A Sobolev space theory for the time-fractional stochastic partial differential equations driven by Levy processes
aa r X i v : . [ m a t h . A P ] J un A SOBOLEV SPACE THEORY FOR THE TIME-FRACTIONALSTOCHASTIC PARTIAL DIFFERENTIAL EQUATIONS DRIVENBY L´EVY PROCESSES
KYEONG-HUN KIM AND DAEHAN PARK
Abstract.
We present an L p -theory ( p ≥
2) for time-fractional stochasticpartial differential equations driven by L´evy processes of the type ∂ αt u = d X i,j =1 a ij u x i x j + f + ∞ X k =1 ∂ βt Z t ( d X i =1 µ ik u x i + g k ) dZ ks given with nonzero intial data. Here ∂ αt and ∂ βt are the Caputo fractionalderivatives, α ∈ (0 , , β ∈ (0 , α +1 /p ), and { Z kt : k = 1 , , · · · } is a sequence ofindependent L´evy processes. The coefficients are random functions dependingon ( t, x ). We prove the uniqueness and existence results in Sobolev spaces,and obtain the maximal regularity of the solution. introduction Let { W kt : k ∈ , , · · · } and { Z kt : k = 1 , , · · · } be sequences of independentone dimensional Brownian motions and d -dimensional L´evy processes respectively.In this article we present an L p -theory ( p ≥
2) for the time-fractional stochasticpartial differential equation (SPDE) defined on R d : ∂ αt u = a ij u x i x j + b i u x i + cu + f + ∂ β t Z t (cid:0) µ ik u x i + ν k u + g k (cid:1) dW ks + ∂ β t Z t (cid:0) ¯ µ irk u x i + ¯ ν rk u + h rk (cid:1) dZ rks , t > ,u (0) = u , ∂ t u (0)1 α> = v . (1.1)Here ∂ αt , ∂ β t , ∂ β t are the Caputo fractional derivatives, α ∈ (0 , , β < α + 1 / , β < α + 1 /p, (1.2)and Einstein’s summation convention is used in (1.1) for the repeated indices i, j ∈ { , , · · · , d } , r ∈ { , , · · · , d } and k = 1 , , · · · . The coefficients dependon ( ω, t, x ) and initial data depend on ( ω, x ). The conditions on β and β in (1.2)are necessary and will be discussed later (see Remark 2.9).Equation (1.1) is understood by its integral form (see Defintion 2.7), and thistype of SPDE naturally arises, for instance, when one describes the random effects Mathematics Subject Classification.
Key words and phrases.
Stochastic partial differential equations, Time-fractional derivatives,L´evy processes, Maximal L p -regularity.The authors were supported by the National Research Foundation of Korea(NRF) grant fundedby the Korea government(MSIT) (No. NRF-2020R1A2C1A01003354). on transport of particles subject to sticking and trapping or particles in mediumwith thermal memory. See [4] for a detailed derivation of such equations. Note thatif α = β = β = 1, then (1.1) becomes the classical second-order parabolic SPDE.This article is a natural continuation of [10], where a Sobolev space theory isintroduced for the equation driven only by Wiener processes. We generalize theresult of [10] in the following two aspects:(i) The equation is more general, that is, it is driven by L´evy processes.(ii) Non-zero initial data condition is imposed.Both (i) and (ii) cause new technical difficulties, and we resolve these based onLittlewood-Paley theory in harmonic analysis together with some stochastic calcu-lus related to L´evy processes.To the best of our knowledge, the regularity result for the time fractional SPDEwas firstly introduced in [5, 6, 7]. The authors in [5, 6, 7] applied H ∞ -functionalcalculus technique to obtain a maximal regularity for the mild solution to theintegral equation U ( t ) + Z t ( t − s ) α − AU ( s ) ds = Z t ( t − s ) β − G ( s ) dW s , (1.3)where W t is a Brownian motion, and A is the generator of a bounded analyticsemigroup and is assumed to admit a bounded H ∞ -calculus on L p . Quite recently,non-linear SPDE of type (1.3) with non-linear term A ( U ) in place of AU was studiedin [14] in the Gelfand triple setting with the restriction α < β < ( α + 1 / ∨ L -theory was introduced in [4] for the equationdriven only by Wiener processes, and the result of [4] was extended in [10] for p ≥ t, x ) together with non-zero initial data. Wedo not impose unnecessary algebraic conditions on α, β , β , and most importantlyour equation is driven by more general processes, that is L´evy processes. Howeverour results do not cover those in [5, 6, 7, 14] because the operator A can belong toquite large class of operators.For the deterministic counterpart of our result we refer e.g to [8, 11, 22]. Wealso refer to [3, 12, 13] for the classical case α = β = β = 1.This article is organized as follows. In Section 2 we introduce stochastic calculusrelated to L´evy processes, preliminary results on the fractional calculus, and someproperties of the solution space, and we present our main result, Theorem 2.13. InSection 3 we use Littlewood-Paley theory to obtain key estimates for solutions. InSection 4 we prove our main result.Finally we introduce notation used in this article. We use “:=” to denote adefinition. As usual, R d stands for the d -dimensional Euclidean space of points x = ( x , . . . , x d ). N denotes the set of natural numbers and N + = { } ∪ N . For i = 1 , , · , d and multi-index a = ( a , · · · , a d ), where a i ∈ N + , we set D i u = u x i = ∂∂x i u, D a u = D a . . . D a d d u, | a | = a + · · · + a d . We also use D mx or D m to denote arbitarry m -th order partial derivative withrespect to x . For a, b ∈ R , a ∨ b := max( a, b ) and a + := a ∨
0. By F ( f ) or ˆ f we denote IME-FRACTIONAL SPDES DRIVEN BY L´EVY PROCESSES 3 the Fourier transform of f . C ∞ c ( R d ) denotes the set of infinitely differentiablefunctions with compact support in R d , S ( R d ) is the class of Schwartz functionson R d , and D = D ( R d ) is the class of tempered distributions. For p ∈ [1 , ∞ ], ameasure space ( X, A , µ ), a normed vector space B with norm k · k B , L p ( X, A ; B )is the set of B -valued ¯ A -measurable functions f satisfying k f k L p ( X, A ; B ) = (cid:18)Z X k f ( x ) k pB dµ (cid:19) /p , where ¯ A is the completion of A with respect to µ . We say X is a version of Y in B if k X − Y k B = 0. If we write N = N ( a, b, · · · ), this means that the constant N depends only on a, b, · · · . Throughout the article, for functions depending on( ω, t, x ), the argument ω ∈ Ω will be usually omitted.2. main results
First we introduce some definitions and facts related to the fractional calculus.For more detail, see e.g. [1, 16, 18, 19]. For α > ϕ ∈ L ((0 , T )), theRiemann-Liouville fractional integral of order α is defined by I αt ϕ ( t ) := 1Γ( α ) Z t ( t − s ) α − ϕ ( s ) ds, t ≤ T. For any p ∈ [1 , ∞ ], we easily have k I α ϕ k L p ((0 ,T )) ≤ N ( α, p, T ) k ϕ k L p ((0 ,T )) . (2.1)It is also easy to check that if ϕ ∈ L p ((0 , T ); B ) and α > /p then I αt ϕ ( t ) is acontinuous function satisfying I αt ϕ (0) = 0. In particular if ϕ is bounded, then I αt ϕ ( t ) is continuous for any α >
0. The similar statements hold if ϕ ( t ) is an L p ( R d )-valued (or Banach space-valued) function.Let n be the integer such that n − ≤ α < n . If ϕ is ( n − ddt ) n − I n − αt ϕ is absolutely continuous on [0 , T ], then the Riemann-Liouvillefractional derivative D αt ϕ and the Caputo fractional derivative ∂ αt ϕ are defined by D αt ϕ := (cid:18) ddt (cid:19) n (cid:0) I n − αt ϕ (cid:1) ,∂ αt ϕ := D αt ϕ ( t ) − n − X k =0 t k k ! ϕ ( k ) (0) ! . (2.2)One can easily check that for any α, β ≥ I α + β ϕ ( t ) = I α I β ϕ ( t ) , D α D β ϕ = D α + β ϕ, (2.3)and D α I β ϕ = ( D α − β ϕ if α > βI β − α ϕ if α ≤ β . (2.4)For p > γ ∈ R , let H γp = H γp ( R d ) denote the class of all tempereddistributions u on R d such that k u k H γp := k (1 − ∆) γ/ u k L p < ∞ , (2.5)where (1 − ∆) γ/ u = F − (cid:16) (1 + | ξ | ) γ/ F ( u ) (cid:17) . KYEONG-HUN KIM AND DAEHAN PARK
The action of u on φ ∈ S ( R d ), which is denoted by ( u, φ ), is defined by( u, φ ) := ((1 − ∆) γ/ u, (1 − ∆) − γ/ φ ) . (2.6)It is well-known that if γ = 0 , , , · · · , then H γp = W γp := { u : D a u ∈ L p ( R d ) , | a | ≤ γ } , H − γp = ( H γp/ ( p − ) ∗ . Let l denote the set of all sequences a = ( a , a , · · · ) such that | a | l := ∞ X k =1 | a k | ! / < ∞ . By H γp ( l ) = H γp ( R d , l ) we denote the class of all l -valued tempered distributions v = ( v , v , · · · ) on R d such that k v k H γp ( l ) := k| (1 − ∆) γ/ v | l k L p < ∞ . Also we write h = ( h , . . . , h d ) ∈ H γp ( l , d ) if k h k H γp ( l ,d ) := d X r =1 k h r k H γp ( l ) < ∞ . Let (Ω , F , P ) be a complete probability space and { F t , t ≥ } be an increasingfiltration of σ -fields F t ⊂ F , each of which contains all ( F , P )-null sets. Weassume that an independent family of one-dimensional Wiener processes { W kt } k ∈ N and d -dimensional L´evy processes { Z kt } k ∈ N relative to the filtration { F t , t ≥ } are given on Ω. By P we denote the predictable σ -field generated by F t , i.e., P isthe smallest σ -field containing sets of the type A × ( s, t ], where s < t and A ∈ F s .For p > γ ∈ R denote H γp ( T ) := L p (cid:0) Ω × (0 , T ) , P ; H γp (cid:1) , L p ( T ) = H p ( T ) , H γp ( T, l ) := L p (cid:0) Ω × (0 , T ) , P ; H γp ( l ) (cid:1) , L p ( T, l ) = H p ( T, l ) , H γp ( T, l , d ) := L p (cid:0) Ω × (0 , T ) , P ; H γp ( l , d ) (cid:1) , L p ( T, l , d ) = H p ( T, l , d ) . Also, for l -valued functions h , we write h ∈ H ∞ c ( T, l ) if h k = 0 for all large k , andeach h k is of the type h k ( t, x ) = n X i =1 ( τ i − ,τ i ] ( t ) g ik ( x ) , where τ i are bounded stopping times, τ i ≤ τ i +1 , and g ik ∈ C ∞ c ( R d ). The space H ∞ c ( T, l , d ) is defined similarly. By [13, Theorem 3.10], H ∞ c ( T, l ) is dense in H γp ( T, l ). Similarly, the space L c of the functions g of the form g ( ω, x ) = n X i =1 A i ( ω ) g i ( x ) , A i ∈ F , g i ∈ C ∞ c ( R d ) (2.7)is dense in L p (Ω , F ; H γp ).For t ≥ A ∈ B ( R d \ { } ), denote N k ( t, A ) := { ≤ s ≤ t : ∆ Z ks := Z ks − Z ks − ∈ A } ˜ N k ( t, A ) := N k ( t, A ) − tν k ( A ) , IME-FRACTIONAL SPDES DRIVEN BY L´EVY PROCESSES 5 where ν k ( A ) := E N k (1 , A ) is the L´evy measure of Z kt . Set m p ( k ) := (cid:18)Z R d | z | p ν k ( dz ) (cid:19) /p . If m ( k ) < ∞ , then by L´evy-Itˆo decomposition, there exist a vector a k =( a k , . . . , a d k ), a non-negative definite d × d matrix b k , and d -dimensional Wienerprocess ˜ W kt such that Z kt = a k t + b k ˜ W kt + Z R d z ˜ N k ( t, dz ) =: a k t + b k ˜ W kt + ˜ Z kt (i.e. Z rkt = a rk t + P d l =1 b rlk ˜ W lt + R R d z r ˜ N k ( t, dz )).In this article we assume the following. Assumption 2.1. (i) p ∈ [2 , ∞ ) and m p := sup k (cid:0) m ( k ) ∨ m p ( k ) (cid:1) < ∞ . (ii) For each k , Z kt = ( Z k , . . . , Z d k ) is a pure jump process with no diffusion anddrift parts (i.e. a k = 0 and b k = 0). Remark 2.2.
