Super-convergence analysis on exponential integrator for stochastic heat equation driven by additive fractional Brownian motion
aa r X i v : . [ m a t h . NA ] J u l Super-convergence analysis on exponential integrator forstochastic heat equation driven by additive fractionalBrownian motion
Jialin Hong and Chuying Huang
Abstract.
In this paper, we consider the strong convergence order of theexponential integrator for the stochastic heat equation driven by an additivefractional Brownian motion with Hurst parameter H ∈ ( , H ∈ ( , L p (Ω)-estimate of the Skorohod integral and the smoothing effect of the Laplacianoperator.
1. Introduction
The fractional Brownian motion (fBm) with Hurst parameter H ∈ (0 ,
1) isa family of Gaussian processes, which extends the standard Brownian motion( H = ). In particular, if H ∈ ( , H ∈ ( ,
4, 5, 7, 12, 13 ] and references therein. These applications motivate numericalresearches about stochastic differential equations driven by additive fBms, amongwhich the strong convergence analysis for numerical schemes is an important part.In general, the strong convergence order of a numerical approximation for astochastic differential equation is restricted by the regularity of the solution. Ifthe order is consistent with the regularity of the solution of the original equation,then the strong convergence order is called optimal. It is a natural and interestingquestion whether the order can exceed the regularity. In particular, If the strongconvergence order in temporal direction is larger than the exponent of temporal
Mathematics Subject Classification. primary 60H35; secondary 60H07; 60H15.
Key words and phrases. infinite dimensional fractional Brownain motion, super-convergentin time, Malliavin calculus, exponential integrator, stochastic heat equation.Authors are funded by National Natural Science Foundation of China (NO. 11971470 andNO. 11871068).
H¨older continuity of the solution, then we say that the numerical scheme is super-convergent in time. In finite dimensional cases, there have been several super-convergence results on numerical schemes for stochastic differential equations drivenby additive fBms with Hurst parameter H ∈ ( , ] and in multi-dimensional cases [ ], while the exponent of temporalH¨older continuity of the solution is not larger than the Hurst parameter of the fBm.For equations with singular drifts, [ ] proves the strong order one of accuracy ofthe backward Euler scheme and applies the scheme to numerically solve the Cox–Ingersoll–Ross interest model driven by an fBm. To our best knowledge, however,there is no super-convergence result in temporal direction on numerical schemesfor stochastic partial differential equations (SPDEs) driven by infinite dimensionalfBms.The goal of this paper is to investigate the super-convergence analysis on theexponential integrator approximating the mild solution of the stochastic heat equa-tion (SHE) driven by an infinite dimensional fBm ( dX t = ∆ X t dt + F ( X t ) dt + dW Q t , t ∈ (0 , T ] ,X = u ∈ V. (1)The infinite dimensional fBm W Q is defined by W Q t := ∞ X i =1 Q f i β it , t ∈ [0 , T ] , (2)where { β i } ∞ i =1 is a sequence of identically distributed and independent scalar fBmswith Hurst parameter H ∈ ( , { f i } ∞ i =1 is an orthonormal basis of another sep-arable Hilbert sapce U and Q ∈ L ( U, V ) is a self-adjoint, nonnegative definite andbounded linear operator. Denote by { S t } t ≥ the analytic semigroup generated by − ∆. Then the mild solution reads X t = S t X + Z t S t − s F ( X s ) ds + Z t S t − s dW Q s , t ∈ [0 , T ] , (3)where the stochastic integral is defined by the fractional calculus [ ].In this paper, we focus on the case that ∆ is the Dirichlet Laplacian and V = L (0 ,
1) equipped with the inner product h g, ˜ g i V := R g ( x )˜ g ( x ) dx . Then theeigensystem of A := − ∆ is { λ i , e i } ∞ i =1 with λ i = i π and e i ( x ) = √ iπx ),where { e i } ∞ i =1 forms an orthonormal basis of V . Defining U := Q U , we denoteby ˙ V θ the domain of A θ endowed with the norm k x k ˙ V θ := (cid:13)(cid:13)(cid:13) A θ x (cid:13)(cid:13)(cid:13) V , x ∈ ˙ V θ , θ ∈ R and by ( L , h· , ·i L ) the space of Hilbert–Schmidt operators from U to V equippedthe inner product h Φ , Φ i L := ∞ X i =1 h Φ Q f i , Φ Q f i i V . Our assumptions on (1) are stated as follows.
Assumption . There exists some γ such that u ∈ ˙ V H + γ − . UPER-CONVERGENCE ON EXPONENTIAL INTEGRATOR FOR SHE DRIVEN BY FBM 3
Assumption . The operator F : V → V is a Nemytskiioperator associated with a function f ∈ C ( R , R ) such that F ( X )( x ) = f ( X ( x )) , x ∈ (0 , , X ∈ V , and sup x ∈ R | f ′ ( x ) | + sup x ∈ R | f ′′ ( x ) | + sup x ∈ R | f ′′′ ( x ) | < ∞ . Assumption . There exists some γ such that (cid:13)(cid:13)(cid:13) A γ − (cid:13)(cid:13)(cid:13) L < ∞ . In the following, we formulate our main result for a fully discrete scheme con-struted by spectral Galerkin method and exponential integrator.
Theorem . Suppose that X is the mild solution of (1) and that X K,M,N isdefined by scheme (9) . Under Assumptions 1-3 with γ > max { − H, } , it holdsthat sup n =0 , ··· ,N (cid:13)(cid:13)(cid:13) X t n − X K,M,Nt n (cid:13)(cid:13)(cid:13) L (Ω; V ) ≤ CN − + CM − + C (cid:13)(cid:13) Q (Id U − P K ) (cid:13)(cid:13) L ( U, ˙ V − H ) , where Id U is the identity operator on U and P K is the project operator from U onto U K := span { f i : i = 1 , · · · , K } . Here and in the rest of the paper, we use C as a generic constant which isindependent of integers K, M, N in (7)-(9) and may be different from line to line.Since the exponent of temporal H¨older continuity of the solution proved in Section3 is the same as the Hurst parameter H , Theorem 1 indicates that the exponentialintegrator is super-convergent in time with strong order one of accuracy. We alsoremark that the exponential integrator does not require the CFL-type conditionappearing in most of explicit methods for SPDEs.As far as we know, Theorem 1 is the first super-convergence result in temporaldirection on full discretizations for SPDEs driven by infinite dimensional fBms withHurst parameter H ∈ ( , γ ≤ max { − H, } ,one can obtain the optimal strong convergence order in temporal direction basedon [ ]. For equations with additive noise which is fractional in space and whitein time, we refer to [
1, 2 ] and references therein for optimal error analysis onnumerical approximations. As H tends to , the parameter γ goes to 2 whichcoincides with the assumption on the SHEs driven by infinite dimensional standardBrownian motions for the strong order one of accuracy of the exponential integrator[
8, 9, 11 ].Compared with the standard Brownian setting, the main diffuculty in the super-convergence analysis on full discretizations for SHEs driven by infinite dimensionalfBms lies in that the fBm is neither a Markov process nor a semi-martingale suchthat the Burkholder–Davis–Gundy inequality is unavailable. As a consequence,we need to take a different strategy to estimate the terms J and J originatedfrom the stochastic Taylor’s expansion in Lemma 8. For term J which involves astochastic integral with respect to the fBm, we utilize the Malliavin calculus to sumup the accumulated errors first and then take the expectation, instead of to estimatethe strong order of accuracy of the local error first and then do the summation.For term J , to prove the temporal regularity of the mild solution in L (Ω; V ), wecombine the L p (Ω)-estimate of the Skorohod integral with respect to the fBm andthe smoothing effect of the Laplacian operator to overcome the difficulty from the JIALIN HONG AND CHUYING HUANG dependency of increments of fBm, i.e., to eliminate the influence of the kernel φ ofthe covariance of fBm.The paper is structured as follows. In Section 2, we introduce the Malliavincalculus with respect to the fBm. In Section 3, we show the regularity of the mildsolution of (1). In Section 4, we prove the optimal strong convergence order of thespectral Galerkin method and a priori estimates for the approximate mild solutionobtained by the spatial semi-discretization. In Section 5, we establish the super-convergence result in temporal direction on the exponential integrator. Section 6gives a conclusion and future works.
2. Preliminaries on Malliavin calculus
This section introduces the definition of the fBm and the associated Mallavincalculus. For more details, we refer to [
3, 15, 16 ].The K -dimensional fBm { B Kt = ( β t , · · · , β Kt ) } t ∈ [0 ,T ] with Hurst parameter H ∈ ( ,
1) is a centered Gaussian process with continuous sample paths and thecovariance E (cid:2) β it β js (cid:3) := 12 (cid:0) t H + s H − | t − s | H (cid:1) { i } ( j ) = (cid:18)Z t Z s φ ( u, v ) dudv (cid:19) { i } ( j ) , where φ ( u, v ) := α H | u − v | H − , α H := H (2 H − > i, j = 1 · · · , K , and { i } isthe indicator function. Define an inner product h· , ·i H by h (cid:0) [0 ,t ] , · · · , [0 ,t K ] (cid:1) , (cid:0) [0 ,s ] , · · · , [0 ,s K ] (cid:1) i H := K X i =1 E (cid:2) β it i β is i (cid:3) , and let the Hilbert space ( H , h· , ·i H ) be the closure of the space of all R K -valuedstep functions on [0 , T ] with respect to h· , ·i H . Then by extending the mapping (cid:0) [0 ,t ] , · · · , [0 ,t K ] (cid:1) P Ki =1 β it i , we obtain an isometry map ϕ B K ( ϕ ), which isfrom H to the Gaussian space associated with B K .For the random variable Y = y (cid:0) B K ( ϕ ) , · · · , B K ( ϕ M ) (cid:1) , (4)where ϕ , · · · , ϕ M ∈ H and y : R M → R is bounded with bounded derivatives ofany order, the Malliavin derivative of Y is an H -valued random variable defined by D t Y := M X i =1 ∂y∂x i (cid:0) B K ( ϕ ) , · · · , B K ( ϕ M ) (cid:1) ϕ it , t ∈ [0 , T ] . In particular, we denote by ( DY ) i the Malliavin derivative of Y with respect to β i , i = 1 , · · · , K . For p ≥
1, define D ,p as the Sobolev space which is the closure ofthe set containing random variables in the form of (4) with the norm k Y k D ,p := (cid:16) E (cid:2) | Y | p (cid:3) + E (cid:2) k DY k p H (cid:3)(cid:17) p . The chain rule holds so that for ˜ f with bounded derivative and Y ∈ D , , D ˜ f ( Y ) = ˜ f ′ ( Y ) DY.
