moderate deviations for a fractional stochastic heat equation with...
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Stochastics and DynamicsVol. 17, No. 4 (2017) 1750025 (23 pages)c© World Scientific Publishing CompanyDOI: 10.1142/S0219493717500253
Moderate deviations for a fractional stochastic heatequation with spatially correlated noise
Yumeng Li∗,¶, Ran Wang†,‖, Nian Yao‡,∗∗ and Shuguang Zhang§,††
∗School of Statistics and Mathematics,Zhongnan University of Economics and Law,
Wuhan, Hubei 430073, P. R. China
†School of Mathematics and Statistics,Wuhan University, Wuhan, Hubei 430072, P. R. China
‡School of Mathematics and Statistics,Shenzhen University, Shenzhen,
Guangdong 518060, P. R. China
§Department of Statistics and Finance,University of Science and Technology of China,
Hefei, Anhui 230026, P. R. China¶[email protected]
‖[email protected]∗∗[email protected]††[email protected]
Received 5 September 2015Accepted 6 May 2016Published 5 July 2016
In this paper, we study the Moderate Deviation Principle for a perturbed stochastic heatequation in the whole space R
d, d ≥ 1. This equation is driven by a Gaussian noise, whitein time and correlated in space, and the differential operator is a fractional derivative
operator. The weak convergence method plays an important role.
Keywords: Fractional derivative operator; stochastic heat equation; moderate deviationprinciple; weak convergence method.
AMS Subject Classification: 60H15, 60F05, 60F10
1. Introduction
Since the work of Freidlin and Wentzell [16], the theory of small perturbationlarge deviations for stochastic (partial) differential equation has been extensivelydeveloped (see [7, 11]). The large deviation principle (LDP) for stochastic reaction-diffusion equations driven by the space-time white noise was first obtained by Frei-dlin [15] and later by Sowers [25], Chenal and Millet [6], Carrai and Rockner [5] and
‡Corresponding author
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other authors. An LDP for a stochastic heat equation driven by a Gaussian noise,white in time and correlated in space was proved by Marquez-Carreras and Sarra[23]. Recently, El Mellali and Mellouk proved an LDP for a fractional stochasticheat equation driven by a spatially correlated noise in [13].
Like the large deviations, the moderate deviation problems arise in the theory ofstatistical inference quite naturally. The moderate deviation principle (MDP) canprovide us with the rate of convergence and a useful method for constructing asymp-totic confidence intervals, see [14, 17, 19, 24, 28] and references therein. Results onthe MDP for processes with independent increments were obtained in De Acosta[10], Ledoux [21] and so on. The study of the MDP estimates for other processeshas been carried out as well, e.g., Wu [29] for Markov processes, Guillin and Liptser[18] for diffusion processes, Wang and Zhang [27] for stochastic reaction-diffusionequations in R, Budhiraja et al. [4] for stochastic differential equations with jumps,and references therein.
In this paper, we study the MDP for the fractional stochastic heat equation inspatial dimension R
d driven by a spatially correlated noise. In [22], we studied theMDP for a perturbed stochastic heat equations defined on [0, T ]×[0, 1]d, driven by aspatially correlated noise. In that paper, the method is the exponential approxima-tion theorem (see [11, Theorem 4.2.13]), which needs some exponential estimates.However, due to the lack of good regularity properties of the Green function for thefractional heat equation, it is difficult to get those exponential estimates. Insteadof proving exponential estimates, we will use the weak convergence approach (see[3]) in this paper.
Now, let us give the fractional stochastic heat equation∂uε
∂t(t, x) = Dα
δ uε(t, x) + b(uε(t, x)) +
√εσ(uε(t, x))F (t, x),
uε(0, x) = 0,(1)
where ε > 0, (t, x) ∈ [0, T ] × Rd, d ≥ 1, α = (α1, . . . , αd), δ = (δ1, . . . , δd) and
we will assume that αi ∈ ] 0, 2]\1 and |δi| ≤ minαi, 2 − δi, i = 1, . . . , d, F isthe “formal” derivative of the Gaussian perturbation and Dα
δ denotes a nonlocalfractional differential operator on R
d defined by
Dαδ :=
d∑i=1
Dαi
δi.