One can also consider equation (1.1) with general Z kt , without As-sumption 2.1 (ii), by rewriting it into the form of the equation driven by a set ofBrownian motions { W kt } ∪ { ˜ W kt } and L´evy processes ˜ Z kt . See (2.1) in [12] for thedetail.Due to the assumption m ( k ) < ∞ , Z kt is a square integrable martingale, and thestochastic integral against Z rkt ( r = 1 , · · · , d ) can be easily understood as follows.For functions h of the type h = P mi =1 a i ( τ i ,τ i +1 ] ( t ), where τ i are bounded stoppingtimes, τ i ≤ τ i +1 , and a i are bounded F τ i -measurable random variables, we define(Λ h ) t := Z t h dZ rks := m X i =0 a i ( Z rkτ i +1 ∧ t − Z rkτ i ∧ t ) . Then Λ h becomes a square integrable martingale with c´adl´ag sample paths, andone can easily check E sup t ≤ T | (Λ h ) t | ≤ c ( k ) k h k L (Ω × [0 ,T ]) . Therefore, the stochastic integral can be continuously extended to all h ∈ L (Ω × [0 , T ] , P ; R ), and R t h dZ rkt becomes a square integrable martingale with c´adl´agsample paths. Furthermore, if h = h in L (Ω × [0 , T ] , P ; R ), then Z t h dZ rkt = Z t h dZ rkt , ∀ t ≤ T ( a.s. ) . This is because both are c´adl´ag processes.
Remark 2.3.
For any h = ( h , . . . , h d ) ∈ L (Ω × [0 , T ] , P ; R d ) with a predictableversion ¯ h , M kt = Z t h · dZ ks := d X r =1 Z t h r dZ rkt = d X r =1 Z t ¯ h r dZ rkt KYEONG-HUN KIM AND DAEHAN PARK is a square integrable martingale with the quadratic variation (see e.g. [17])[ M k ] t = d X r,l =1 Z t Z R d z r z l ¯ h rs ¯ h ls N k ( ds, dz ) . (2.8)By [3, Lemma 2.5] (or see [15, Lemma 1]) we have E ∞ X k =1 Z T Z R d | z | | ¯ h k ( s ) | N k ( ds, dz ) ! p/ ≤ N ( p, m p ) E Z T ∞ X k =1 | h k ( s ) | ds ! p/ + Z T ∞ X k =1 | h k ( s ) | p ds , (2.9)where | h k ( s ) | = P d r =1 | h rk ( s ) | . Since ∞ X k =1 | a k | p ≤ ∞ X k =1 | a k | ! p/ , (cid:18)Z t | h | ds (cid:19) p/ ≤ t p/ − Z t | h | p/ ds, (recall that p ≥ E ∞ X k =1 Z T Z R d | z | | ¯ h k ( s ) | N k ( ds, dz ) ! p/ ≤ N d X r =1 E k h r k pL p ([0 ,T ]; l ) , (2.10)where N = N ( p, m p , d , T ). Therefore by the Burkholder-Davis-Gundy inequality,(2.8), and (2.10), E " sup s ≤ t (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∞ X k =1 M ks (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) p ≤ N d X r =1 E k h r k pL p ([0 ,T ]; l ) . (2.11) Remark 2.4. If g ∈ H γp ( T, l ), and h ∈ H γp ( T, l , d ), then the series ∞ X k =1 Z t ( g k ( s, · ) , φ ) dW ks , ∞ X k =1 Z t ( h k ( s, · ) , φ ) · dZ ks are well-defined due to Assumption 2.1 and Remark 2.3. Indeed, using Remark 2.3one can show (see [13, Remark 3.2] for detail) d X r =1 ∞ X k =1 Z T ( h rk , φ ) ds ≤ N ( φ, m p , d , T ) k h k p H γp ( T,l ,d ) . Therefore, the series ∞ X k =1 Z t ( h k ( s, · ) , φ ) · dZ ks converges in probability uniformly on [0 , T ], and it is a square integrable martingaleon [0 , T ], which is c´adl´ag. The same argument holds for P ∞ k =1 R t ( g k ( s, · ) , φ ) dW ks ,which is a continuous martingale on [0 , T ].We say that X t = Y t for almost all t ≤ T at once if P (cid:0) { ω : X t ( ω ) = Y t ( ω ) , a.e. t ≤ T } (cid:1) = 1 , and X t = Y t for all t ≤ T at once if P (cid:0) { ω : X t ( ω ) = Y t ( ω ) , ∀ t ≤ T } (cid:1) = 1 . IME-FRACTIONAL SPDES DRIVEN BY L´EVY PROCESSES 7
Lemma 2.5.
Let X kt = W kt or X kt = Z rkt , r ∈ { , · · · , d } , and h ∈ L (Ω × [0 , T ] , P ; l ) .(i) Let α > and h = ( h , h , · · · ) ∈ L ( T, l ) . Then I α ∞ X k =1 Z · h k ( s ) dX ks ! ( t ) = ∞ X k =1 I α (cid:18)Z · h k ( s ) dX ks (cid:19) ( t ) for almost all t ≤ T at once.(ii) Under the assumptions in (i), ∞ X k =1 I α (cid:18)Z · h k ( s ) dX ks (cid:19) ( t ) = 1 α Γ( α ) ∞ X k =1 Z t ( t − s ) α h k ( s ) dX ks a.e. on Ω × [0 , T ] .(iii) If α < / , then D αt ∞ X k =1 Z · h k ( s ) dX ks ! ( t ) = ∞ X k =1 D αt (cid:18)Z · h k ( s ) dX ks (cid:19) ( t )= 1Γ( α ) ∞ X k =1 Z t ( t − s ) − α h k ( s ) dX ks a.e. on Ω × [0 , T ] .Proof. See Lemmas 3.1 and 3.3 in [4] for (i) and (iii). Actually, the case X kt = W kt is proved in [4], and the same argument works for the general case for X kt = Z rkt .(ii) easily follows from the Stochastic Fubuni theorem (see [17, Chapter IV,Theorem 65]). (cid:3) Next, to state our condition on the initial data, we introduce the Besov space.We fix Ψ ∈ S ( R d ) such that its Fourier transform ˆΨ( ξ ) has support in a strip { ξ ∈ R d | ≤ | ξ | ≤ } , ˆΨ( ξ ) > < | ξ | <
2, and X j ∈ Z ˆΨ(2 − j ξ ) = 1 for ξ = 0 . Define ˆΨ j ( ξ ) = ˆΨ(2 − j ξ ) , j = ± , ± , . . . , ˆΨ ( ξ ) = 1 − ∞ X j =1 ˆΨ j ( ξ ) . For a distributions (or a function) f , we denote f j := Ψ j ∗ f .It is known that if u ∈ H γp , then k u k H γp ∼ k u k L p + k ( ∞ X j =1 γj | u j | ) / k L p . (2.12)For 1 < p < ∞ , s ∈ R , we define Besov space B sp = B sp ( R d ) as the collection of alltempered distributions u such that k u k B sp := k u k L p + ∞ X j =1 spj k u j k pL p /p < ∞ . KYEONG-HUN KIM AND DAEHAN PARK
Remark 2.6.
It is well known (e.g. [2, 20]) that C ∞ c ( R d ) is dense in B sp , theinclusion B s p ⊂ B s p holds for s ≤ s , and H sp ⊂ B sp , ≤ p < ∞ . Furthermore, ( − ∆) γ is a bounded operator from B s + γp to B sp , and (1 − ∆) γ/ is anisometry from B s + γp to B sp and from H s + γp to H sp .Fix a small constant κ >
0. Set c := 1 β > / (2 β − α + κ β =1 / , ¯ c := 1 β > /p (2 β − /p ) α + κ β =1 /p . (2.13)Note that 0 ≤ c , ¯ c < c = 0 if β < /
2, and ¯ c = 0 if β < /p . Also set U γ +2 p = L p (Ω , F ; B γ +(2 − /αp ) + p ) , and V γ +2 p = ( L p (Ω , F ; B γ +2 − /α − /αpp ) α > /pL p (Ω , F ; B γ +2 − /αp ) 1 < α ≤ /p. (2.14)Note that if α > /p , then 2 − /α − /αp >
0, and 2 − /α > α > Definition 2.7.