Let δ be the adjoint operator of the derivative operator D . For an H -valuedrandom variable ϕ ∈ L (Ω; H ), if (cid:12)(cid:12) E (cid:2) h ϕ, DY i H (cid:3)(cid:12)(cid:12) ≤ C ( ϕ ) k Y k L (Ω; R ) , ∀ Y ∈ D , , UPER-CONVERGENCE ON EXPONENTIAL INTEGRATOR FOR SHE DRIVEN BY FBM 5 we say ϕ ∈ Dom( δ ). Then δ ( ϕ ) ∈ L (Ω; R ) is defined by the random variablesatisfying E (cid:2) h ϕ, DY i H (cid:3) = E (cid:2) Y δ ( ϕ ) (cid:3) , ∀ Y ∈ D , . (5)Indeed, the definition of the Malliavin derivative can be extended to H -valuedrandom variables. Then the space D ,p can be extended to D ,p ( H ) with the norm k Z k D ,p ( H ) := (cid:16) E (cid:2) k Z k p H (cid:3) + E (cid:2) k DZ k p H⊗H (cid:3)(cid:17) p . According to [ , Proposition 1.3.1], we have D , ( H ) ⊂ Dom( δ ). Moreover, theSkorohod integral of ϕ with respect to fBm is Z T ϕ t δB Kt := δ ( ϕ ) , ϕ ∈ Dom( δ ) , and the integration by parts formula holds that δ ( Y ϕ ) =
Y δ ( ϕ ) − h DY, ϕ i H , if ϕ ∈ Dom( δ ), Y ∈ D , and Y ϕ ∈ L (Ω; H ).The following lemmas are useful for us to deal with the stochastic integrals inthe regularity analysis and the error estimate. Lemma . (see also [ , Lemma 1] ) For g , g ∈ H , it holds that E (cid:2) Y δ ( g ) δ ( g ) (cid:3) = E (cid:2) h D [ h D [ Y ] , g i H ] , g i H (cid:3) + E (cid:2) Y h g , g i H (cid:3) . Proof.
Using (5) and the chain rule for the Malliavin derivative, we obtain E (cid:2) Y δ ( g ) δ ( g ) (cid:3) = E (cid:2) h D [ Y δ ( g )] , g i H (cid:3) = E (cid:2) h D [ Y ] , g i H δ ( g ) (cid:3) + E (cid:2) Y h g , g i H (cid:3) = E (cid:2) h D [ h D [ Y ] , g i H ] , g i H (cid:3) + E (cid:2) Y h g , g i H (cid:3) . (cid:3) Lemma . ( [ , Proposition 1.3.1 and Proposition 1.5.8] ) Let p > , k ϕ k |H| := K X i =1 Z [0 ,T ] | ϕ iu || ϕ iv | φ ( u, v ) dudv, k Dϕ k |H|⊗|H| := K X i,j =1 Z [0 ,T ] (cid:12)(cid:12)(cid:12) (cid:0) D v ϕ iu (cid:1) j (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:0) D v ϕ iu (cid:1) j (cid:12)(cid:12)(cid:12) φ ( v , v ) φ ( u , u ) dv du dv du . If a process ϕ ∈ Dom( δ ) satisfying (cid:13)(cid:13) E [ ϕ ] (cid:13)(cid:13) p |H| + E (cid:2) k Dϕ k p |H|⊗|H| (cid:3) < ∞ , then E "(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z T ϕ t δB Kt (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) p ≤ C ( H, p ) (cid:16)(cid:13)(cid:13) E [ ϕ ] (cid:13)(cid:13) p |H| + E (cid:2) k Dϕ k p |H|⊗|H| (cid:3)(cid:17) . In particular, if K = 1 and p = 2 , it holds that E (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z T ϕ t δB t (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = E " Z [0 ,T ] ϕ u ϕ v φ ( u, v ) dudv + Z [0 ,T ] D v ϕ u D v ϕ u φ ( v , u ) φ ( v , u ) dv du dv du . JIALIN HONG AND CHUYING HUANG
3. Well-posedness and regularity
In this section, we show the regularity of the mild solution of (1), which relieson the parameters γ and H . We begin with several Lemmas concerned aboutthe smoothing effect of the Laplacian operator and the isometry of the stochasticintegral with respect to the fBm. Lemma . ( [ , Lemma B.9] ) For any < t ≤ T , ν ≤ ≤ µ , ≤ α ≤ and x ∈ V , it holds that k A ν k L ( V ) ≤ C, k A − α ( S t − Id V ) k L ( V ) ≤ Ct α , k A µ S t k L ( V ) ≤ Ct − µ , (cid:13)(cid:13)(cid:13)(cid:13) A α Z ts S t − σ xdσ (cid:13)(cid:13)(cid:13)(cid:13) V ≤ C | t − s | − α k x k V . Lemma . ( [ , Lemma 3.6] ) There exists some constant C = C ( H ) such thatfor any ≤ ρ ≤ H , ≤ s < t ≤ T and x ∈ V , Z ts Z ts h A ρ S t − u x, A ρ S t − v x i V φ ( u, v ) dudv ≤ C ( t − s ) H − ρ ) k x k V . Lemma . For
Φ : [0 , T ] → L , it holds that E (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)Z T Φ t dW Q t (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) V = ∞ X i =1 Z T Z T h Φ u Q f i , Φ v Q f i i V φ ( u, v ) dudv, i.e., E (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)Z T Φ t dW Q t (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) V = Z T Z T h Φ u , Φ v i L φ ( u, v ) dudv. Proof.
Using the definiton of W Q and the Parseval equality, we have E (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)Z T Φ t dW Q t (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) V = ∞ X j =1 ∞ X i =1 E (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z T h Φ t Q f i , e j i V dβ it (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . Thanks to Lemma 2, we obtain E (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)Z T Φ t dW Q t (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) V = ∞ X j =1 ∞ X i =1 Z T Z T h Φ u Q f i , e j i V h Φ v Q f i , e j i V φ ( u, v ) dudv = ∞ X i =1 Z T Z T h Φ u Q f i , Φ v Q f i i V φ ( u, v ) dudv = Z T Z T h Φ u , Φ v i L φ ( u, v ) dudv. (cid:3) The following theorem provides the optimal regularity of the mild solution of(1).
UPER-CONVERGENCE ON EXPONENTIAL INTEGRATOR FOR SHE DRIVEN BY FBM 7
Theorem . Let Assumptions 1-3 be satisfied with γ ∈ (1 , − H ] . Then (1) admits a unique mild solution such that k X t k L (Ω; ˙ V H + γ − ) ≤ C (cid:16) k u k ˙ V H + γ − (cid:17) . Moreover, for any µ ∈ [0 , H + γ − , it holds that k X t − X s k L (Ω; ˙ V µ ) ≤ C (cid:16) k u k ˙ V H + γ − (cid:17) | t − s | min (cid:8) H + γ − − µ ,H (cid:9) . Proof.
