge to 1 point- wisely. Takingϕs(x):=χε(s)ζR(x)φ(x)in the definition of generalized solutions, we have

ε⁻¹ _∞

0

R^duρ

(· −t )/ε ζRφ=

_∞

0

R^dχεuL^∗0(ζRφ)+ _∞

0

R^dF χεζRφ.

Lettingε→0 andR→ ∞and using the dominated convergence theorem, we immediately have (ii).

(ii)⇒(i) and (iii). Fixt >0. By the chain rule, one sees that for anyh∈C¹([0, t]),

R^du_th_tφ= _t

0

R^du_sh_sφ+ _t

0

R^du_sh_sL^∗0φ+ _t

0

R^dF_sh_sφ.

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By the linearity and a density argument, we obtain (i) and also have for allϕ∈C¹([0, t];C_b^∞(R^d)),

R^du_tϕ_t= t

0

R^du_s∂_sϕ_s+ t

0

R^du_sL^∗₀ϕ_s+ t

0

R^dF_sϕ_s.

Now for anyφ∈C_b^∞(R^d), takingϕ_s(x)=Tt−sφ(x)∈C¹([0, t];C_b^∞(R^d))in the above equality, we obtain

R^du_tφ= − _t

0

R^du_sL0ϕ_s+ _t

0

R^du_sL^∗₀ϕ_s+ _t

0

R^dF_sTt−sφ= _t

0

R^dTt−sF_sφ, where we have used the symmetry ofTt. Thus, (iii) follows by the arbitrariness ofφ.

(iii)⇒(ii). By Fubini’s theorem, we have _t

0

R^du_sL^∗0φds= _t

0

R^d

s

0 Ts−rF_rdr

L^∗0φds= _t

0

R^d

t

r L^∗0Ts−rφds

F_rdr

= _t

0

R^d(Tt−rφ−φ)F_rdr=

R^du_tφ− _t

0

R^dF_rφdr.

(ii) now follows.

We have the following existence-uniqueness result about the generalized solution of Eq. (3.7).

Theorem 3.6. Forp >1,α∈(1,2),γ∈ [1, α)andq >_α₋^α_γ,assume thatf ∈L^q_loc(R+;L^p(R^d)).Then,there exists a unique generalized solutionu∈C([0,∞);H^γp)to PIDE(3.7).Moreover,

utγ ,p≤CtfL^q([0,t];L^p), ∀t≥0, (3.8)

whereCt≥0is a continuous increasing function oftwithCt=O(t¹⁻^{γ /α}⁻^1/q)ast→0.

Proof. Letu⁽⁰⁾≡0. Forn∈N, defineu⁽ⁿ⁾recursively by u⁽ⁿ⁾_t (x)=

t 0

Tt−s

κ∇u⁽ⁿ_s ⁻¹⁾+fs

(x)ds, ∀t≥0. (3.9)

By (i) of Lemma2.5and Hölder’s inequality, we have forq >_α₋^α_γ, u⁽ⁿ⁾_t

γ ,p≤C _t

0

(t−s)⁻^{γ /α}

κ∇u⁽ⁿ_s ⁻¹⁾

p+ f_sp

ds

≤C t

0

(t−s)⁻^{qγ /((q}⁻^1)α)ds

(q−1)/q t 0

κ^q∇u⁽ⁿ_s ⁻¹⁾^q

p+ fs^qp

ds 1/q

, which yields that

u⁽ⁿ⁾_t ^q

γ ,p≤Ct^(q(α⁻^{γ )}⁻^α)/α t

0

u⁽ⁿ_s ⁻¹⁾^q

1,pds+ _t

0 f_s^qpds

≤Ct^(q(α⁻^{γ )}⁻^α)/α t

0

u⁽ⁿ_s ⁻¹⁾^q

γ ,pds+ _t

0 f_s^qpds

. By Gronwall’s inequality, we obtain that for allt≥0,

sup

n∈N

u⁽ⁿ⁾_t ^q

γ ,p≤C_t _t

0 f_s^qpds. (3.10)