Let p ≥ γ ∈ R . We write u ∈ H γ +2 p ( T ) if u ∈ H γ +2 p ( T )and there exist f ∈ H γp ( T ) , g ∈ H γ + c p ( T, l ) , h ∈ H γ +¯ c p ( T, l , d ) , u ∈ U γ +2 p , and v ∈ V γ +2 p such that u satisfies ∂ αt u ( t, x ) = f ( t, x ) + ∂ β t ∞ X k =1 Z t g k ( s, x ) dW ks + ∂ β t ∞ X k =1 Z t h k ( s, x ) · dZ ks , t ∈ (0 , T ] u (0 , · ) = u , u ′ (0 , · )1 α> = v (2.15)in the sense of distributions. In other words, for any φ ∈ S ( R d ), the equality( u ( t ) − u − tv α> , φ ) = I αt ( f, φ ) + ∞ X k =1 I α − β t Z t ( g k ( s ) , φ ) dW ks + ∞ X k =1 I α − β t Z t ( h k ( s ) , φ ) · dZ ks (2.16)holds a.e. on Ω × [0 , T ], (here I α − β i t := D β i − αt if β i > α ). In this case, we write D u = f, S c u = g, S d u = h. The norm in H γ +2 p ( T ) is defined by k u k H γ +2 p ( T ) = k u k H γ +2 p ( T ) + k D u k H γp ( T ) + k S c u k H γ + c p ( T,l ) + k S d u k H γ +¯ c p ( T,l ,d ) + k u (0) k pU γ +2 p + 1 α> k u t (0) k pV γ +2 p . Remark 2.8.
Note that, since β < α + 1 / β < α + 1 /p , the right hand sideof (2.16) makes sense due to Lemma 2.5. IME-FRACTIONAL SPDES DRIVEN BY L´EVY PROCESSES 9
Remark 2.9. If β > α + 1 /p then (2.16) does not make sense. For simplicity, let u = v = 0 , f = 0 and g = 0. Then taking I β − αt to (2.16) we get I β − αt ( u ( t ) , φ ) = ∞ X k =1 Z t ( h k ( s ) , φ ) · dZ ks . Since ( u ( t ) , φ ) ∈ L p ([0 , T ]) (a.s.) and β − α > /p , the left hand side above isa continuous process. However, the right hand side is only c´adl´ag process. Thenecessity of condition β < α + 1 / Proposition 2.10.
Let u ∈ H γ +2 p ( T ) , f ∈ H γp ( T ) , g ∈ H γ + c p ( T, l ) , h ∈ H γ +¯ c p ( T, l , d ) , u ∈ U γ +2 p , and v ∈ V γ +2 p .Then the following are equivalent;(i) u ∈ H γ +2 p ( T ) and (2.15) holds in the sense of Definition 2.7.(ii) For any constant Λ such that Λ ≥ max( α, β , β ) and Λ > p ,I Λ t u has an H γp -valued c´adl´ag version in H γp ( T ) , still denoted by I Λ − αt u , suchthat for any φ ∈ S ( R d ) , the equality ( I Λ − αt u − I Λ − αt ( u + tv α> ) , φ )= I Λ t ( f, φ ) + ∞ X k =1 I Λ − β t Z t ( g k ( s, · ) , φ ) dW ks + ∞ X k =1 I Λ − β t Z t ( h k ( s, · ) , φ ) · dZ ks (2.17) holds for all t ∈ [0 , T ] at once.Proof. Considering (1 − ∆) γ/ u in place of u , we may assume γ = 0.1. Suppose (2.17) holds for all t at once. Then by applying D Λ − αt to (2.17) andusing (2.3), (2.16), and Lemma 2.5, we find that (2.15) holds for a.e. ( ω, t ).2. Suppose (2.15) holds a.e. ( ω, t ). Since I Λ − αt ( u + 1 α> tv ) is a continuous L p -valued process, we may assume u = v = 0.Take a nonnegative funtion ζ ∈ C ∞ c ( R d ) with unit integral. For each n > ζ n ( x ) = n − d ζ ( nx ). For any tempered distribution v , define v ( n ) ( x ) := v ∗ ζ n ( x ). Then v ( n ) is infinitely differentiable function with respect to x .Plugging φ = ζ n ( · − x ) in (2.15) and applying I Λ − αt to both sides of (2.15), foreach x we get( I Λ − αt ( u ) ( n ) )( t, x ) = ( I Λ t f ( n ) )( t, x ) + ∞ X k =1 I Λ − β t Z t ( g k ) ( n ) ( s, x ) dW ks + ∞ X k =1 I Λ − β t Z t ( h k ) ( n ) ( s, x ) · dZ ks (2.18)a.e. on Ω × [0 , T ]. Note that since Λ > /p , I Λ t f ( n ) is a continuous L p -valued functions. Also, thestochastic integrals ∞ X k =1 Z t ( g k ) ( n ) ( s, x ) dW ks , ∞ X k =1 Z t ( h k ) ( n ) ( s, x ) · dZ ks are L p -valued c´adl´ag functions, and in particular they are bounded on [0 , T ] (a.s.).Therefore, the right hand side of (2.18) is an L p -valued c´adl´ag process, and con-sequently the left hand side has an L p -valued c´adl´ag version, still denoted by I Λ − αt ( u ) ( n ) .By (2.1) with p = ∞ and (2.11), E sup t ≤ T (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) I Λ − β t ∞ X k =1 Z t ( h k ) ( n ) ( s, · ) · dZ ks (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) pL p ≤ N Z R d E sup t ≤ T (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) I Λ − β t ∞ X k =1 Z t ( h k ) ( n ) ( s, x ) · dZ ks (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) p dx ≤ N Z R d E sup t ≤ T (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∞ X k =1 Z t ( h k ) ( n ) ( s, x ) · dZ ks (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) p dx ≤ N E Z T k h ( n ) ( s, · ) k pL p ( l ,d ) ds. We handle two other terms on the right hand side of (2.18) similarly, and get E sup t ≤ T (cid:13)(cid:13)(cid:13) I Λ − α ( u ) ( n ) ( t, · ) (cid:13)(cid:13)(cid:13) pL p ≤ N (cid:0) k ( f ) ( n ) k p L p ( T ) + k g ( n ) k p L p ( T,l ) + k h ( n ) k p L p ( T,l ,d ) (cid:1) . (2.19)Considering (2.19) corresponding to I Λ − αt ( u ) ( n ) − I Λ − αt ( u ) ( m ) , we find that I Λ − αt ( u ) ( n ) is a Cauchy sequence in L p (Ω; D ([0 , T ]; L p )), where D ([0 , T ]; L p ) is a space of L p -valued c´adl´ag functions. Let w denote the limit in this space. Then since I Λ − αt ( u ) ( n ) → I Λ − α u in L p ( T ), we conclude w = I Λ − αt u a.e on Ω × [0 , T ], and w is an L p -valued c´adl´ag version of I Λ − αt u . This proves that (2.17) holds for all t atonce because both sides are read-valued c´adl´ag processes. (cid:3) Theorem 2.11.
Let p ≥ , γ ∈ R and T ∈ (0 , ∞ ) .(i) For any ν ∈ R , the map (1 − ∆) ν/ : H γ +2 p ( T ) → H γ − ν +2 p ( T ) is an isometry.(ii) H γ +2 p ( T ) is a Banach space.(iii) Let u ∈ H γ +2 p ( T ) with u (0) = u t (0) = 0 . Then for any t ≤ T , k u k p H γp ( t ) ≤ N Z t ( t − s ) θ − (cid:16) k D u k p H γp ( s ) + k S c u k p H γp ( s,l ) + k S d u k p H γp ( s,l ,d ) (cid:17) ds, where θ := min { α, α − β ) + 1 , p ( α − β ) + 2 } , and the constant N depends onlyon α, β , β , d, d , p and T .Proof. (i) This easily follows from the fact that(1 − ∆) γ/ : H µp → H µ − γp , (1 − ∆) γ/ : B µp → B µ − γp is an isometry.(ii) The assertion can be proved by repeating the proof of [10, Theorem 2.1],which is proved for the equation driven only by Wiener processes.(iii) This is introduced in [10, Theorem 2.1] when S d u = 0, and for the generalcase we repeat the proof in [10]. By the result of (i) we may assume γ = 0. IME-FRACTIONAL SPDES DRIVEN BY L´EVY PROCESSES 11
We take notations from the proof of Proposition 2.10. Then, from (2.16), foreach x ∈ R d we get u ( n ) ( t, x ) = I αt f ( n ) ( t, x ) + ∞ X k =1 I α − β t Z t ( g k ) ( n ) ( s, x ) dW ks (2.20)+ ∞ X k =1 I α − β t Z t ( h k ) ( n ) ( s, x ) · dZ ks a.e. on Ω × [0 , T ]. Since u ( n ) → u in L p ( T ), to prove (2.20), it is enough to estimate k u ( n ) k L p ( t ) . For this, we only estimate the last term in (2.20) because other termsare estimated in the proof of [10, Theorem 2.1].Let ¯ h be a predictable version of h . By Burkerholder-Davis-Gundy inequality,(2.10) and Lemma 2.5 (cid:13)(cid:13)(cid:13)(cid:13) I α − β t Z · ( h k ) ( n ) ( s, x ) · dZ ks (cid:13)(cid:13)(cid:13)(cid:13) p L p ( t ) ≤ N Z R d Z t E ∞ X k =1 Z s Z R | z | | ( s − r ) α − β (¯ h k ) ( n ) ( r, x ) | N k ( dr, dz ) ! p/ dsdx ≤ N Z t Z s ( s − r ) p ( α − β ) k h ( n ) ( r ) k p L p ( l ,d ) drds ≤ N Z t ( t − s ) p ( α − β )+1 k h ( s ) k p L p ( s,l ,d ) ds ≤ N Z t ( t − s ) θ − k h ( s ) k p L p ( s,l ,d ) ds. This is what we needed, and (iii) is proved. (cid:3)
Take κ ′ ∈ (0 , r ≥
0, set B r = L ∞ ( R d ) if r = 0 C r − , ( R d ) if r = 1 , , , . . .C r + κ ′ ( R d ) otherwise , where C r + κ ′ ( R d ) and C r − . ( R d ) are H¨older space and Zygmund space respectively.We use B r ( l ) for l -valued analogue. It is known that for u ∈ H γp , and v ∈ H γp ( l ) k au k H γp ≤ N ( d, p, κ ′ , γ ) | a | B | γ | k u k H γp , k bv k H γp ( l ) ≤ N ( d, p, κ ′ , γ ) | b | B | γ | ( l ) k v k H γp ( l ) . (2.21) Assumption 2.12. (i) All the coefficients are
P ⊗ B ( R d )-measurable functions.(ii) The coefficients µ i , ν, ¯ µ ir , ¯ ν r are l -valued functions, where i = 1 , , · · · , d and r = 1 , , · · · , d .(iii) There exists a constant 0 < δ < ω, t, x ) δ | ξ | ≤ a ij ( t, x ) ξ i ξ j ≤ δ − | ξ | , ∀ ξ ∈ R d . (2.22)(iv) The coefficients a ij ( ω, t, x ) is uniformly continuous in ( t, x ), uniformly on Ω.(v) For each ω, t, i, j, r , | a ij ( t, · ) | B | γ | + | b i ( t, · ) | B | γ | + | c ( t, · ) | B | γ | + | µ i ( t, · ) | B | γ + c | ( l ) + | ν ( t, · ) | B | γ + c | ( l ) + | ¯ µ ir ( t, · ) | B | γ +¯ c | ( l ) + | ¯ ν r ( t, · ) | B | γ +¯ c | ( l ) ≤ δ − . (v) µ i = 0 if β ≥ / α/
2, and ¯ µ ir = 0 if β ≥ /p + α/ Here is the main result of this article.