Here Dαi
δidenotes the fractional differential derivative w.r.t. the ith coordinate
defined via its Fourier transform F by
F(Dαδ φ)(ξ) = −|ξ|αi exp
(−ıδiπ2 sgnξ
)F(φ)(ξ),
with ı2 +1 = 0. The noise F (t, x) is a martingale measure in the sense of Walsh [26]and Dalang [8], which will be defined with details in the sequel. The coefficients
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b and σ : R → R are given functions. From now on, we shall refer to Eq. (1) asEq
αδ,ε(d, b, σ). See Sec. 2 for details.As the parameter ε tends to zero, the solutions uε of (1) will tend to the solution
of the deterministic equation defined by∂u0
∂t(t, x) = Dα
δ u0(t, x) + b(u0(t, x)),
u0(0, x) = 0.
(2)
In this paper we shall investigate deviations of uε from the deterministic solutionu0, as ε decreases to 0, that is, the asymptotic behavior of the trajectories,
1√ελ(ε)
(uε − u0)(t, x), (t, x) ∈ [0, T ]× Rd,
where λ(ε) is some deviation scale, which strongly influences the asymptoticbehavior.
The case λ(ε) = 1/√ε provides some large deviations estimates. Under suitable
assumptions, El Mellali and Mellouk proved that the law of the solution uε satisfiesa large deviation principle on the Holder space in [13].
If λ(ε) is identically equal to 1, we are in the domain of the central limit theorem.To fill in the gap between the central limit theorem scale [λ(ε) = 1] and the
large deviations scale [λ(ε) = 1/√ε], we will study moderate deviations, that is
when the deviation scale satisfies
λ(ε) → +∞,√ελ(ε) → 0 as ε→ 0. (3)
The moderate deviation principle enables us to refine the estimates obtainedthrough the central limit theorem. It provides the asymptotic behavior for P(‖uε −u0‖ ≥ δ
√ελ(ε)) while the central limit theorem gives asymptotic bounds for
P(‖uε − u0‖ ≥ δ√ε). Throughout this paper, we assume (3) is in place.
The rest of this paper is organized as follows. In Sec. 2, the precise framework isstated. In Sec. 3, the Skeleton equation is studied. It is proved that the solution isa continuous map from the level set into the Holder space. Section 4 is devoted tothe proof of the moderate deviation principle by the weak convergence approach.We give some precise estimates of the fundamental solution G in the Appendix.
Throughout the paper, Cp is a positive constant depending on the parame-ter p, and C,C1, . . . are constants depending on no specific parameter (except Tand the Lipschitz constants), whose value may be different from line to line byconvention.
For any T > 0,K ⊂ Rd, β = (β1, β2), let Cβ([0, T ]×K; Rd) be the Holder space
equipped with the norm defined by
‖f‖β,K := sup(t,x)∈[0,t]×K
|f(t, x)| + sups=t∈[0,T ]
supx =y∈K
|f(t, x) − f(s, y)||t− s|β1 + |x− y|β2
. (4)
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Since Cβ([0, T ] ×K; Rd) is not separable, we consider the space Cβ′,0([0, T ] ×K; Rd) of Holder continuous functions f with the degree β′
i < βi, i = 1, 2 such that
limδ→0+
(sup
|t−s|+|x−y|<δ
|f(t, x) − f(s, y)||t− s|β′
1 + |x− y|β′2
)= 0,
and Cβ′,0([0, T ]×K; Rd) is a Polish space containing Cβ([0, T ]×K; Rd). From nowon, let Eβ([0, T ] ×K; Rd) := Cβ,0([0, T ]×K; Rd), where β = (β1, β2).
2. Framework
In this section, let us give the framework taken from Boulanba et al. [2] and ElMellali and Mellouk [13].
2.1. The operator Dαδ
In one-dimensional space, the operator Dαδ is a closed, densely defined operator on
L2(R) and it is the infinitesimal generator of a semigroup which is in general notsymmetric and not a contraction. It is self-adjoint only when δ = 0 and in this case,it coincides with the fractional power of the Laplacian.