Under Assumptions 1-3, the existence and uniqueness of the mild so-lution in L (Ω; V ) follows from Lemma 5 and the Gronwall’s inequality. In thesequel, we concentrate on the proof for the regularity of the solution.Suppose 0 ≤ s < t ≤ T . By means of the formulation (3) and Assumption 1,we get k X t k L (Ω; ˙ V H + γ − ) ≤k u k ˙ V H + γ − + (cid:13)(cid:13)(cid:13)(cid:13)Z t S t − s F ( X s ) ds (cid:13)(cid:13)(cid:13)(cid:13) L (Ω; ˙ V H + γ − ) + (cid:13)(cid:13)(cid:13)(cid:13)Z t S t − s dW Q s (cid:13)(cid:13)(cid:13)(cid:13) L (Ω; ˙ V H + γ − ) . Applying Lemmas 4-5 with ρ = H , we derive from Assumption 3 that (cid:13)(cid:13)(cid:13)(cid:13)Z t S t − s dW Q s (cid:13)(cid:13)(cid:13)(cid:13) L (Ω; ˙ V H + γ − ) = Z t Z t h A H S t − u A γ − , A H S t − v A γ − i L φ ( u, v ) dudv = ∞ X i =1 Z t Z t h A H S t − u A γ − Q f i , A H S t − v A γ − Q f i i V φ ( u, v ) dudv ≤ C (cid:13)(cid:13)(cid:13) A γ − (cid:13)(cid:13)(cid:13) L . If γ ∈ (1 , − H ), Lemma 3 and Assumption 2 lead to (cid:13)(cid:13)(cid:13)(cid:13)Z t S t − s F ( X s ) ds (cid:13)(cid:13)(cid:13)(cid:13) L (Ω; ˙ V H + γ − ) ≤ Z t (cid:13)(cid:13)(cid:13) A H + γ − S t − s F ( X s ) (cid:13)(cid:13)(cid:13) L (Ω; V ) ds (6) ≤ C Z t | t − s | − H + γ − k F ( X s ) k L (Ω; V ) ds ≤ C (cid:18)Z t | t − s | − H + γ − ds (cid:19) (cid:16) k u k ˙ V H + γ − (cid:17) ≤ C (cid:16) k u k ˙ V H + γ − (cid:17) . Then k X t k L (Ω; ˙ V H + γ − ) ≤ C (cid:16) k u k ˙ V H + γ − (cid:17) . Considering the temporal regularity, we have k X t − X s k L (Ω; ˙ V µ ) ≤k ( S t − s − Id V ) X s k L (Ω; ˙ V µ ) + (cid:13)(cid:13)(cid:13)(cid:13)Z ts S t − σ F ( X σ ) dσ (cid:13)(cid:13)(cid:13)(cid:13) L (Ω; ˙ V µ ) + (cid:13)(cid:13)(cid:13)(cid:13)Z ts S t − σ dW Q σ (cid:13)(cid:13)(cid:13)(cid:13) L (Ω; ˙ V µ ) . JIALIN HONG AND CHUYING HUANG
Applying Lemma 4 with ρ = µ +1 − γ ∈ [0 , H ], i.e., µ ∈ [ γ − , H + γ − (cid:13)(cid:13)(cid:13)(cid:13)Z ts S t − σ dW Q σ (cid:13)(cid:13)(cid:13)(cid:13) L (Ω; ˙ V µ ) = ∞ X i =1 Z ts Z ts h A µ +1 − γ S t − u A γ − Q f i , A µ +1 − γ S t − v A γ − Q f i i V φ ( u, v ) dudv ≤ C | t − s | min (cid:8) H + γ − − µ, H (cid:9) (cid:13)(cid:13)(cid:13) A γ − (cid:13)(cid:13)(cid:13) L . Lemma 3 implies that for any µ ∈ [0 , (cid:13)(cid:13)(cid:13)(cid:13)Z ts S t − σ F ( X σ ) dσ (cid:13)(cid:13)(cid:13)(cid:13) L (Ω; ˙ V µ ) ≤ Z ts (cid:13)(cid:13)(cid:13) A µ S t − σ F ( X σ ) (cid:13)(cid:13)(cid:13) L (Ω; V ) dσ ≤ C | t − s | − µ (cid:16) k u k ˙ V H + γ − (cid:17) . Combining with k ( S t − s − Id V ) X s k L (Ω; ˙ V µ ) = (cid:13)(cid:13)(cid:13) A µ A − H + γ − ( S t − s − Id V ) A H + γ − X s (cid:13)(cid:13)(cid:13) L (Ω; V ) ≤ C | t − s | H + γ − − µ k X s k L (Ω; ˙ V H + γ − ) , we have k X t − X s k L (Ω; ˙ V µ ) ≤ C (cid:16) k u k ˙ V H + γ − (cid:17) | t − s | min (cid:8) H + γ − − µ ,H (cid:9) , for γ ∈ (1 , − H ) and µ ∈ [0 , H + γ − γ = 3 − H , it suffices to revise the estimate in (6). The previousarguments yield a priori estimates that k X t k L (Ω; ˙ V ) ≤ C (cid:16) k u k ˙ V (cid:17) , k X t − X s k L (Ω; ˙ V ) ≤ C (cid:16) k u k ˙ V (cid:17) | t − s | . Taking advantage of Lemma 3 and a priori estimates above, we deduce (cid:13)(cid:13)(cid:13)(cid:13)Z t S t − s F ( X s ) ds (cid:13)(cid:13)(cid:13)(cid:13) L (Ω; ˙ V ) ≤ (cid:13)(cid:13)(cid:13)(cid:13) A Z t S t − s F ( X t ) ds (cid:13)(cid:13)(cid:13)(cid:13) L (Ω; V ) + (cid:13)(cid:13)(cid:13)(cid:13) A Z t S t − s ( F ( X t ) − F ( X s )) ds (cid:13)(cid:13)(cid:13)(cid:13) L (Ω; V ) ≤ C k F ( X t ) k L (Ω; V ) + C Z t ( t − s ) − k F ( X t ) − F ( X s ) k L (Ω; V ) ds ≤ C (cid:16) k u k ˙ V (cid:17) (cid:18) Z t ( t − s ) − ds (cid:19) ≤ C (cid:16) k u k ˙ V (cid:17) , from which we complete the proof. (cid:3) UPER-CONVERGENCE ON EXPONENTIAL INTEGRATOR FOR SHE DRIVEN BY FBM 9
4. Spatial semi-discretizaiton
In this section, we study the spatial semi-discretization for (1). First, basedon the K -dimensional subspace U K := span { f i : i = 1 , · · · , K } ⊂ U , we truncatethe infinite dimensional fBm to obtain the SHE driven by the K -dimensional fBm.We give the estimates for Malliavin derivatives of its mild solution and the strongconvergence order associated with the truncation. Next, we apply the spectralGalerkin method to spatially discretize the SHE driven by the K -dimensional fBmand show the optimal strong convergence rate of the spectral Galerkin method,which coincides with the optimal spatial regularity of the solution of (1). Further-more, in preparation for proving that the exponential integrator is super-convergentin time in the next section, we derive the temporal regularity of the mild solutionin L (Ω; V ) as a priori estimate.Let W Q ,Kt := K X i =1 Q f i β it be the truncation of the infinite dimensional fBm. Consider the mild solution X Kt of the SHE driven by the K -dimensional fBm ( dX Kt = − AX Kt dt + F ( X Kt ) dt + dW Q ,Kt , t ∈ (0 , T ] ,X K = u . (7)For any x ∈ (0 , t ∈ (0 , T ] and p ≥
1, one knows from [ , Proposition 3.3 andLemma 3.4] and [ , Proposition 7] that the random variable X Kt ( x ) ∈ D ,p . Theestimates for the Malliavin derivatives are given as follows. Lemma . Let Assumptions 1-2 be satisfied with γ > . Denote Ψ i t,r ( x ) := (cid:0) D r X Kt ( x ) (cid:1) i and Ψ i ,i t,r ,r ( x ) := (cid:0) D r (cid:0) D r X Kt ( x ) (cid:1) i (cid:1) i , for ≤ r , r ≤ t and ≤ i , i ≤ K . Then for any ≤ r < , there exists some constant C = C ( T, F, r ) such that (cid:13)(cid:13) Ψ i t,r (cid:13)(cid:13) ˙ V r ≤ C (cid:13)(cid:13) Q f i (cid:13)(cid:13) ˙ V r , (cid:13)(cid:13) Ψ i ,i t,r ,r (cid:13)(cid:13) V ≤ C (cid:13)(cid:13) Q f i (cid:13)(cid:13) V (cid:13)(cid:13) Q f i (cid:13)(cid:13) V . Proof.
Based on [ , Proposition 3.3 and Lemma 3.4] and [ , Proposition 7],we have that Ψ i t,r , Ψ i ,i t,r ,r satisfy the linear equations Ψ i t,r = S t − r Q f i + Z tr S t − s (cid:0) F ′ ( X Ks ) Ψ i s,r (cid:1) ds,Ψ i ,i t,r ,r = Z tr ∨ r S t − s (cid:0) F ′′ ( X Ks ) Ψ i s,r Ψ i s,r (cid:1) ds + Z tr ∨ r S t − s (cid:0) F ′ ( X Ks ) Ψ i ,i s,r ,r (cid:1) ds, respectively. The uniform boundedness of { S t } t ≥ , Lemma 3 and Assumption 2lead to (cid:13)(cid:13) Ψ i t,r (cid:13)(cid:13) ˙ V r ≤ (cid:13)(cid:13) Q f i (cid:13)(cid:13) ˙ V r + C Z tr (cid:13)(cid:13) A r S t − s (cid:0) F ′ ( X Ks ) Ψ i s,r (cid:1)(cid:13)(cid:13) V ds ≤ (cid:13)(cid:13) Q f i (cid:13)(cid:13) ˙ V r + C Z tr | t − s | − r (cid:13)(cid:13) F ′ ( X Ks ) Ψ i s,r (cid:13)(cid:13) V ds ≤ (cid:13)(cid:13) Q f i (cid:13)(cid:13) ˙ V r + C Z tr | t − s | − r (cid:13)(cid:13) Ψ i s,r (cid:13)(cid:13) V ds. Then the Gronwall’s inequality yields (cid:13)(cid:13) Ψ i t,r (cid:13)(cid:13) ˙ V r ≤ C (cid:13)(cid:13) Q f i (cid:13)(cid:13) ˙ V r . For the second derivative, utilizing Lemma 3 and the Sobolev embedding ˙ V α ֒ → L ∞ ([0 , < α <
1, we have (cid:13)(cid:13) S t − s (cid:0) F ′′ ( X Ks ) Ψ i s,r Ψ i s,r (cid:1)(cid:13)(cid:13) V = (cid:13)(cid:13) A α S t − s A − α (cid:0) F ′′ ( X Ks ) Ψ i s,r Ψ i s,r (cid:1)(cid:13)(cid:13) V ≤| t − s | − α (cid:13)(cid:13) F ′′ ( X Ks ) Ψ i s,r Ψ i s,r (cid:13)(cid:13) ˙ V − α ≤| t − s | − α Z (cid:12)(cid:12) f ′′ ( X Ks ( x )) Ψ i s,r ( x ) Ψ i s,r ( x ) (cid:12)(cid:12) dx ≤| t − s | − α (cid:13)(cid:13) Ψ i s,r (cid:13)(cid:13) V (cid:13)(cid:13) Ψ i s,r (cid:13)(cid:13) V . Therefore, we obtain (cid:13)(cid:13) Ψ i ,i t,r ,r (cid:13)(cid:13) V ≤ C Z tr ∨ r | t − s | − α (cid:13)(cid:13) Ψ i s,r (cid:13)(cid:13) V (cid:13)(cid:13) Ψ i s,r (cid:13)(cid:13) V ds + C Z tr ∨ r (cid:13)(cid:13) Ψ i ,i s,r ,r (cid:13)(cid:13) V ds ≤ C Z tr ∨ r | t − s | − α (cid:13)(cid:13) Q f i (cid:13)(cid:13) V (cid:13)(cid:13) Q f i (cid:13)(cid:13) V ds + C Z tr ∨ r (cid:13)(cid:13) Ψ i ,i s,r ,r (cid:13)(cid:13) V ds ≤ C (cid:13)(cid:13) Q f i (cid:13)(cid:13) V (cid:13)(cid:13) Q f i (cid:13)(cid:13) V + C Z tr ∨ r (cid:13)(cid:13) Ψ i ,i s,r ,r (cid:13)(cid:13) V ds ≤ C (cid:13)(cid:13) Q f i (cid:13)(cid:13) V (cid:13)(cid:13) Q f i (cid:13)(cid:13) V , where the Gronwall’s inequality is used in the last inequality. (cid:3) Denote P K as the project operator from U onto U K . We give the strong erroranalysis for the SHE driven by the K -dimensional fBm. Theorem . Let X and X K be mild solutions of (1) and (7) , respectively.Under Assumptions 1-3 with γ > , it holds that sup t ∈ [0 ,T ] (cid:13)(cid:13) X t − X Kt (cid:13)(cid:13) L (Ω; V ) ≤ C (cid:13)(cid:13) Q (Id U − P K ) (cid:13)(cid:13) L ( U, ˙ V − H ) . Proof.