Theorem 2.13.
Let γ ∈ R , p ≥ , and T < ∞ . Suppose Assumption 2.12 holdand α ∈ (0 , , β < α + 1 / , β < α + 1 /p. Then for any u ∈ U α,γ +2 p , v ∈ V α,γ +2 p , f ∈ H γp ( T ) , g ∈ H γ + c ′ p ( T, l ) and h ∈ H γ +¯ c ′ p ( T, l , d ) , equation (1.1) has a unique solution u in the class H γ +2 p ( T ) in thesense of Definition 2.7. Moreover, k u k H γ +2 p ( T ) ≤ N (cid:0) k u k U γ +2 p + 1 α> k v k V γ +2 p + k f k H γp ( T ) + k g k H γ + c p ( T,l ) + k h k H γ +¯ c p ( T,l ,d ) (cid:1) , (2.23) where the constant N depends only on α, β , β , d, d , p, δ, γ, κ , and T . Remark 2.14. If α ∈ (0 ,
1] then Assumption 2.12(iv) can be relaxed and replacedby the uniformly continuity in x , uniformly on Ω × [0 , T ]. Assumption 2.12(iv) isinherited from a result on the deterministic equation, [11, Theorem 2.10]. Howeverif α ∈ (0 ,
1) then the continuity in t can be completely dropped for the deterministicequation (see [8]). 3. Key estimates
In this section we study the convolution operators of the type((1 − ∆) a D bt p ) ∗ f, where a, b ∈ R and p ( t, x ) is the fundamental solution to the time fractional heatequation ∂ αt u = ∆ u . To explain the necessity of such study, let us consider ∂ αt u = ∆ u + ∂ βt Z t h ( s ) dZ t , t > u (0) = 1 α> u t (0) = 0 , where Z t is a L´evy process. It turns out that for the solution u and c ≥ k (1 − ∆) c/ u k p L p ( T ) ≤ N (cid:13)(cid:13)(cid:13) Z t (cid:12)(cid:12)(cid:12)(cid:16) (1 − ∆) c/ D β − αt p (cid:17) ∗ h ( s ) (cid:12)(cid:12)(cid:12) p ds (cid:13)(cid:13)(cid:13) L (Ω × [0 ,T ]; L ( R d )) . Thus, for the estimations of solutions, we need to handle the right hand side of theabove inequality. If non-zero initial condition is given, this also leads to the similarsituation.Now, let 0 < α < p ( t, x ) be the fundamental solution to the equation ∂ αt u = ∆ u, u (0 , x ) = u ( x ) , α> ∂ t u (0 , x ) = 0 . (3.1)That is, p ( t, x ) is the function such that, under appropriate smoothness assumptionon u , u = p ( t, · ) ∗ u is the solution to (3.1). For β < α + , we define q α,β ( t, x ) = ( I α − βt p ( t, x ) α ≥ β,D β − αt p ( t, x ) α < β. Below we list some properties of p and q α,β . Lemma 3.1.
Let < α < and β < α + . IME-FRACTIONAL SPDES DRIVEN BY L´EVY PROCESSES 13 (i) For any t > , and x = 0 , ∂ αt p ( t, x ) = ∆ p ( t, x ) , ∂∂t p ( t, x ) = ∆ q α, ( t, x ) , (3.2) and ∂∂t p ( t, x ) → as t ↓ . Moreover, ∂∂t p ( t, · ) is integrable in R d uniformlyon t ∈ [ ε, T ] for any ε > .(ii) For f ∈ C ∞ c ( R d ) , the convolution Z R d p ( t, x − y ) f ( y ) dy converges to f ( x ) uniformly as t ↓ .(iii) For any m ∈ N + , there exist constants c = c ( α, d, m ) and N = N ( α, d, m ) such that if R := | x | t − α ≥ , then | D mx p ( t, x ) | ≤ N t − α ( d + m )2 exp {− c | x | − α t − α − α } , (3.3) and if R < , then | D mx p ( t, x ) | ≤ N | x | − d − m ( R + R log R d =2 ,m =0 + R / d =1 ,m =0 ) . (3.4) (iv) It holds that F{ D σt q α,β ( t, · ) } = t α − β − σ E α, α − β − σ ( − t α | ξ | ) , (3.5) where E a,b , a > is the Mittag-Leffler function defined as E a,b ( z ) = ∞ X k =0 z k Γ( ak + b ) , z ∈ C . (v) For any σ, γ ≥ , there exists a constant N = N ( α, β, σ, γ, d ) such that | D σt ( − ∆) γ/ q α,β (1 , x ) | + | D σt ( − ∆) γ/ ∂ t q α,β (1 , x ) | ≤ N ( | x | − d +2 − γ ∧ | x | − d − γ ) (3.6) if d ≥ , and | D σt ( − ∆) γ/ q α,β (1 , x ) | + | D σt ( − ∆) γ/ ∂ t q α,β (1 , x ) |≤ N ( | x | − γ (1 + log | x | γ =1 ) ∧ | x | − − γ ) (3.7) if d = 1 .(vi) For any σ, γ ≥ , the following scaling property holds: D σt ( − ∆) γ/ q α,β ( t, x ) = t − σ − α ( d + γ )2 + α − β q α,β (1 , t − α x ) . (3.8) Proof.
For (i), (iv), (v), and (vi), see [10, Lemma 3.1]. Also see [11, Lemma 3.1]for (iii), and see [10, Corollary 3.2] for (ii). (cid:3)
Lemma 3.2.
Let < a < and b < a + 1 . Then there exist constants η > , η ∈ R , η ∈ ( − , which depded only on a such that for any v > , E a,b ( − v ) = 1 πa Z ∞ r − ba exp ( − r /a η )[ r sin ( ψ − η a ) + v sin ( ψ )] r + 2 rvη + v dr, (3.9) where ψ = ψ ( r ) = r /a sin ( η ) + ( η ( a + 1 − b )) . Proof.
The proof is based on [9, Chapter 4]. Since 0 < a <
2, we can choose aconstant η satisfying a π < η < ( π ∧ aπ ). Then by using formula (4.7.13) in [9], forany v > < λ < v , we have E a,b ( − v ) = 1 πa Z ∞ λ r − ba exp ( r /a cos ( η/a ))[ r sin ( ψ − η ) + v sin ( ψ )] r + 2 rv cos ( η ) + v dr + Z η − η G ( a, b, λ, φ, v ) dφ, (3.10)where ψ = ψ ( r ) = r /a sin ( η/a ) + ( η ( a + 1 − b ) /a ) ,G = λ − b ) /a πa exp ( λ /a cos ( φ/a )) e iν λe iφ + v , and ν = λ /a sin ( φ/a ) + φ (1 + (1 − b ) /a ). Since b − < a , by the dominatedconvergence theorem, if we let λ ↓
0, the second integral in (3.10) goes to zero.Also since a π < η < ( π ∧ aπ ), cos( η/a ) has negative value, and | cos ( η ) | 6 = 1.Therefore, as λ goes to zero, the first integral in (3.10) converges to the integral overpositive real line with the same integrand by the dominated convergence theorem.Therefore, to finish the proof, it is enough to take η = − cos ( η/a ) , η = η/a and η = cos ( η ). (cid:3) Remark 3.3. If b = 1, then we have E a, ( − v ) = sin aππ Z ∞ r a − r a + 2 r a cos ( aπ ) + 1 exp ( − rv /a ) rdr. (3.11)(see e.g. [9, Exercise 3.9.5]). Lemma 3.4.
Let α ∈ (0 , and β < α + 1 / . Then there exist constants N and m > , m ∈ R , m ∈ R , m ∈ R , m ∈ ( − , , depending only on α, β , such that for any µ ∈ R F{ ( − ∆) µ/ q α,β ( t, · ) } ( ξ )= N | ξ | µ + β − αα Z ∞ exp ( − m t | ξ | α r )[ r α sin ( ˜ ψ + m ) + sin ( ˜ ψ + m )] r α − r α m + 1 r β − dr, (3.12) where ˜ ψ = ˜ ψ ( r ) = m t | ξ | α r .Proof. By the definition of fractional Laplacian and (3.5), we have F{ ( − ∆) µ/ q α,β ( t, · ) } ( ξ ) = t α − β | ξ | µ E α, α − β ( − t α | ξ | ) . By (3.9) with a = α, b = 1 + α − β , and the change of variables r → vr , for any v > E α, α − β ( − v )= 1 πα Z ∞ r β − αα exp ( − r /α η )[ r sin ( ψ − η α ) + v sin ( ψ )] r + 2 rvη + v dr = 1 πα Z ∞ v β − αα r β − αα exp ( − v /α r /α η )[ r sin ( ψ − η α ) + sin ( ψ )] r + 2 rη + 1 dr, IME-FRACTIONAL SPDES DRIVEN BY L´EVY PROCESSES 15 where ψ = ψ ( r ) = v /α r /a sin ( η ) + η β . By the change of variables r → r α , E α, α − β ( − v )= 1 πα Z ∞ v β − αα r β − α exp ( − v /α rη )[ r α sin ( ψ ′′ − η α ) + sin ( ψ ′ )] r α + 2 r α η + 1 αr α − dr = N Z ∞ v β − αα exp ( − v /α rη )[ r α sin ( ψ − η α ) + sin ( ψ )] r α + 2 r α η + 1 r β − dr, where ψ = ψ ( r ) = v /α r sin ( η ) + η β . Putting v = t α | ξ | , we have F{ ( − ∆) µ/ q α,β ( t, · ) } ( ξ )= N | ξ | µ + β − αα Z ∞ exp ( − η t | ξ | α r )[ r α sin ( ψ − η α ) + sin ( ψ )] r α + 2 r α η + 1 r β − dr, due to (3.5) and (3.14), where ψ = ψ ( r ) = sin ( η ) t | ξ | α r + η β .Finally, for (3.12) we take m = η , m = sin ( η ) , m = η β, m = m − αη , m = η . The lemma is proved. (cid:3)
For each j = 0 , , . . . and c >
0, denote q c,jα,β ( t, x ) = Ψ j ∗ ( − ∆) c q α,β ( t, x )= F − { ˆΨ(2 − j · ) F{ ( − ∆) c q α,β } ( t, · ) } ( x )= 2 jd F − { ˆΨ( · ) F{ ( − ∆) c q α,β } ( t, j · ) } (2 j x )=: 2 jd ¯ q c,jα,β ( t, j x ) . (3.13) Lemma 3.5.
Assume p ≥ , < α < , p < β < α + 1 p , (3.14) denote c := α +1 /p − β ) α > . Then for any constants ε, δ satisfying p < β − α ε, β − α < p − δ < p , (3.15) we have k q c + ε,jα,β ( t, · ) k L ≤ N (2 δα j + εj t − p + δ ∧ t − p − αε ) , (3.16) where N = N ( α, β, ε, δ, d, p ) .Proof. Put c = c + ε . The by (3.15), 0 < c <
2. Due to (3.8), we easily get k ( − ∆) c q α,β ( t, · ) k L ≤ N t − αc + α − β . (3.17)Recall that the convolution operator is bounded in L p for any p ≥
1, that is k f ∗ g k L p ≤ k f k L k g k L p . Thus, (3.17) together with the first equality in (3.13)yields k q c ,jα,β ( t, · ) k L ≤ N t − p − αε . This and the equality k q c ,jα,β ( t, · ) k L = k ¯ q c ,jα,β ( t, · ) k L show that it remains to prove k ¯ q c ,jα,β ( t, · ) k L ≤ N δα j + εj t − p + δ . By definition (see (3.13)) F (¯ q c ,jα,β )( t, ξ ) = ˆΨ( ξ ) F{ ( − ∆) c q α,β } ( t, j ξ ) . (3.18)Thus |F (¯ q c ,jα,β )( t, ξ ) | = | ˆΨ( ξ ) ||F{ ( − ∆) c q α,β ( t, · ) } (2 j ξ ) |≤ N ≤| ξ |≤ |F{ ( − ∆) c q α,β ( t, · ) } (2 j ξ ) | . (3.19)By (3.12) with µ = c , |F{ ( − ∆) c q α,β ( t, · ) } (2 j ξ ) | (3.20) ≤ N | j ξ | αp + ε Z ∞ exp ( − m t | j ξ | α r )( | r α sin ( ψ + m ) | + | sin ( ψ + m ) | ) r α − r α m + 1 r β − dr, where ψ = m t | j ξ | α r . Note that for any polynomial Q of degree m and c > N ( c, m ) such that Q ( r ) exp ( − cr ) ≤ N r − p + δ r > . (3.21)Applying this inequality with Q ( r ) = 1 and c = m to (3.20), we have |F (¯ q c ,jα,β )( t, ξ ) | ≤ N ≤| ξ |≤ | j ξ | αp + ε × (cid:18)Z ( t | j ξ | α r ) − p + δ r β − dr + Z ∞ ( t | j ξ | α r ) − p + δ r β − r − α dr (cid:19) ≤ N δα j + εj t − p + δ ≤| ξ |≤ . For the second inequality above we used β − α < p − δ < β .Similarly, using (3.18), (3.12) and following the above computations, for anymulti-index γ we get | D γξ F (¯ q c ,jα,β )( t, ξ ) | ≤ N δα j + εj t − p + δ ≤| ξ |≤ . Hence, we have k ¯ q c ,jα,β ( t, · ) k L = Z R d (1 + | x | d ) − (1 + | x | d ) | ¯ q c ,jα,β ( t, x ) | dx ≤ N Z R d (1 + | x | d ) − sup ξ | (1 + ∆ dξ ) F (¯ q c ,jα,β )( t, ξ )) | dx ≤ N δα j + εj t − p + δ . For the first inequality above we used the fact that if F ( f ) has compact support,then | f ( x ) | = |F − ( F ( f ))( x ) | ≤ kF ( f ) k L ≤ N sup ξ |F ( f ) | . The lemma is proved. (cid:3)
The following result will be used later to study the regularity relation betweenthe solutions and free terms in stochastic parts.
IME-FRACTIONAL SPDES DRIVEN BY L´EVY PROCESSES 17
Theorem 3.6.
Let (3.14) and (3.15) hold, and denote c := α +1 /p − β ) α . Thenthere exists a constant N depdending only on α, β, d, p, ε, δ , and T such that forany g ∈ C ∞ c ((0 , ∞ ) × R d ) Z T Z t Z R d (cid:12)(cid:12)(cid:12) ( − ∆) c ε q α,β ( t − s, x ) ∗ g ( s )( x ) (cid:12)(cid:12)(cid:12) p dxdsdt ≤ N Z T k g ( t, · ) k pB εp dt. (3.22) Proof.
Denote c = c + ε and Q ( t, x ) := ( − ∆) c q α,β ( t, x ). By (2.12), Z T Z t Z R d | Q ( t − s ) ∗ g ( s )( x ) | p dxdsdt ≤ N Z T Z t Z R d | Ψ ∗ ( Q ( t − s ) ∗ g ( s ))( x ) | p + | ∞ X j =1 | Ψ j ∗ ( Q ( t − s ) ∗ g ( s ))( x ) | p/ dxdsdt. Note that ˆΨ j = ˆΨ j ( ˆΨ j − + ˆΨ j + ˆΨ j +1 ) , j = 1 , , . . . , ˆΨ = ˆΨ ( ˆΨ + ˆΨ ) . (3.23)Using this and the relation F ( f ∗ f ) = F ( f ) F ( f ), we get ∞ X j =1 | Ψ j ∗ ( Q ( t − s ) ∗ g ( s ))( x ) | = ∞ X j =1 | j +1 X i = j − Q i ( t − s ) ∗ g j ( s )( x ) | , | Ψ ∗ ( Q ( t − s ) ∗ g ( s ))( x ) | = | q c , α,β ( t − s ) ∗ g ( s )( x ) + q c , α,β ( t − s, · ) ∗ g ( s )( x ) | . Therefore, Z T Z t Z R d | Q ( t − s ) ∗ g ( s )( x ) | p dxdsdt ≤ N Z T Z t Z R d | q c , α,β ( t − s ) ∗ g ( s )( x ) | p dxdsdt + N Z T Z t Z R d | q c , α,β ( t − s ) ∗ g ( s )( x ) | p dxdsdt + N Z T Z t Z R d (cid:0) ∞ X j =1 | j +1 X i = j − q c ,iα,β ( t − s ) ∗ g j ( s )( x ) | (cid:1) p/ dxdsdt. (3.24)By (3.16), the first two integrals on the right hand side of (3.24) are bounded by N Z T Z t ( t − s ) − δp k g ( s, · ) k pL p dtds ≤ N ( T ) Z T k g ( t, · ) k pL p dt. (3.25)By Minkowski’s inequality and Fubini’s thoerem, the third integral is bounded by N Z T Z Ts (cid:0) ∞ X j =1 | K j ( t − s ) | k g j ( s, · ) k L p (cid:1) p/ dtds, where K j ( t − s ) = (2 δα j + εj ( t − s ) − p + δ ∧ ( t − s ) − p − α ε ). If p = 2 then ∞ X j =1 Z T Z Ts | K j ( t − s ) | k g j ( s, · ) k L dtds ≤ N Z T ∞ X j =1 Z s +2 − α j s δjα +2 εj ( t − s ) − δ k g j ( s, · ) k L dtds + N Z T ∞ X j =1 Z ∞ s +2 − α j ( t − s ) − − αε k g j ( s, · ) k L dtds = N Z T ∞ X j =1 εj k g j ( s, · ) k L ds. This proves the theorem if p = 2.If p >
2, then Z T Z Ts (cid:0) ∞ X j =1 | K j ( t − s ) | k g j ( s, · ) k L p (cid:1) p/ dtds ≤ N Z T Z Ts (cid:0) ∞ X j =1 J ( t, s, j ) | K j ( t − s ) | k g j ( s, · ) k L p (cid:1) p/ dtds + Z T Z Ts (cid:0) ∞ X j =1 J c ( t, s, j ) | K j ( t − s ) | k g j ( s, · ) k L p (cid:1) p/ dtds, where J = { ( t, s, j ) | j ( t − s ) α ≤ } . By (3.15), if ( t, s, j ) ∈ J , then K j ( t − s ) =2 δjα + εj ( t − s ) − p + δ . Therefore, by H¨older’s inequality, we have ∞ X j =1 J | K j ( t − s ) | k g j ( s, · ) k L p = ∞ X j =1 J aj − aj δjα +2 εj ( t − s ) − p +2 δ k g j ( s, · ) k L p ≤ ( t − s ) − p +2 δ (cid:0) X j ∈ J ( t,s ) aqj (cid:1) /q (cid:0) X j ∈ J ( t,s ) − apj δpjα + pεj k g j ( s, · ) k pL p (cid:1) /p , where q = pp − , a ∈ (0 , δα ), and J ( t, s ) = { j = 1 , , . . . | ( t, s, j ) ∈ J } . Note that (cid:0) X j ∈ J ( t,s ) aqj (cid:1) /q ≤ N ( p )( t − s ) − αa . IME-FRACTIONAL SPDES DRIVEN BY L´EVY PROCESSES 19
Thus we get Z T Z Ts (cid:0) ∞ X j =1 J ( t, s, j ) | K j ( t − s ) | k g j ( s, · ) k L p (cid:1) p/ dtds ≤ N Z T ∞ X j =1 Z s +2 − jα s ( t − s ) − pδ − pαa − apj δpjα + pεj k g j ( s, · ) k pL p dtds ≤ N Z T ∞ X j =1 pεj k g j ( t, · ) k pL p dt. (3.26)Next we consider the remaining part: ∞ X j =1 J c | K j ( t − s ) | k g j ( s, · ) k L p = ∞ X j =1 J c bj − bj ( t − s ) − p − αε k g j ( s, · ) k L p ≤ ( t − s ) − p − αε (cid:0) X j / ∈ J ( t,s ) bqj (cid:1) /q (cid:0) X j / ∈ J ( t,s ) − bpj k g j ( s, · ) k pL p (cid:1) /p , where q = pp − , and b ∈ ( − ε, (cid:0) X j / ∈ J ( t,s ) bqj (cid:1) /q ≤ N ( p )( t − s ) − αb . Therefore it follows that Z T Z Ts (cid:0) ∞ X j =1 J ( t, s, j ) | K j ( t − s ) | k g j ( s, · ) k L p (cid:1) p/ dtds ≤ N Z T ∞ X j =1 Z ∞ s +2 − jα ( t − s ) − − αbp − αεp − bpj k g j ( s, · ) k pL p dtds ≤ N Z T ∞ X j =1 pεj k g j ( t, · ) k pL p dt. (3.27)Combining (3.25), (3.26) and (3.27) we get (3.22) for p >
2. Hence, the theorem isproved. (cid:3)
The next part of this section is related to the non-zero initial value problem ∂ αt u = ∆ u, t > u (0) = u , α> u t (0) = 0 . The solution is given in the form of p ( t, · ) ∗ u , and we study the regularity of thisconvolution.Define p j ( t, x ) = (Ψ j ( · ) ∗ p ( t, · ))( x ) = F − ( ˆΨ(2 − j · )ˆ p ( t, · ))( x )= 2 jd F − ( ˆΨ( · )ˆ p ( t, j · ))(2 j x ) := 2 jd ¯ p j ( t, j x ) . (3.28) Lemma 3.7.