The error has the following decomposition (cid:13)(cid:13) X t − X Kt (cid:13)(cid:13) L (Ω; V ) ≤ (cid:13)(cid:13)(cid:13)(cid:13)Z t S t − s (cid:0) F ( X s ) − F ( X Ks ) (cid:1) ds (cid:13)(cid:13)(cid:13)(cid:13) L (Ω; V ) + (cid:13)(cid:13)(cid:13)(cid:13)Z t S t − s dW Q s − Z t S t − s dW Q ,Ks (cid:13)(cid:13)(cid:13)(cid:13) L (Ω; V ) . Using Lemmas 4-5 with ρ = H , we get E "(cid:13)(cid:13)(cid:13)(cid:13)Z t S t − s dW Q s − Z t S t − s dW Q ,Ks (cid:13)(cid:13)(cid:13)(cid:13) V = ∞ X i = K +1 Z t Z t h S t − u Q f i , S t − v Q f i i V φ ( u, v ) dudv UPER-CONVERGENCE ON EXPONENTIAL INTEGRATOR FOR SHE DRIVEN BY FBM 11 = ∞ X i = K +1 Z t Z t h A H S t − u A − H Q f i , A H S t − v A − H Q f i i V φ ( u, v ) dudv ≤ C ∞ X i = K +1 (cid:13)(cid:13) A − H Q f i (cid:13)(cid:13) V = C (cid:13)(cid:13) Q (Id U − P K ) (cid:13)(cid:13) L ( U, ˙ V − H ) . Hence, (cid:13)(cid:13) X t − X Kt (cid:13)(cid:13) L (Ω; V ) ≤ C Z t (cid:13)(cid:13) X s − X Ks (cid:13)(cid:13) L (Ω; V ) ds + C (cid:13)(cid:13) Q (Id U − P K ) (cid:13)(cid:13) L ( U, ˙ V − H ) , from which we conclude the result by the Gronwall’s inequality. (cid:3) In the next step, we spatially discretize (7) by the spectral Galerkin method.More precisely, denoting by P M the project operator from V onto V M := span { e i : i = 1 , · · · , M } ⊂ V and A M := AP M , we obtain ( dX K,Mt = − A M X K,Mt dt + P M F ( X K,Mt ) dt + P M dW Q ,Kt , t ∈ (0 , T ] ,X K,M = P M u . (8)The optimal strong convergence rate of the spectral Galerkin method is proved asfollows. Theorem . Suppose that X K and X K,M are mild solutions of (7) and (8) ,respectively. Under Assumptions 1-3 with γ ∈ (1 , − H ] , it holds that sup t ∈ [0 ,T ] (cid:13)(cid:13)(cid:13) X Kt − X K,Mt (cid:13)(cid:13)(cid:13) L (Ω; V ) ≤ C (cid:16) k u k ˙ V H + γ − (cid:17) M − (2 H + γ − . Proof.
Define by { S Mt } t ≥ the analytic semigroup generated by − A M . Wedecompose the error by (cid:13)(cid:13)(cid:13) X Kt − X K,Mt (cid:13)(cid:13)(cid:13) L (Ω; V ) ≤ k (Id V − P M ) u k L (Ω; V ) + (cid:13)(cid:13)(cid:13)(cid:13)Z t (cid:0) S t − s − S Mt − s (cid:1) F ( X Ks ) ds (cid:13)(cid:13)(cid:13)(cid:13) L (Ω; V ) + (cid:13)(cid:13)(cid:13)(cid:13)Z t (cid:0) S Mt − s F ( X Ks ) − S Mt − s F ( X K,Ms ) (cid:1) ds (cid:13)(cid:13)(cid:13)(cid:13) L (Ω; V ) + (cid:13)(cid:13)(cid:13)(cid:13)Z t (cid:0) S t − s − S Mt − s (cid:1) dW Q ,Ks (cid:13)(cid:13)(cid:13)(cid:13) L (Ω; V ) ≤ C k u k ˙ V H + γ − M − (2 H + γ − + Z t (cid:13)(cid:13)(cid:0) S t − s − S Mt − s (cid:1) F ( X Ks ) (cid:13)(cid:13) L (Ω; V ) ds + Z t (cid:13)(cid:13)(cid:13) X Ks − X K,Ms (cid:13)(cid:13)(cid:13) L (Ω; V ) ds + (cid:13)(cid:13)(cid:13)(cid:13)Z t (cid:0) S t − s − S Mt − s (cid:1) dW Q ,Ks (cid:13)(cid:13)(cid:13)(cid:13) L (Ω; V ) . By Lemma 3 and Theorem 2, we get Z t (cid:13)(cid:13)(cid:0) S t − s − S Mt − s (cid:1) F ( X Ks ) (cid:13)(cid:13) L (Ω; V ) ds ≤ Z t (cid:13)(cid:13)(cid:0) S t − s − S Mt − s (cid:1) F ( X Kt ) (cid:13)(cid:13) L (Ω; V ) ds + Z t (cid:13)(cid:13)(cid:0) S t − s − S Mt − s (cid:1)(cid:0) F ( X Kt ) − F ( X Ks ) (cid:1)(cid:13)(cid:13) L (Ω; V ) ds ≤ Z t (cid:13)(cid:13) (Id V − P M ) F ( X Kt ) (cid:13)(cid:13) L (Ω; V ) ds + Z t (cid:13)(cid:13)(cid:13) A H + γ − S t − s (Id V − P M ) A − (2 H + γ − (cid:0) F ( X Kt ) − F ( X Ks ) (cid:1)(cid:13)(cid:13)(cid:13) L (Ω; V ) ds ≤ CM − (2 H + γ − (cid:13)(cid:13) F ( X Kt ) (cid:13)(cid:13) L (Ω; ˙ V H + γ − ) + CM − (2 H + γ − Z t ( t − s ) − (2 H + γ − (cid:13)(cid:13) F ( X Kt ) − F ( X Ks ) (cid:13)(cid:13) L (Ω; V ) ds ≤ CM − (2 H + γ − . According to Lemma 5 and [ , Lemma 4.1], we have (cid:13)(cid:13)(cid:13)(cid:13)Z t (cid:0) S t − s − S Mt − s (cid:1) dW Q ,Ks (cid:13)(cid:13)(cid:13)(cid:13) L (Ω; V ) = K X i =1 Z t Z t h (cid:0) S t − u − S Mt − u (cid:1) Q f i , (cid:0) S t − v − S Mt − v (cid:1) Q f i i V φ ( u, v ) dudv ≤ C ∞ X i =1 M − H + γ − (cid:13)(cid:13) Q f i (cid:13)(cid:13) V γ − ≤ CM − H + γ − (cid:13)(cid:13)(cid:13) A γ − (cid:13)(cid:13)(cid:13) L . Then the strong convergence order of the spectral Galerkin method in spatial di-rection is a consequence of the Gronwall’s inequality. (cid:3)
Remark . Suppose U = V , K = M and W Q t = P ∞ i =1 η i e i β it . One can utilize P M to perform the spectral Galerkin method in terms of the noise W Q and thesolution X at the same time, with the spatial convergence rate O (cid:0) M − (2 H + γ − (cid:1) . The following lemma gives the estimates for the fourth-order moment of the sto-chastic integral with respect to the K -dimensional fBm. Based on these estimates,we obtain the temporal regularity of X K,M in L (Ω; V ). Lemma . Let Assumption 3 be satisfied with γ > − H . Then (cid:13)(cid:13)(cid:13)(cid:13)Z t S Mt − s dW Q ,Ks (cid:13)(cid:13)(cid:13)(cid:13) L (cid:18) Ω; ˙ V H + γ − (cid:19) ≤ C, (cid:13)(cid:13)(cid:13)(cid:13)Z ts S Mt − σ dW Q ,Kσ (cid:13)(cid:13)(cid:13)(cid:13) L (Ω; V ) ≤ C | t − s | . Proof.