Let p > , < α < and α = 1 . Then there exists a constant N depending only on α, d such that k p j ( t, · ) k L ≤ N (2 − jα t − ∧ , t > . (3.29) Proof.
Let R := | x | t − α . Then by (3.3), and (3.4), Z R d | p ( t, x ) | dx = Z R ≥ | p ( t, x ) | dx + Z R< | p ( t, x ) | dx ≤ N Z R ≥ t − αd exp {− c | x | − α t − α − α } dx + N Z R< | x | − d ( R + R log R d =2 + R / d =1 ) dx. By using change of variables and the relation r ν | log r | ≤ N ( ν ) 0 < r ≤ , ν > k p ( t, · ) k L ≤ N . Due to this and the relation k p j ( t, · ) k L = k ¯ p j ( t, · ) k L , itonly remains to show k ¯ p j ( t, · ) k L ≤ N − jα t − . By definition (see (3.28)) F (¯ p j )( t, ξ ) = ˆΨ( · )ˆ p ( t, j ξ ) . (3.30)Since q α,α := D α − αt p = p , by (3.5) and (3.11), we have |F ¯ p j ( t, ξ ) | ≤ N ≤| ξ |≤ Z r α − exp ( − jα | ξ | α tr ) rdr + N ≤| ξ |≤ Z ∞ r − α − exp ( − jα | ξ | α tr ) rdr. (3.31)Note that for any polynomial Q of degree m and constant c >
0, we have Q ( r ) e − cr ≤ N ( c, m ) r − . This and (3.31) easily yield |F ¯ p j ( t, ξ ) | ≤ N − jα t − ≤| ξ |≤ . Similarly, using (3.30) and following above computations, for any multi-index γ weget | D γξ F ¯ p j ( t, ξ ) | ≤ N ( α, γ, d )2 − jα t − ≤| ξ |≤ . Therefore, we finally have k ¯ p j ( t, · ) k L = Z R d (1 + | x | d ) − (1 + | x | d ) | ¯ p j ( t, x ) | dx ≤ N Z R d (1 + | x | d ) − sup ξ | (1 + ∆ dξ ) F (¯ p j )( t, ξ ) | dx ≤ N − jα j t − . The lemma is proved. (cid:3)
Theorem 3.8.
Let, p > , < α < and f ∈ C ∞ c ( R d ) . Then we have Z T Z R d | p ( t, · ) ∗ f | p dxdt ≤ N k f k pB − αpp , (3.32) where the constant N depends only on α, d, T . IME-FRACTIONAL SPDES DRIVEN BY L´EVY PROCESSES 21
Proof.
Since the case α = 1 is a classical result, we assume α = 1. By (3.23), andthe relation F ( f ∗ f ) = F ( f ) F ( f ), Z T Z R d | p ( t, · ) ∗ f | p dxdt ≤ N Z T ( k p ( t, · ) k L + k p ( t, · ) k L ) p k f k pL p dt + N Z T (cid:0) ∞ X j =1 j +1 X i = j − k p i ( t, · ) k L k f j k L p (cid:1) p dt. By (3.29), Z T ( k p ( t, · ) k L + k p ( t, · ) k L ) p k f k pL p dt ≤ N ( T ) k f k pL p , (3.33)and Z T (cid:0) ∞ X j =1 j +1 X i = j − k p i ( t, · ) k L k f j k L p (cid:1) p dt ≤ N Z T (cid:0) ∞ X j =1 (2 − jα t − ∧ k f j k L p (cid:1) p dt. Observe that Z T (cid:0) ∞ X j =1 (2 − jα t − ∧ k f j k L p (cid:1) p dt ≤ Z T (cid:0) ∞ X j =1 J ( t, j ) k f j k L p (cid:1) p dt + Z T (cid:0) ∞ X j =1 J c ( t, j )2 − jα t − k f j k L p (cid:1) p dt, where J = { ( t, j ) | jα t ≤ } . By H¨older’s inequality, Z T (cid:0) ∞ X j =1 J k f j k L p (cid:1) p dt = Z T (cid:0) X j ∈ J ( t ) − jα a jα a k f j k L p (cid:1) p dt ≤ Z T (cid:0) X j ∈ J ( t ) − jα aq (cid:1) p/q (cid:0) X j ∈ J ( t ) jα ap k f j k pL p (cid:1) dt, where a ∈ ( − p , q = pp − , and J ( t ) = { j = 1 , , . . . | ( t, j ) ∈ J } . Since X j ∈ J ( t ) − jα aq ≤ N ( q, a ) t aq , we have Z T (cid:0) ∞ X j =1 J k f j k L p (cid:1) p dt ≤ N ∞ X j =1 Z − jα t ap jα ap k f j k pL p dt ≤ N ∞ X j =1 − jα k f j k pL p . (3.34) By H¨older’s inequality again, for b ∈ ( − , − p ) and q = pp − , Z T (cid:0) ∞ X j =1 J c − jα t − k f j k L p (cid:1) p dt = Z T (cid:0) X j / ∈ J ( t ) − jα b jα b − jα t − k f j k L p (cid:1) p dt ≤ Z T t − p (cid:0) X j / ∈ J ( t ) − jα bq − jα q (cid:1) p/q (cid:0) X j / ∈ J ( t ) jα bp k f j k pL p (cid:1) dt. Since X j / ∈ J ( t ) − jα ( b +1) q ≤ N ( q, b ) t ( b +1) q , we have Z T (cid:0) ∞ X j =1 J c − jα t − k f j k L p (cid:1) p dt ≤ N ∞ X j =1 Z ∞ − jα t − p t ( b +1) p jα bp k f j k pL p dt = N ∞ X j =1 − jα k f j k pL p . (3.35)Combining (3.33), (3.34) and (3.35), we have (3.32). The theorem is proved. (cid:3) The last part of this section is related to the non-zero initial date problem of thetype ∂ αt u = ∆ u, t > u (0 , x ) = 0 , α> ∂ t u (0 , x ) = v ( x ) . (3.36)Let α >
1. Then using Lemma 3.1, each x = 0, one can check that P ( t, x ) := q α,α − = Z t p ( s, x ) ds is well defined and becomes a fundamental solution to (3.36).For j = 0 , , , . . . define P j ( t, x ) = (Ψ j ( · ) ∗ ( − ∆) ε/ P ( t, · ))( x )= F − ( ˆΨ(2 − j · ) F (( − ∆) ε/ P )( t, · ))( x )= 2 jd F − ( ˆΨ( · ) F (( − ∆) ε/ P )( t, j · ))(2 j x ):= 2 jd ¯ P j ( t, j x ) . (3.37) Lemma 3.9.
Let α ∈ (1 , . Then, for any δ ∈ (0 , α ) , there exists a constant N depending only on α, d, p, ε, δ such that for any t > , k P j ( t, · ) k L ≤ N (2 − j + δα j t − α + δ ∧ t ) . (3.38) Proof.
By (3.6), (3.7) and (3.8), we easily get k P ( t, · ) k L ≤ N t.
Therefore, it suffices to show that k ¯ P j ( t, · ) k L ≤ N − j + δα j t − α + δ . IME-FRACTIONAL SPDES DRIVEN BY L´EVY PROCESSES 23
By definition, F ( ¯ P j )( t, ξ ) = ˆΨ( ξ ) F ( P )( t, j ξ ) . (3.39)Also, by Lemma 3.2 and Lemma 3.4 with µ = 0, β = α −
1, we have |F{ P ( t, · ) } (2 j ξ ) |≤ N | j ξ | − α Z ∞ exp ( − m t | j ξ | α r )( | r α sin ( ψ + m ) | + | sin ( ψ + m ) | ) r α − r α m + 1 r α − dr, where ψ = m t | j ξ | α r . Note that for any polynomial Q of degree m , and c > Q ( r ) exp ( − cr ) ≤ N ( c, m ) r − α + δ , r > . (3.40)This with the condition δ ∈ (0 , α ) gives |F ( ¯ P j )( t, ξ ) | ≤ / ≤| ξ |≤ |F ( P )( t, j ξ ) |≤ N / ≤| ξ |≤ − jα (cid:0) Z ( t | j ξ | α r ) − α + δ r α − dr + Z ∞ ( t | j ξ | α r ) − α + δ r − dr (cid:1) ≤ N − j + δα j t − α + δ ≤| ξ |≤ . Using (3.39) and similar computations above, we also get for any multi-index γ | D γξ F ( ¯ P j ) | ≤ N − j + δα j t − α + δ ≤| ξ |≤ . Therefore, we have k ¯ P j ( t, · ) k L = Z R d (1 + | x | d ) − (1 + | x | d ) | ¯ P j ( t, x ) | dx ≤ N Z R d (1 + | x | d ) − sup ξ | (1 + ∆ dξ ) F ( ¯ P j )( t, ξ ) | dx ≤ N − j + δα j t − α + δ . The lemma is proved. (cid:3)
Theorem 3.10.