By the definition of W Q ,K , we have (cid:13)(cid:13)(cid:13)(cid:13)Z t S Mt − s dW Q ,Ks (cid:13)(cid:13)(cid:13)(cid:13) L (cid:18) Ω; ˙ V H + γ − (cid:19) = (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) K X j =1 Z t A H + γ − S Mt − s Q f j dβ js (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L (Ω; V ) = (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) M X i =1 K X j =1 Z t h A H + γ − S Mt − s Q f j , e i i V dβ js e i (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L (Ω; V )UPER-CONVERGENCE ON EXPONENTIAL INTEGRATOR FOR SHE DRIVEN BY FBM 13 ≤ M X i =1 (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) K X j =1 Z t h A H + γ − S Mt − s Q f j , e i i V dβ js e i (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L (Ω; V ) ≤ M X i =1 (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) K X j =1 Z t h A H + γ − S Mt − s Q f j , e i i V dβ js (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L (Ω; R ) . Denoting B Ks = ( β s , · · · , β Ks )and ϕ s = (cid:16) h A H + γ − S Mt − s Q f , e i i V , · · · , h A H + γ − S Mt − s Q f K , e i i V (cid:17) ∈ R K , we derive from Lemma 2 that (cid:13)(cid:13)(cid:13)(cid:13)Z t S Mt − s dW Q ,Ks (cid:13)(cid:13)(cid:13)(cid:13) L (cid:18) Ω; ˙ V H + γ − (cid:19) ≤ M X i =1 (cid:13)(cid:13)(cid:13)(cid:13)Z t ϕ s δB Ks (cid:13)(cid:13)(cid:13)(cid:13) L (Ω; R ) ≤ C M X i =1 " K X j =1 Z t Z t (cid:12)(cid:12)(cid:12) h A H + γ − S Mt − u Q f j , e i i V (cid:12)(cid:12)(cid:12) (cid:12)(cid:12)(cid:12) h A H + γ − S Mt − v Q f j , e i i V (cid:12)(cid:12)(cid:12) φ ( u, v ) dudv . Under Assumption 3, we obtain K X j =1 Z t Z t (cid:12)(cid:12)(cid:12) h A H + γ − S Mt − u Q f j , e i i V (cid:12)(cid:12)(cid:12) (cid:12)(cid:12)(cid:12) h A H + γ − S Mt − v Q f j , e i i V (cid:12)(cid:12)(cid:12) φ ( u, v ) dudv ≤ ∞ X j =1 (cid:13)(cid:13)(cid:13) A γ − Q f j (cid:13)(cid:13)(cid:13) V Z t Z t λ H − γ +12 i e − λ i ( t − u ) e − λ i ( t − v ) φ ( u, v ) dudv = (cid:13)(cid:13)(cid:13) A γ − (cid:13)(cid:13)(cid:13) L Z t Z t λ H − γ +12 i e − λ i ( t − u ) e − λ i ( t − v ) φ ( u, v ) dudv. Hence, (cid:13)(cid:13)(cid:13)(cid:13)Z t S Mt − s dW Q ,Ks (cid:13)(cid:13)(cid:13)(cid:13) L (cid:18) Ω; ˙ V H + γ − (cid:19) ≤ C (cid:13)(cid:13)(cid:13) A γ − (cid:13)(cid:13)(cid:13) L M X i =1 " Z t Z t λ H − γ +12 i e − λ i ( t − u ) e − λ i ( t − v ) φ ( u, v ) dudv . Similarly, (cid:13)(cid:13)(cid:13)(cid:13)Z ts S Mt − σ dW Q ,Kσ (cid:13)(cid:13)(cid:13)(cid:13) L (Ω; V ) ≤ C (cid:13)(cid:13)(cid:13) A γ − (cid:13)(cid:13)(cid:13) L M X i =1 " Z ts Z ts λ − ( γ − i e − λ i ( t − u ) e − λ i ( t − v ) φ ( u, v ) dudv . It suffices to show ∞ X i =1 " Z t Z t λ H − γ +12 i e − λ i ( t − u ) e − λ i ( t − v ) φ ( u, v ) dudv ≤ C and ∞ X i =1 " Z ts Z ts λ − ( γ − i e − λ i ( t − u ) e − λ i ( t − v ) φ ( u, v ) dudv ≤ C | t − s | . Indeed, since H ∈ ( , p > H − p > − p + q = 1, we obtain Z t Z t e − λ i ( t − u ) e − λ i ( t − v ) φ ( u, v ) dudv = Z t e − λ i ( t − v ) (cid:20)Z t e − λ i ( t − u ) φ ( u, v ) du (cid:21) dv = Z t e − λ i ( t − v ) (cid:20)Z t e − λ i ( t − u ) q du (cid:21) q (cid:20)Z t φ ( u, v ) p du (cid:21) p dv ≤ C Z t e − λ i ( t − v ) (cid:20)Z t e − λ i ( t − u ) q du (cid:21) q dv ≤ C λ i (cid:18) qλ i (cid:19) q . Let α = γ − (3 − H ). Taking 0 < ǫ < min (cid:8) H − , α (cid:9) and p = − H + ǫ , we have p >
1, (2 H − p > − q = 2 H − − ǫ , which leads to ∞ X i =1 " Z t Z t λ H − γ +12 i e − λ i ( t − u ) e − λ i ( t − v ) φ ( u, v ) dudv ≤ C ∞ X i =1 i H − γ +12 i (cid:18) i (cid:19) q = C ∞ X i =1 i − ǫ − α ≤ C. Similarly, based on Z ts Z ts e − λ i ( t − u ) e − λ i ( t − v ) φ ( u, v ) dudv = Z ts e − λ i ( t − v ) (cid:20)Z ts e − λ i ( t − u ) φ ( u, v ) du (cid:21) dv = Z ts e − λ i ( t − v ) (cid:20)Z ts e − λ i ( t − u ) q du (cid:21) q (cid:20)Z ts φ ( u, v ) p du (cid:21) p dv ≤ C Z ts e − λ i ( t − v ) (cid:20)Z ts e − λ i ( t − u ) q du (cid:21) q dv ≤ C | t − s | (cid:18) qλ i (cid:19) q , we obtain ∞ X i =1 " Z ts Z ts λ − ( γ − i e − λ i ( t − u ) e − λ i ( t − v ) φ ( u, v ) dudv ≤ C ∞ X i =1 | t − s | i − ( γ − (cid:18) i (cid:19) q ≤ C ∞ X i =1 | t − s | i − ǫ − α ≤ C | t − s | . UPER-CONVERGENCE ON EXPONENTIAL INTEGRATOR FOR SHE DRIVEN BY FBM 15
This finishes the proof. (cid:3)
Theorem . Under Assumptions 1-3 with γ ∈ (3 − H, − H ] , the mildsolution of (8) satisfies (cid:13)(cid:13)(cid:13) X K,Mt (cid:13)(cid:13)(cid:13) L (cid:18) Ω; ˙ V H + γ − (cid:19) ≤ C (cid:16) k u k ˙ V H + γ − (cid:17) , (cid:13)(cid:13)(cid:13) X K,Mt − X K,Ms (cid:13)(cid:13)(cid:13) L (Ω; V ) ≤ C (cid:16) k u k ˙ V H + γ − (cid:17) | t − s | . Proof.
According to Lemma 7, we have (cid:13)(cid:13)(cid:13)(cid:13)Z t S Mt − s dW Q ,Ks (cid:13)(cid:13)(cid:13)(cid:13) L (Ω; V ) ≤ C. Then the Gronwall’s inequality yields (cid:13)(cid:13)(cid:13) X K,Mt (cid:13)(cid:13)(cid:13) L (Ω; V ) ≤ C (cid:16) k u k ˙ V H + γ − (cid:17) . It follows from Lemma 3 and Lemma 7 that for γ ∈ (3 − H, − H ), (cid:13)(cid:13)(cid:13) X K,Mt (cid:13)(cid:13)(cid:13) L (cid:18) Ω; ˙ V H + γ − (cid:19) ≤k u k ˙ V H + γ − + (cid:13)(cid:13)(cid:13)(cid:13)Z t S Mt − s F ( X K,Ms ) ds (cid:13)(cid:13)(cid:13)(cid:13) L (cid:18) Ω; ˙ V H + γ − (cid:19) + (cid:13)(cid:13)(cid:13)(cid:13)Z t S Mt − s dW Q ,Ks (cid:13)(cid:13)(cid:13)(cid:13) L (cid:18) Ω; ˙ V H + γ − (cid:19) ≤k u k ˙ V H + γ − + Z t (cid:13)(cid:13)(cid:13) A H + γ − S Mt − s F ( X K,Ms ) (cid:13)(cid:13)(cid:13) L (Ω; V ) ds + (cid:13)(cid:13)(cid:13)(cid:13)Z t S Mt − s dW Q ,Ks (cid:13)(cid:13)(cid:13)(cid:13) L (cid:18) Ω; ˙ V H + γ − (cid:19) ≤k u k ˙ V H + γ − + Z t | t − s | − H + γ − (cid:13)(cid:13) F ( X K,Ms ) (cid:13)(cid:13) L (Ω; V ) ds + (cid:13)(cid:13)(cid:13)(cid:13)Z t S Mt − s dW Q ,Ks (cid:13)(cid:13)(cid:13)(cid:13) L (cid:18) Ω; ˙ V H + γ − (cid:19) ≤ C (cid:16) k u k ˙ V H + γ − (cid:17) . Furthermore, we get from Lemma 7 that (cid:13)(cid:13)(cid:13) X K,Mt − X K,Ms (cid:13)(cid:13)(cid:13) L (Ω; V ) ≤ (cid:13)(cid:13)(cid:13)(cid:0) S Mt − s − Id V (cid:1) X K,Ms (cid:13)(cid:13)(cid:13) L (Ω; V ) + (cid:13)(cid:13)(cid:13)(cid:13)Z ts S Mt − σ F ( X K,Mσ ) dσ (cid:13)(cid:13)(cid:13)(cid:13) L (Ω; V ) + (cid:13)(cid:13)(cid:13)(cid:13)Z ts S Mt − σ dW Q ,Kσ (cid:13)(cid:13)(cid:13)(cid:13) L (Ω; V ) ≤ C | t − s | H + γ − ∧ (cid:13)(cid:13)(cid:13) X K,Ms (cid:13)(cid:13)(cid:13) L (cid:18) Ω; ˙ V H + γ − ∧ (cid:19) + C | t − s | (cid:16) k u k L (Ω; V ) (cid:17) + C | t − s | ≤ C (cid:16) k u k ˙ V H + γ − (cid:17) | t − s | . If γ = 5 − H , using Lemma 3, we get (cid:13)(cid:13)(cid:13)(cid:13)Z t S Mt − s F ( X K,Ms ) ds (cid:13)(cid:13)(cid:13)(cid:13) L (cid:18) Ω; ˙ V H + γ − (cid:19) ≤ (cid:13)(cid:13)(cid:13)(cid:13)Z t S Mt − s F ( X K,Mt ) ds (cid:13)(cid:13)(cid:13)(cid:13) L (cid:18) Ω; ˙ V H + γ − (cid:19) + (cid:13)(cid:13)(cid:13)(cid:13)Z t S Mt − s (cid:0) F ( X K,Mt ) − F ( X K,Ms ) (cid:1) ds (cid:13)(cid:13)(cid:13)(cid:13) L (cid:18) Ω; ˙ V H + γ − (cid:19) ≤ C (cid:13)(cid:13)(cid:13) X K,Mt (cid:13)(cid:13)(cid:13) L (Ω; V ) + Z t | t − s | − H + γ − (cid:13)(cid:13)(cid:13) X K,Mt − X K,Ms (cid:13)(cid:13)(cid:13) L (Ω; V ) ds ≤ C (cid:16) k u k ˙ V H + γ − (cid:17) , from which we complete the proof. (cid:3)
5. Full discretization
In this section, we apply the exponential integrator to (8) to construct a fullydiscrete scheme, which is ( X K,M,Nt n := S Mh X K,M,Nt n − + S Mh F ( X K,M,Nt n − ) h + S Mh ∆ W Q ,Kn ,X K,M,N := P M u , (9)where n = 1 , · · · , N , h := TN , N ∈ N + and ∆ W Q ,Kn := W Q ,Kt n − W Q ,Kt n − .Now we are in the position to prove that the exponential integrator is super-convergent in time, together with Lemma 8, where the Malliavin calculus is anessential tool. Theorem . Suppose that X K,M is the mild solution of (8) and that X K,M,N is defined by scheme (9) . Under Assumptions 1-3 with γ > max { − H, } , itholds that sup n =0 , ··· ,N (cid:13)(cid:13)(cid:13) X K,Mt n − X K,M,Nt n (cid:13)(cid:13)(cid:13) L (Ω; V ) ≤ C (cid:16) k u k ˙ V H + γ − (cid:17) N − . Proof.
We rewrite scheme (9) as X K,M,Nt n = S Mt n P M u + n − X i =0 Z t i +1 t i S Mt n − t i F ( X K,M,Nt i ) ds + n − X i =0 Z t i +1 t i S Mt n − t i dW Q ,Ks . Introducing the notation ⌊ t ⌋ := max { t n : t n ≤ t, n = 0 , · · · , N } , we decompose theerror into four parts (cid:13)(cid:13)(cid:13) X K,Mt n − X K,M,Nt n (cid:13)(cid:13)(cid:13) L (Ω; V ) ≤ (cid:13)(cid:13)(cid:13)(cid:13)Z t n (cid:16) S Mt n − s F ( X K,Ms ) − S Mt n −⌊ s ⌋ F ( X K,M,N ⌊ s ⌋ ) (cid:17) ds (cid:13)(cid:13)(cid:13)(cid:13) L (Ω; V ) + (cid:13)(cid:13)(cid:13)(cid:13)Z t n (cid:0) S Mt n − s − S Mt n −⌊ s ⌋ (cid:1) dW Q ,Ks (cid:13)(cid:13)(cid:13)(cid:13) L (Ω; V ) ≤ (cid:13)(cid:13)(cid:13)(cid:13)Z t n S Mt n − s (cid:16) F ( X K,Ms ) − F ( X K,M ⌊ s ⌋ ) (cid:17) ds (cid:13)(cid:13)(cid:13)(cid:13) L (Ω; V ) + (cid:13)(cid:13)(cid:13)(cid:13)Z t n (cid:0) S Mt n − s − S Mt n −⌊ s ⌋ (cid:1) F ( X K,M ⌊ s ⌋ ) ds (cid:13)(cid:13)(cid:13)(cid:13) L (Ω; V ) + (cid:13)(cid:13)(cid:13)(cid:13)Z t n S Mt n −⌊ s ⌋ (cid:16) F ( X K,M ⌊ s ⌋ ) − F ( X K,M,N ⌊ s ⌋ ) (cid:17) ds (cid:13)(cid:13)(cid:13)(cid:13) L (Ω; V )UPER-CONVERGENCE ON EXPONENTIAL INTEGRATOR FOR SHE DRIVEN BY FBM 17 + (cid:13)(cid:13)(cid:13)(cid:13)Z t n (cid:0) S Mt n − s − S Mt n −⌊ s ⌋ (cid:1) dW Q ,Ks (cid:13)(cid:13)(cid:13)(cid:13) L (Ω; V ) = : J + J + J + J . The estimate for J is postponed to Lemma 8 which gives J ≤ C (cid:16) k u k ˙ V H + γ − (cid:17) h. For the second part J , note that J ≤ (cid:13)(cid:13)(cid:13)(cid:13)Z t n (cid:0) S Mt n − s − S Mt n −⌊ s ⌋ (cid:1)(cid:16) F ( X K,Mt n − ) − F ( X K,M ⌊ s ⌋ (cid:17) ds (cid:13)(cid:13)(cid:13)(cid:13) L (Ω; V ) + (cid:13)(cid:13)(cid:13)(cid:13)Z t n (cid:0) S Mt n − s − S Mt n −⌊ s ⌋ (cid:1) F ( X K,Mt n − ) ds (cid:13)(cid:13)(cid:13)(cid:13) L (Ω; V ) = : J + J . Using the temporal regularity shown in Theorem 2, we get J ≤ C Z t n | t n − s | − | s − ⌊ s ⌋| (cid:13)(cid:13)(cid:13) F ( X K,Mt n − ) − F ( X K,M ⌊ s ⌋ ) (cid:13)(cid:13)(cid:13) L (Ω; V ) ds ≤ C (cid:16) k u k ˙ V (cid:17) Z t n | t n − s | − | s − ⌊ s ⌋|| t n − s | H ds ≤ C (cid:16) k u k ˙ V (cid:17) h. Together with Z t n e − λ j ( t n − s ) − e − λ j ( t n −⌊ s ⌋ ) ds ≤ λ j h Z t n e − λ j ( t n − s ) ds ≤ Ch, the Parseval equality leads to (cid:13)(cid:13)(cid:13)(cid:13)Z t n (cid:0) S Mt n − s − S Mt n −⌊ s ⌋ (cid:1) F ( X K,Mt n − ) ds (cid:13)(cid:13)(cid:13)(cid:13) V = M X j =1 h Z t n (cid:0) S Mt n − s − S Mt n −⌊ s ⌋ (cid:1) F ( X K,Mt n − ) ds, e j i V = M X j =1 (cid:18)Z t n h (cid:0) S Mt n − s − S Mt n −⌊ s ⌋ (cid:1) F ( X K,Mt n − ) , e j i V ds (cid:19) ≤ ∞ X j =1 (cid:18)Z t n e − λ j ( t n − s ) − e − λ j ( t n −⌊ s ⌋ ) ds (cid:19) h F ( X K,Mt n − ) , e j i V ≤ Ch (cid:13)(cid:13)(cid:13) F ( X K,Mt n − ) (cid:13)(cid:13)(cid:13) V . Then we obtain J ≤ C (cid:16) k u k ˙ V (cid:17) h. For the third part J , it holds that J ≤ Ch n − X i =0 (cid:13)(cid:13)(cid:13) X K,Mt i − X K,M,Nt i (cid:13)(cid:13)(cid:13) L (Ω; V ) . In order to deal with J , we deduce from Lemma 5 and [ , Lemma 4.8] that J = K X i =1 Z t n Z t n h (cid:0) S Mt n − u − S Mt n −⌊ u ⌋ (cid:1) Q f i , (cid:0) S Mt n − v − S Mt n −⌊ v ⌋ (cid:1) Q f i i V φ ( u, v ) dudv ≤ Ch H + γ − (cid:13)(cid:13)(cid:13) A γ − (cid:13)(cid:13)(cid:13) L . Gathering the above estimates and using h = TN , we concludesup n =0 , ··· ,N (cid:13)(cid:13)(cid:13) X K,Mt n − X K,M,Nt n (cid:13)(cid:13)(cid:13) L (Ω; V ) ≤ C (cid:16) k u k ˙ V H + γ − (cid:17) N − . (cid:3) Lemma . Let the assumptions be satisfied as in Theorem 6, then J = (cid:13)(cid:13)(cid:13)(cid:13)Z t n S Mt n − s (cid:16) F ( X K,Ms ) − F ( X K,M ⌊ s ⌋ ) (cid:17) ds (cid:13)(cid:13)(cid:13)(cid:13) L (Ω; V ) ≤ C (cid:16) k u k ˙ V H + γ − (cid:17) h. Proof.
Applying the Taylor’s expansion leads to F ( X K,Ms ) − F ( X K,M ⌊ s ⌋ )= F ′ ( X K,M ⌊ s ⌋ ) (cid:16) X K,Ms − X K,M ⌊ s ⌋ (cid:17) + Z Z θF ′′ ((1 − ˜ θ ) X K,M ⌊ s ⌋ + ˜ θ ( θX K,Ms + (1 − θ ) X K,M ⌊ s ⌋ )) (cid:16) X K,Ms − X K,M ⌊ s ⌋ (cid:17) d ˜ θdθ = F ′ ( X K,M ⌊ s ⌋ ) (cid:0) S Ms −⌊ s ⌋ − Id V (cid:1) X K,M ⌊ s ⌋ + F ′ ( X K,M ⌊ s ⌋ ) Z s ⌊ s ⌋ S Ms − σ F ( X K,Mσ ) dσ + F ′ ( X K,M ⌊ s ⌋ ) Z s ⌊ s ⌋ S Ms − σ dW Q ,Kσ + Z Z θF ′′ ((1 − ˜ θ ) X K,M ⌊ s ⌋ + ˜ θ ( θX K,Ms + (1 − θ ) X K,M ⌊ s ⌋ )) (cid:16) X K,Ms − X K,M ⌊ s ⌋ (cid:17) d ˜ θdθ. Therefore, J ≤ (cid:13)(cid:13)(cid:13)(cid:13)Z t n S Mt n − s F ′ ( X K,M ⌊ s ⌋ ) (cid:0) S Ms −⌊ s ⌋ − Id V (cid:1) X K,M ⌊ s ⌋ ds (cid:13)(cid:13)(cid:13)(cid:13) L (Ω; V ) + (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)Z t n S Mt n − s F ′ ( X K,M ⌊ s ⌋ ) Z s ⌊ s ⌋ S Ms − σ F ( X K,Mσ ) dσds (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L (Ω; V ) + (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)Z t n S Mt n − s F ′ ( X K,M ⌊ s ⌋ ) Z s ⌊ s ⌋ S Ms − σ dW Q ,Kσ ds (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L (Ω; V ) + (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) Z t n S Mt n − s Z Z θF ′′ ((1 − ˜ θ ) X K,M ⌊ s ⌋ + ˜ θ ( θX K,Ms + (1 − θ ) X K,M ⌊ s ⌋ )) × (cid:16) X K,Ms − X K,M ⌊ s ⌋ (cid:17) d ˜ θdθds (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L (Ω; V ) = : J + J + J + J . In the following, we show the estimates for the four terms separately.