Let α ∈ (1 , and h ∈ C ∞ c ( R d ) . Then there exists a constant N = N ( α, d, p, T ) such that Z T Z R d | ( P ( t ) ∗ f )( x ) | p dxdt ≤ N k h k pB − αp − αp , if α > p (3.41) and Z T Z R d | ( P ( t ) ∗ f )( x ) | p dxdt ≤ N k h k pB − αp , if < α ≤ /p. (3.42) Proof.
Case 1 . Let α > /p . Then by assumption on α , we can take δ ∈ (0 , α )such that α − − δ − p > , − δα < . (3.43) By (3.23) and the relation F ( h ∗ h ) = F ( h ) F ( h ), Z T Z R d | ( P ( t ) ∗ h )( x ) | p dxdt ≤ N Z T ( k P ( t, · ) k L + k P ( t, · ) k L ) p k h k pL p dt + N Z T (cid:0) ∞ X j =1 j +1 X i = j − k P i ( t, · ) k L k h j k L p (cid:1) p dt. Note that (3.38) with (3.43) easily yields Z T ( k P ( t, · ) k L + k P ( t, · ) k L ) p k h k pL p dt ≤ N ( T ) k h k pL p . (3.44)Also by (3.43), Z T (cid:0) ∞ X j =1 j +1 X i = j − k P i ( t, · ) k L k h j k L p (cid:1) p dt ≤ N Z T (cid:0) ∞ X j =1 L j ( t ) k h j k L p (cid:1) p dt ≤ N Z T (cid:0) ∞ X j =1 J ( t, j ) L j ( t ) k h j k L p (cid:1) p dt + N Z T (cid:0) ∞ X j =1 J c ( t, j ) L j ( t ) k h j k L p (cid:1) p dt, where J := { ( t, j ) | j t α ≥ } , and L j ( t ) := (2 − j + δα j t − α + δ ∧ t ) = ( − j + δα j t − α + δ : ( t, j ) ∈ Jt : ( t, j ) / ∈ J. By H¨older’s inequality, Z T (cid:0) ∞ X j =1 J L j ( t ) k h j k L p (cid:1) p dt = Z T (cid:0) ∞ X j =1 J − j + δα j t − α − δ − bjα bjα k h j k L p (cid:1) p dt ≤ Z T t (1 − α + δ ) p (cid:0) X j ∈ J ( t ) − bqjα (cid:1) p/q (cid:0) X j ∈ J ( t ) − pj + δα pj pbjα k h j k pL p (cid:1) dt, (3.45)where b ∈ (0 , α − − p − δ ), q = pp − , and J ( t ) = { j = 1 , , . . . | ( t, j ) ∈ J } . Since (cid:0) X j ∈ J ( t ) − qbα j (cid:1) p/q ≤ N ( α, p ) t bp , IME-FRACTIONAL SPDES DRIVEN BY L´EVY PROCESSES 25 we have Z T (cid:0) ∞ X j =1 J L j ( t ) k h j k L p (cid:1) p dt ≤ N Z T t (1 − α + δ ) p t bp (cid:0) X j ∈ J ( t ) − pj + δα pj pbjα k h j k pL p (cid:1) dt ≤ N ∞ X j =1 Z ∞ − jα t (1 − α + δ ) p t bp − pj + δα pj pbjα k h j k pL p ≤ N ∞ X j =1 − pjα − jα k h j k pL p . (3.46)Again by H¨older’s inequality, Z T (cid:0) ∞ X j =1 J c L j ( t ) k h j k L p (cid:1) p dt = Z T (cid:0) X j / ∈ J ( t ) t − ajα ajα k h j k L p (cid:1) p dt ≤ Z T t p (cid:0) X j / ∈ J ( t ) − aqjα (cid:1) p/q (cid:0) X j / ∈ J ( t ) pajα k h j k pL p (cid:1) dt, where a ∈ ( − − p , q = pp − . Since (cid:0) X j / ∈ J ( t ) − aqjα (cid:1) p/q ≤ N ( α, p ) t ap , we have Z T (cid:0) ∞ X j =1 J c L j ( t ) k h j k L p (cid:1) p dt ≤ N Z T t p + ap (cid:0) X j / ∈ J ( t ) pajα k h j k pL p (cid:1) dt ≤ N ∞ X j =1 Z − jα t p + ap pajα k h j k pL p dt ≤ N ∞ X j =1 − pjα − jα k h j k pL p . (3.47)Combining (3.44), (3.46), and (3.47), we get (3.41). The theorem is proved. Case 2 . Let 1 ≤ α < /p . This time, we choose δ, b > α − − δ > , b ∈ (0 , α − − δ ) , (3.48) and repeat the proof of Case 1. The only difference is we need to replace (3.46) bythe following: Z T (cid:0) ∞ X j =1 J L j ( t ) k h j k L p (cid:1) p dt ≤ N Z T t (1 − α + δ ) p t bp (cid:0) X j ∈ J ( t ) − pj + δα pj pbjα k h j k pL p (cid:1) dt ≤ N Z T t (1 − α + δ − /p ) p t bp (cid:0) X j ∈ J ( t ) − pj + δα pj pbjα k h j k pL p (cid:1) dt ≤ N ∞ X j =1 Z ∞ − jα t (1 − α + δ − /p ) p t bp − pj + δα pj pbjα k h j k pL p ≤ N ∞ X j =1 − pjα k h j k pL p . On the other hand, (3.47) still holds without any changes, and this certainly implies Z T (cid:0) ∞ X j =1 J c L j ( t ) k h j k L p (cid:1) p dt ≤ N ∞ X j =1 − pjα k h j k pL p . Hence, Case 2 is also proved. (cid:3) Proof of Theorem 2.13
Lemma 4.1.
Let < α < , < p < ∞ and γ ∈ R . Then for any u ∈ U α,γ +2 p , v ∈ V α,γ +2 p and f ∈ H γp ( T ) the equation ∂ αt u = ∆ u + f, t > , x ∈ R d ; u (0) = u , α> u ′ (0) = v (4.1) has a unique solution u ∈ H γ +2 p ( T ) , and moreover k u k H γ +2 p ( T ) ≤ N (cid:0) k u k U α,γ +2 p + 1 α> k v k V α,γ +2 p + k f k H γp ( T ) (cid:1) , (4.2) where the constant N depends only on α, d, γ , and T .Proof. Due to Remark 2.6, it is enough to prove the lemma for a particular γ , andtherefore we assume γ = − u = v = 0 due to [10, Theorem 2.3](or [11, Theorem 2.10]), from which the uniqueness result follows. Furthermore,considering u − v , where v is the solution to the equation with u = v = 0 takenfrom [10, Theorem 2.3], we may assume that f = 0.Now, let u , v ∈ L c and define u ( t, x ) := ( p ( t, · ) ∗ u ( · ))( x ) + 1 α> ( P ( t, · ) ∗ v ( · ))( x ) . Then by Lemma 3.1 (or see [11, Lemma 3.5] for more detail), u satisfies equation(4.1), and u ∈ H np ( T ) for any n ∈ R , since u , v ∈ L c . Moreover, for this solutionestimate (4.2) holds with γ = − u n , v n ∈ L c such that u n → u in U α, p and v n → v in V α, p , andfor each n let u n denote the solution to the equations with initial data u n and v n .Then estimate (4.2) corresponding to u n − u m , where n, m ∈ N , shows that u n is aCauchy sequence in H p ( T ), which is a Banach space. Now it is easy to check that IME-FRACTIONAL SPDES DRIVEN BY L´EVY PROCESSES 27 the limit of the Cauchy sequence becomes a solution to the equation with initialdata u and v , and the estimate also follows. The lemma is proved. (cid:3) Lemma 4.2.
Let α ∈ (0 , , β < α + 1 /p and h ∈ H ∞ c ( T ) . Denote u ( t, x ) := Z t (cid:18)Z R d q α,β ( t − s, x − y ) h k ( s, y ) dy (cid:19) · dZ ks . (4.3) Then u ∈ H p ( T ) and satisfies ∂ αt u = ∆ u + ∂ β t Z t h k ( s, x ) · dZ ks , t > , x ∈ R d ; u (0) = u ′ (0)1 α> = 0 (4.4) in the sense of Definition 2.7.Proof. It is enough to repeat the proof of [4, Lemma 3.10], which deals with theequation driven by Brownian motions. (cid:3)
Next we prove a version of Theorem 2.13 for the model equation ∂ αt u = ∆ u + f + ∂ β t Z t g k ( s, x ) dW ks + ∂ β t Z t h k ( s, x ) · dZ ks , t > , x ∈ R d ,u (0) = u , ∂ t u (0)1 α> = v , (4.5) Theorem 4.3.
Let γ ∈ R , p ≥ , β < α + 1 / and β < α + 1 /p . Then, for any u ∈ U α,γ +2 p , v ∈ V α,γ +2 p , f ∈ H γp ( T ) , g ∈ H γ + c p ( T, l ) and h ∈ H γ +¯ c p ( T, l , d ) ,equation (4.5) has a unique solution u in the class H γ +2 p ( T ) , and for this solutionit holds that k u k H γ +2 p ( T ) ≤ N (cid:0) k u k U α,γ +2 p + 1 α> k v k V α,γ +2 p + k f k H γp ( T ) + k g k H γ + c p ( T,l ) + k h k H γ +¯ c p ( T,l ,d ) (cid:1) , (4.6) where N = N ( α, β , β , d, d , p, γ, T ) .Proof. Due to Remark 2.6 it is enough to prove the lemma for γ = 0. The unique-ness follows from Lemma 4.1.Recall that the lemma holds if h = 0 and u = v = 0 by [10, Theorem 2.3], andit holds if f = 0 , g = 0 , h = 0 by Lemma 4.1. By the linearity of the equation, if h = 0 then the existence and the desired estimate is easily obtained by combining[10, Theorem 2.3] and Lemma 4.1. The case h = 0 is proved.Furthermore, by the result for the case h = 0 and the linearity of the equation,to finish the proof of the lemma, we only need to prove the existence result andestimate (4.6), provided that u = v = 0 , f = 0 and g = 0. We divide the proof ofthis into following three cases. Case 1 . Let β > /p .If h ∈ H ∞ c ( T, l , d ), we define u ∈ H p ( T ) as in (4.3) such that it becomes asolution to equation (4.4). Denote c := α +1 /p − β ) α and take a small constant ε ∈ (0 , c ) satisfying (3.15) with β in place of β , and set v := ( − ∆) (2 − c − ε ) / u, ¯ h := ( − ∆) (2 − c − ε ) / h. By Burkholder-Davis-Gundy inequality, and (2.9) k ∆ u k p L p ( T ) = k ( − ∆) ( c + ε ) / v k p L p ( T ) ≤ N E Z R d Z T Z t ∞ X k =1 (cid:12)(cid:12)(cid:12) ( − ∆) c ε q α,β ( t − s, · ) ∗ ¯ h k ( s, · ) (cid:12)(cid:12)(cid:12) ( x ) ds ! p dtdx + N E Z R d Z T Z t ∞ X k =1 (cid:12)(cid:12)(cid:12) ( − ∆) c ε q α,β ( t − s, · ) ∗ ¯ h k ( s, · ) (cid:12)(cid:12)(cid:12) p ( x ) dsdtdx. By [10, Theorem 3.1] we have E Z R d Z T Z t ∞ X k =1 (cid:12)(cid:12)(cid:12) ( − ∆) c ε q α,β ( t − s, · ) ∗ ¯ h k ( s, · ) (cid:12)(cid:12)(cid:12) ( x ) ds ! p dtdx ≤ N k ¯ h k p L p ( T,l ,d ) , where the constant N depends only on α, β , d, d , and p . Also by Theorem 3.6 andRemark 2.6 E Z R d Z T Z t ∞ X k =1 (cid:12)(cid:12)(cid:12) ( − ∆) c ε q α,β ( t − s, · ) ∗ ¯ h k ( s, · ) (cid:12)(cid:12)(cid:12) p ( x ) dsdtdx ≤ N E d X r =1 ∞ X k =1 Z T k ¯ h rk ( t, · ) k pB εp dt ≤ N E d X r =1 ∞ X k =1 Z T k h rk ( t, · ) k pH − c p dt, where the constant N depends only on α, β , d, d , and p . The above estimationsand the inequality P ∞ k =1 | a k | p ≤ (cid:0)P ∞ k =1 | a k | (cid:1) p/ yield k ∆ u k p L p ( T ) ≤ N k h k p H − c p ( T,l ,d ) . (4.7)Also, due to (2.20) and the inequality k · k L p ( s ) ≤ k · k L p ( T ) for s ≤ T , we have k u k p L p ( T ) ≤ N Z T ( T − s ) θ − (cid:16) k ∆ u k p H γp ( T ) + k h k p L p ( T,l ,d ) (cid:17) ds ≤ N ( k ∆ u k p L p ( T ) + k h k p L p ( T,l ,d ) ) ≤ N k h k p H − c p ( T,l ,d ) . This, (4.7) and the inequality k u k H p ≤ k u k L p + k ∆ u k L p yield estimate (4.6).For general h ∈ H ¯ c p ( T, l , d ), it is enough to repeat the approximation argumentused in the proof of Lemma 4.1. Case 2 . Let β = 1 /p . The argument used in Case 1 shows that to prove theexistence result and estimate (4.6) we may assume h ∈ H ∞ c ( T, l , d ). In this casethe existence is a consequence of Lemma 4.2.Let u ∈ H p ( T ) be the solution to the equation. Take κ > κ ′ = κα/ β ′ = 1 /p + κ ′ > /p . Then using stochastic Fubini’s theorem and ∂ β t = ∂ β ′ t I β ′ − β t , we conclude that u satisfies ∂ αt u = ∆ u + ∂ β ′ t Z t ¯ h k ( s ) · dZ ks , t > , where ¯ h = I β ′ − β t h . Thus, by the result of Case 1 and (2.1) k u k H p ( T ) ≤ N k ¯ h k H β ′ − /pαp ( T,l ,d ) ≤ N k h k H ¯ c p ( T,l ,d ) . Hence the case β = 1 /p is also proved. IME-FRACTIONAL SPDES DRIVEN BY L´EVY PROCESSES 29
Case 3 . Let β < /p . As in Case 2, we only need to prove estimate (4.6),provide that h ∈ H ∞ c ( T, l , d ) and the solution u already exists.Put ¯ f ( t, x ) =: 1Γ( α ) Z t ( t − s ) − β h k ( x ) · dZ ks . Then by Burkerholder-Davis-Gundy inequality and (2.10), k ¯ f k p L p ( T ) ≤ E Z T Z t ( t − s ) − β p | h ( s, · ) | pL p ( l ,d ) dsdt ≤ N k h k p L p ( T,l ,d ) . (4.8)Note that by Lemma 2.5, u satisfies ∂ αt u = ∆ u + ¯ f , t > u (0) = 1 α> u t (0) = 0 . Therefore, estimate (4.6) follows from (4.8) and Lemma 4.1. The theorem is proved. (cid:3)
Proof of theorem 2.13 .Due to the method of continuity (see e.g. [10, Lemma 5.1]) and Theorem 4.3we only need to prove that a priori estimate (2.23) holds, provided that a solution u ∈ H γ +2 p ( T ) to equation (1.1) already exists. Step 1.
Assume u = v = 0. Denote¯ g k := µ ik u x i + ν k u + g k , ¯ h k := ¯ µ ik u x i + ¯ ν k u + h k . Recall that c , ¯ c <
2. By Assumption 2.12, µ = 0 if c ≥
1, and ¯ µ = 0 if ¯ c ≥ k ¯ g k H γ + c p ( t,l ) ≤ N c < k u x k H γ + c p ( T ) + N k u k H γ + c p ( T ) + k g k H γ + c p ( T,l ) ≤ N c < k u k H γ + c p ( T ) + N k u k H γ + c p ( T ) + k g k H γ + c p ( T,l ) . The similar estimate holds for ¯ h . Using these and the embedding inequality k u k H γ + δp ≤ ε k u k H γ +2 p + N ( δ, ε ) k u k H γp , δ ∈ (0 , , ε > , we get, for any ε > t ≤ T , k ¯ g k H γ + c p ( t,l ) + k ¯ h k H γ +¯ c p ( t,l ,d ) (4.9) ≤ ε k u k H γ +2 p ( t ) + N k u k H γp ( t ) + k g k H γ + c p ( t,l ) + k h k H γ +¯ c p ( t,l ,d ) < ∞ . Recall ∂∂x i : H νp → H ν − p is a bounded operator for any ν ∈ R . Using this, (2.21)and Assumption 2.12, we easily have k D u k H γp ( T ) + k S c u k H γ + c p ( T,l ) + k S d u k H γ +¯ c p ( T,l ,d ) ≤ N k u k H γ +2 p ( T ) + k f k H γp ( T ) + k g k H γ + c p ( T,l ) + k h k H γ +¯ c p ( T,l ,d ) . (4.10)Due to Theorem 4.3 and (4.9), we can define v ∈ H γ +2 p ( T ) as the solution toequation (4.5) with ¯ g and ¯ h in place of g and h , respectively. Furthermore, for each t ≤ T we have k v k H γ +2 p ( t ) ≤ N k f k H γp ( t ) + N k ¯ g k H γ + c p ( t,l ) + N k ¯ h k H γ +¯ c p ( t,l ,d ) . Note that ¯ u := u − v ∈ H γ +2 p ( T ) satisfies ∂ αt ¯ u = a ij ¯ u x i x j + b i ¯ u x i + c ¯ u + ¯ f , t > u (0) = 1 α> ¯ u t (0) = 0 , where ¯ f := ( a ij − δ ij ) v x i x j + b i v x i + cv + f. By Lemma 4.1, (4.11) and (4.9), for each t ≤ T k u k H γ +2 p ( t ) ≤ k u − v k H γ +2 p ( t ) + k v k H γ +2 p ( t ) ≤ N ε k u k H γ +2 p ( t ) + N k u k H γp ( t ) + N k f k H γp ( t ) + N k g k H γ + c p ( t,l ) + N k h k H γ +¯ c p ( t,l ,d ) . Therefore, k u k p H γ +2 p ( t ) ≤ N k u k p H γp ( t ) + N k f k H γp ( t ) + N k g k H γ + c p ( t,l ) + N k h k H γ +¯ c p ( t,l ,d ) . (4.11)Combining this, (4.10) and (2.20), we get for each t ≤ T k u k p H γp ( t ) ≤ N Z t ( t − s ) θ − k u k p H γp ( s ) ds + N k f k H γp ( T ) + N k g k H γ + c p ( T,l ) + N k h k H γ +¯ c p ( T,l ,d ) . (4.12)We use (4.12) and Gronwall’s inequality (see [21]) to estimate k u k p H γp ( T ) . Then,applying this estimate to (4.11) and using (4.10), we get a priori estimate (2.23) if u = v = 0. Step 2.
General case. Let v ∈ H γ +2 p ( T ) denote the solution to equation (4.5)taken from Theorem 4.3. Then ¯ u := u − v ∈ H γ +2 p ( T ) satisfies equation (1.1) with u = v = 0, ˜ f , ˜ g and ˜ h , where˜ f := ( a ij − δ ij ) v x i x j + b i v x i + cv, ˜ g k := µ ik v x i + ν k v, ˜ h rk := ¯ µ rk v x i + ¯ ν rk v. By the result of Step 1, k u − v k H γ +2 p ( T ) ≤ N k ˜ f k H γp ( T ) + N k ˜ g k H γ + c p ( T,l ) + N k ˜ h k H γ +¯ c p ( T,l ,d ) ≤ N k v k H γ +2 p ( T ) . (4.13)For the second inequality above we used the calculations in Step 1 (see (4.9)).Combining (4.13) with the estimate for v , that is (4.6), we finally get a prioriestimate (2.23) for u . Hence, the theorem is proved. References [1] D. Baleanu,
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