UPER-CONVERGENCE ON EXPONENTIAL INTEGRATOR FOR SHE DRIVEN BY FBM 19
By Lemma 3, we have J ≤ Z t n (cid:13)(cid:13)(cid:13) A − (cid:0) S Ms −⌊ s ⌋ − Id V (cid:1) AX K,M ⌊ s ⌋ (cid:13)(cid:13)(cid:13) L (Ω; V ) ds ≤ Ch Z t n (cid:13)(cid:13)(cid:13) X K,M ⌊ s ⌋ (cid:13)(cid:13)(cid:13) L (Ω; ˙ V ) ds ≤ C (cid:16) k u k ˙ V H + γ − (cid:17) h. Since Assumption 2 implies that F is linear growth, we get J ≤ C Z t n Z s ⌊ s ⌋ (cid:13)(cid:13) F ( X K,Mσ ) (cid:13)(cid:13) L (Ω; V ) dσds ≤ C (cid:16) k u k ˙ V H + γ − (cid:17) h. Based on Theorem 5 and arguments in Lemma 6, we take < r < J ≤ C Z t n | t n − s | − r (cid:13)(cid:13)(cid:13) X K,Ms − X K,M ⌊ s ⌋ (cid:13)(cid:13)(cid:13) L (Ω; V ) ds ≤ C (cid:16) k u k ˙ V H + γ − (cid:17) Z t n | t n − s | − r | s − ⌊ s ⌋| ds ≤ C (cid:16) k u k ˙ V H + γ − (cid:17) h. It remains to consider the third term J which can be rewritten as J = (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)Z t n S Mt n − s F ′ ( X K,M ⌊ s ⌋ ) Z s ⌊ s ⌋ S Ms − σ dW Q ,Kσ ds (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L (Ω; V ) = E "(cid:13)(cid:13)(cid:13)(cid:13) Z t n S Mt n − s F ′ ( X K,M ⌊ s ⌋ ) Z s ⌊ s ⌋ S Ms − σ dW Q ,Kσ ds (cid:13)(cid:13)(cid:13)(cid:13) V . The Parseval equality leads to J = E " h Z t n S Mt n − s F ′ ( X K,M ⌊ s ⌋ ) Z s ⌊ s ⌋ S Ms − σ dW Q ,Kσ ds , (10) Z t n S Mt n − s F ′ ( X K,M ⌊ s ⌋ ) Z s ⌊ s ⌋ S Ms − σ dW Q ,Kσ ds i V = E " M X i =1 h Z t n S Mt n − s F ′ ( X K,M ⌊ s ⌋ ) Z s ⌊ s ⌋ S Ms − σ dW Q ,Kσ ds , e i i V × h Z t n S Mt m − s F ′ ( X K,M ⌊ s ⌋ ) Z s ⌊ s ⌋ S Ms − σ dW Q ,Kσ ds , e i i V = E " M X i =1 Z t n Z t n h S Mt n − s F ′ ( X K,M ⌊ s ⌋ ) Z s ⌊ s ⌋ S Ms − σ dW Q ,Kσ , e i i V × h S Mt n − s F ′ ( X K,M ⌊ s ⌋ ) Z s ⌊ s ⌋ S Ms − σ dW Q ,Kσ , e i i V ds ds = E " M X i =1 Z t n Z t n h F ′ ( X K,M ⌊ s ⌋ ) Z s ⌊ s ⌋ S Ms − σ dW Q ,Kσ , e − λ i ( t n − s ) e i i V × h F ′ ( X K,M ⌊ s ⌋ ) Z s ⌊ s ⌋ S Ms − σ dW Q ,Kσ , e − λ i ( t n − s ) e i i V ds ds = E " M X i =1 Z t n Z t n e − λ i ( t n − s ) e − λ i ( t n − s ) h F ′ ( X K,M ⌊ s ⌋ ) Z s ⌊ s ⌋ S Ms − σ dW Q ,Kσ , e i i V × h F ′ ( X K,M ⌊ s ⌋ ) Z s ⌊ s ⌋ S Ms − σ dW Q ,Kσ , e i i V ds ds = M X i =1 Z t n Z t n e − λ i ( t n − s ) e − λ i ( t n − s ) K X j =1 K X j =1 E " h F ′ ( X K,M ⌊ s ⌋ ) × Z s ⌊ s ⌋ S Ms − σ Q f j dβ j σ , e i i V h F ′ ( X K,M ⌊ s ⌋ ) Z s ⌊ s ⌋ S Ms − σ Q f j dβ j σ , e i i V ds ds . The definition of the inner product in terms of the Hilbert space V and the sto-chastic Fubini theorem produce E " h F ′ ( X K,M ⌊ s ⌋ ) Z s ⌊ s ⌋ S Ms − σ Q f j dβ j σ , e i i V h F ′ ( X K,M ⌊ s ⌋ ) Z s ⌊ s ⌋ S Ms − σ Q f j dβ j σ , e i i V (11)= E " Z f ′ ( X K,M ⌊ s ⌋ ( x )) (cid:18) Z s ⌊ s ⌋ S Ms − σ Q f j dβ j σ (cid:19) ( x ) e i ( x ) dx × Z f ′ ( X K,M ⌊ s ⌋ ( x )) (cid:18) Z s ⌊ s ⌋ S Ms − σ Q f j dβ j σ (cid:19) ( x ) e i ( x ) dx = E " Z f ′ ( X K,M ⌊ s ⌋ ( x )) ∞ X l =1 (cid:18) Z s ⌊ s ⌋ h S Ms − σ Q f j , e l i V dβ j σ (cid:19) e l ( x ) e i ( x ) dx × Z f ′ ( X K,M ⌊ s ⌋ ( x )) ∞ X l =1 (cid:18) Z s ⌊ s ⌋ h S Ms − σ Q f j , e l i V dβ j σ (cid:19) e l ( x ) e i ( x ) dx = Z Z ∞ X l =1 ∞ X l =1 E " f ′ ( X K,M ⌊ s ⌋ ( x )) (cid:18) Z s ⌊ s ⌋ h S Ms − σ Q f j , e l i V dβ j σ (cid:19) × f ′ ( X K,M ⌊ s ⌋ ( x )) (cid:18) Z s ⌊ s ⌋ h S Ms − σ Q f j , e l i V dβ j σ (cid:19) e l ( x ) e i ( x ) e l ( x ) e i ( x ) dx dx . According to Lemma 1, we have E " f ′ ( X K,M ⌊ s ⌋ ( x )) (cid:18) Z s ⌊ s ⌋ h S Ms − σ Q f j , e l i V dβ j σ (cid:19) (12) × f ′ ( X K,M ⌊ s ⌋ ( x )) (cid:18) Z s ⌊ s ⌋ h S Ms − σ Q f j , e l i V dβ j σ (cid:19) UPER-CONVERGENCE ON EXPONENTIAL INTEGRATOR FOR SHE DRIVEN BY FBM 21 = E " Z T Z s ⌊ s ⌋ Z T Z s ⌊ s ⌋ (cid:16) D v (cid:16) D u h f ′ ( X K,M ⌊ s ⌋ ( x )) f ′ ( X K,M ⌊ s ⌋ ( x ) i(cid:17) j (cid:17) j × h S Ms − σ Q f j , e l i V φ ( σ , u ) dσ du h S Ms − σ Q f j , e l i V φ ( σ , v ) dσ dv + E " f ′ ( X K,M ⌊ s ⌋ ( x )) f ′ ( X K,M ⌊ s ⌋ ( x )) × (cid:18) Z s ⌊ s ⌋ Z s ⌊ s ⌋ h S Ms − σ Q f j , e l i V h S Ms − σ Q f j , e l i V φ ( σ , σ ) dσ dσ (cid:19) j = j . Substituting (11)-(12) into (10), we obtain J = M X i =1 Z t n Z t n e − λ i ( t n − s ) e − λ i ( t n − s ) K X j =1 K X j =1 Z Z ∞ X l =1 ∞ X l =1 E " f ′ ( X K,M ⌊ s ⌋ ( x )) f ′ ( X K,M ⌊ s ⌋ ( x )) (cid:18) Z s ⌊ s ⌋ h S Ms − σ Q f j , e l i V dβ j σ (cid:19) × (cid:18) Z s ⌊ s ⌋ h S Ms − σ Q f j , e l i V dβ j σ (cid:19) e l ( x ) e i ( x ) e l ( x ) e i ( x ) dx dx ds ds = M X i =1 Z t n Z t n e − λ i ( t n − s ) e − λ i ( t n − s ) K X j =1 K X j =1 Z Z ∞ X l =1 ∞ X l =1 E " Z T Z s ⌊ s ⌋ Z T Z s ⌊ s ⌋ (cid:16) D v (cid:16) D u h f ′ ( X K,M ⌊ s ⌋ ( x )) f ′ ( X K,M ⌊ s ⌋ ( x ) i(cid:17) j (cid:17) j × h S Ms − σ Q f j , e l i V h S Ms − σ Q f j , e l i V φ ( σ , u ) φ ( σ , v ) dσ dudσ dv × e l ( x ) e i ( x ) e l ( x ) e i ( x ) dx dx ds ds + M X i =1 Z t n Z t n e − λ i ( t n − s ) e − λ i ( t n − s ) K X j =1 Z Z ∞ X l =1 ∞ X l =1 E " f ′ ( X K,M ⌊ s ⌋ ( x )) f ′ ( X K,M ⌊ s ⌋ ( x )) (cid:18) Z s ⌊ s ⌋ Z s ⌊ s ⌋ h S Ms − σ Q f j , e l i V × h S Ms − σ Q f j , e l i V φ ( σ , σ ) dσ dσ (cid:19) e l ( x ) e i ( x ) e l ( x ) e i ( x ) dx dx ds ds = : I + I . Noting I = M X i =1 Z t n Z t n e − λ i ( t n − s ) e − λ i ( t n − s ) K X j =1 Z s ⌊ s ⌋ Z s ⌊ s ⌋ E " h F ′ ( X K,M ⌊ s ⌋ ) S Ms − σ Q f j , e i i V × F ′ ( X K,M ⌊ s ⌋ ) S Ms − σ Q f j , e i i V φ ( σ , σ ) dσ dσ ds ds , we have | I | ≤ C M X i =1 Z t n Z t n e − λ i ( t n − s ) e − λ i ( t n − s ) K X j =1 Z s ⌊ s ⌋ Z s ⌊ s ⌋ (cid:13)(cid:13) Q f j (cid:13)(cid:13) V φ ( σ , σ ) dσ dσ ds ds ≤ C k A γ − k L ∞ X i =1 Z t n Z t n e − λ i ( t n − s ) e − λ i ( t n − s ) Z s ⌊ s ⌋ Z s ⌊ s ⌋ φ ( σ , σ ) dσ dσ ds ds . Based on the chain rule of the Malliavin derivative, we get (cid:16) D v (cid:16) D u h f ′ ( X K,M ⌊ s ⌋ ( x )) f ′ ( X K,M ⌊ s ⌋ ( x )) i(cid:17) j (cid:17) j = (cid:16) D v h f ′′ ( X K,M ⌊ s ⌋ ( x )) (cid:16) D u X K,M ⌊ s ⌋ ( x ) (cid:17) j f ′ ( X K,M ⌊ s ⌋ ( x ))+ f ′ ( X K,M ⌊ s ⌋ ( x )) f ′′ ( X K,M ⌊ s ⌋ ( x )) (cid:16) D u X K,M ⌊ s ⌋ ( x ) (cid:17) j i(cid:17) j = f ′′′ ( X K,M ⌊ s ⌋ ( x )) (cid:16) D v X K,M ⌊ s ⌋ ( x ) (cid:17) j (cid:16) D u X K,M ⌊ s ⌋ ( x ) (cid:17) j f ′ ( X K,M ⌊ s ⌋ ( x ))+ f ′′ ( X K,M ⌊ s ⌋ ( x )) (cid:16) D v (cid:16) D u X K,M ⌊ s ⌋ ( x ) (cid:17) j (cid:17) j f ′ ( X K,M ⌊ s ⌋ ( x ))+ f ′′ ( X K,M ⌊ s ⌋ ( x )) (cid:16) D u ( X K,M ⌊ s ⌋ ( x ) (cid:17) j f ′′ ( X K,M ⌊ s ⌋ ( x )) (cid:16) D v X K,M ⌊ s ⌋ ( x ) (cid:17) j + f ′′ ( X K,M ⌊ s ⌋ ( x )) (cid:16) D v X K,M ⌊ s ⌋ ( x ) (cid:17) j f ′′ ( X K,M ⌊ s ⌋ ( x )) (cid:16) D u X K,M ⌊ s ⌋ ( x ) (cid:17) j + f ′ ( X K,M ⌊ s ⌋ ( x )) f ′′′ ( X K,M ⌊ s ⌋ ( x )) (cid:16) D v X K,M ⌊ s ⌋ ( x ) (cid:17) j (cid:16) D u X K,M ⌊ s ⌋ ( x ) (cid:17) j + f ′ ( X K,M ⌊ s ⌋ ( x )) f ′′ ( X K,M ⌊ s ⌋ ( x )) (cid:16) D v (cid:16) D u X K,M ⌊ s ⌋ ( x ) (cid:17) j (cid:17) j = : I (1) x ,x + I (2) x ,x + I (3) x ,x + I (4) x ,x + I (5) x ,x + I (6) x ,x , which implies that I = M X i =1 Z t n Z t n e − λ i ( t n − s ) e − λ i ( t n − s ) K X j =1 K X j =1 Z T Z s ⌊ s ⌋ Z T Z s ⌊ s ⌋ E " Z Z (cid:16) I (1) x ,x + I (2) x ,x + I (3) x ,x + I (4) x ,x + I (5) x ,x + I (6) x ,x (cid:17) × (cid:16) S Ms − σ Q f j (cid:17) ( x ) (cid:16) S Ms − σ Q f j (cid:17) ( x ) e i ( x ) e i ( x ) dx dx × φ ( σ , u ) φ ( σ , v ) dσ dudσ dvds ds . Since for any r > and ϕ , ϕ ∈ ˙ V r , the pointwise multiplication ϕ · ϕ satisfies k ϕ · ϕ k ˙ V r ≤ k ϕ k ˙ V r k ϕ k ˙ V r , it follows from Lemma 6 and Assumption 2 that for any < r < (cid:12)(cid:12)(cid:12)(cid:12)Z Z I (1) x ,x (cid:16) S Ms − σ Q f j (cid:17) ( x ) (cid:16) S Ms − σ Q f j (cid:17) ( x ) e i ( x ) e i ( x ) dx dx (cid:12)(cid:12)(cid:12)(cid:12) ≤ C (cid:13)(cid:13)(cid:13)(cid:16) D v X K,M ⌊ s ⌋ (cid:17) j (cid:13)(cid:13)(cid:13) ˙ V r (cid:13)(cid:13)(cid:13)(cid:16) D u X K,M ⌊ s ⌋ (cid:17) j (cid:13)(cid:13)(cid:13) ˙ V r (cid:13)(cid:13)(cid:13) S Ms − σ Q f j (cid:13)(cid:13)(cid:13) ˙ V r (cid:13)(cid:13)(cid:13) S Ms − σ Q f j (cid:13)(cid:13)(cid:13) ˙ V r UPER-CONVERGENCE ON EXPONENTIAL INTEGRATOR FOR SHE DRIVEN BY FBM 23 ≤ C (cid:13)(cid:13) Q f j (cid:13)(cid:13) V r (cid:13)(cid:13) Q f j (cid:13)(cid:13) V r and (cid:12)(cid:12)(cid:12)(cid:12)Z Z I (2) x ,x (cid:16) S Ms − σ Q f j (cid:17) ( x ) (cid:16) S Ms − σ Q f j (cid:17) ( x ) e i ( x ) e i ( x ) dx dx (cid:12)(cid:12)(cid:12)(cid:12) ≤ C (cid:13)(cid:13)(cid:13)(cid:16) D v (cid:16) D u X K,M ⌊ s ⌋ (cid:17) j (cid:17) j (cid:13)(cid:13)(cid:13) V (cid:13)(cid:13)(cid:13) S Ms − σ Q f j (cid:13)(cid:13)(cid:13) V (cid:13)(cid:13)(cid:13) S Ms − σ Q f j (cid:13)(cid:13)(cid:13) V ≤ C (cid:13)(cid:13) Q f j (cid:13)(cid:13) V (cid:13)(cid:13) Q f j (cid:13)(cid:13) V . Using similar arguments for I (3) , I (4) , I (5) and I (6) , we obtain that for γ > , | I | ≤ C k A γ − k L ∞ X i =1 Z t n Z t n e − λ i ( t n − s ) e − λ i ( t n − s ) × " Z T Z s ⌊ s ⌋ Z T Z s ⌊ s ⌋ φ ( σ , u ) dσ duφ ( σ , v ) dσ dv ds ds . It then suffices to prove ∞ X i =1 Z t n Z t n e − λ i ( t n − s ) e − λ i ( t n − s ) (13) × " Z T Z s ⌊ s ⌋ Z T Z s ⌊ s ⌋ φ ( σ , u ) dσ duφ ( σ , v ) dσ dv ds ds ≤ Ch and ∞ X i =1 Z t n Z t n e − λ i ( t n − s ) e − λ i ( t n − s ) Z s ⌊ s ⌋ Z s ⌊ s ⌋ φ ( σ , σ ) dσ dσ ds ds ≤ Ch . (14)Since Z T Z s ⌊ s ⌋ | σ − v | H − dσ dv = Z s ⌊ s ⌋ Z T | σ − v | H − dvdσ ≤ C ( H, T ) h, we obtain (13) by Z t n e − λ i ( t n − s ) ds = 1 − e − λ i t n λ i ≤ λ i = 1 i π . Defining ⌈ t ⌉ := t i +1 , for t ∈ ( t i , t i +1 ], we get Z t n Z t n Z s ⌊ s ⌋ Z s ⌊ s ⌋ e − λ i ( t n − s ) e − λ i ( t n − s ) | σ − σ | H − dσ dσ ds ds = Z t n Z t n Z ⌈ σ ⌉ σ Z ⌈ σ ⌉ σ e − λ i ( t n − s ) e − λ i ( t n − s ) | σ − σ | H − ds ds dσ dσ = Z t n Z ⌈ σ ⌉ σ e − λ i ( t n − s ) " Z t n Z ⌈ σ ⌉ σ e − λ i ( t n − s ) | σ − σ | H − ds dσ ds dσ = Z t n Z ⌈ σ ⌉ σ e − λ i ( t n − s ) " Z t n | σ − σ | H − " Z ⌈ σ ⌉ σ e − λ i ( t n − s ) ds dσ ds dσ ≤ C ( H, T ) h Z t n e − λ i ( t n − s ) (cid:20) Z s ⌊ s ⌋ dσ (cid:21) ds ≤ C h i π , which is summable with respect to i from 1 to infinity, and then (14) holds.Collecting the above estimates finishes the proof. (cid:3) Remark . If one uses the temporal H¨older continuity of X K,M to estimate J directly, then the convergence order will be restricted by H . Proof of Theorem 1.
Combining Theorems 3, 4 and 6, we obtain the con-clusion of Theorem 1, which is the main result of this paper. (cid:3)
6. Concluding remarks
In this paper, we present the strong convergence order of the exponential in-tegrator for the SHE driven by an infinite dimensional fractional Brownian mo-tion with Hurst parameter H ∈ ( ,
1) is 1 under Assumptions 1-3 with γ > max { − H, } , which establishes the first super-convergence result in temporaldirection on full discretizations for SPDEs driven by infinite dimensional fractionalBrownian motions with Hurst parameter H ∈ ( , H of accuracy in temporal direction isachieved as long as γ ≥
1, we conjecture that the strong convergence order of theexponential integrator when 1 < γ ≤ max { − H, } is between H and 1. Ourfurther work is to investigate the concrete relationship between γ and the strongconvergence order in the case of 1 < γ ≤ max { − H, } . Due to the lack ofthe Burkholder–Davis–Gundy inequality for SPDEs driven by fBms, more effortsshould be paid to develop new techniques to deal with the associated stochasticintegrals.For rougher case H ∈ (0 , ), since the kernel of the covariance of fBm is singular,the rough path theory needs to be employed in numerical analysis. As for generalmultiplicative noises, the estimates for Malliavin derivatives of the exact solutionare more complicated and the strong convergence order of the full discretization isstill unsolved for SPDEs driven by infinite dimensional fBms. We will leave thesetopics as future works. References [1] Y. Cao, J. Hong, and Z. Liu. Approximating stochastic evolution equations with additivewhite and rough noises.
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Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing100190, China; School of Mathematical Sciences, University of Chinese Academy ofSciences, Beijing 100049, China
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