According to [12, 20], Dαδ can be represented for 1 < α < 2, by
Dαδ =
∫ +∞
−∞
φ(x+ y) − φ(x) − yφ′(x)|y|1+α
(κδ−1(−∞,0)(y) + κδ
+1(0,+∞)(y))dy,
and for 0 < α < 1, by
Dαδ =
∫ +∞
−∞
φ(x+ y) − φ(x)|y|1+α
(κδ−1(−∞,0)(y) + κδ
+1(0,+∞)(y))dy,
where κδ− and κδ
+ are two nonnegative constants satisfying κδ− + κδ
+ > 0 and φ isa smooth function for which the integral exists, and φ′ stands for its derivative.This representation identifies it as the infinitesimal generator for a non-symmetricα-stable Levy process.
Let Gα,δ(t, x) denotes the fundamental solution of the equation Eqαδ,1(1, 0, 0),
that is, the unique solution of the Cauchy problem∂u
∂t(t, x) = Dα
δ u0(t, x),
u(0, x) = δ0(x), t > 0, x ∈ R,
where δ0 is the Dirac distribution. Using Fourier’s calculus one gets
Gα,δ(t, x) =12π
∫ ∞
−∞exp
(−ızx− t|z|α exp
(−ıδ π
2sgn(z )
))dz. (5)
The relevant parameters, α called the index of stablity and δ called the skewness,are real numbers satisfying α ∈ ] 0, 2] and |δ| ≤ minα, 2 − α.
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Now, for higher dimension d ≥ 1 and any multi index α = (α1, . . . , αd) andδ = (δ1, . . . , δd), let Gα,δ(t, x) be the Green function of the deterministic equationEq
αδ (d, 0, 0). Clearly,
Gα,δ(t, x) =d∏
i=1
Gαi,δi(t, xi)
=1
(2π)d
∫Rd
exp
(−ı〈ξ, x〉 − t
d∑i=1
|ξi|αi exp(−ıδiπ2 sgn(ξi)
))dξ, (6)
where 〈·, ·〉 stands for the inner product in Rd.
The properties of the Green function Gα,δ(t, x) will be given in the Appendix.
2.2. The driving noise F
Let us explicitly describe here the spatially homogeneous noise, see Dalang [8].Let S(Rd+1) be the space of Schwartz test functions. On a complete prob-
ability space (Ω,G,P), the noise F = F (φ), φ ∈ S(Rd+1) is assumed to bean L2(Ω,G,P)-valued Gaussian process with mean zero and covariance functionalgiven by
J(ϕ, ψ) := E[F (φ)F (ψ)]
=∫
R+
ds
∫Rd
(φ(s, ) ∗ ψ(s, ))(x)Γ(dx)ds, φ, ψ ∈ S(Rd+1),
where ψ(s, x) := ψ(s,−x) and Γ is a nonnegative and nonnegative definite temperedmeasure, therefore symmetric. ∗ denotes the convolution product and stands forthe spatial variable.
Let µ be the spectral measure of Γ, which is also a trivial tempered measure,that is µ = F−1(Γ) and this gives
J(φ, ψ) =∫
R+
ds
∫Rd
µ(dξ)Fφ(s, )(ξ)Fψ(s, )(ξ), (7)
where z is the complex conjugate of z.As in Dalang [8], the Gaussian process F can be extended to a worthy martingale
measure, in the sense of Walsh [26],
M := Mt(A), t ∈ R+, A ∈ Bb(Rd),where Bb(Rd) denotes the collection of all bounded Borel measurable sets in R
d. LetGt be the completion of the σ-field generated by the random variables F (s,A); 0 ≤s ≤ t, A ∈ Bb(Rd).
Then, Boulanba et al. [2] gave a rigorous meaning to the solution of equa-tion Eq
αδ,ε(d, b, σ) by means of a joint measurable and Gt-adapted process
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uε(t, x); (t, x) ∈ R+ × Rd satisfying, for each t ≥ 0 and for almost all x ∈ R
d
the following evolution equation
uε(t, x) =√ε
∫ t
0
∫Rd
Gα,δ(t− s, x− y)σ(uε(s, y))F (dsdy)
+∫ t
0
ds
∫Rd
Gα,δ(t− s, x− y)b(uε(s, y))dy. (8)
In order to prove our main result, we are going to give other equivalent approach tothe solution of Eqα
δ,ε(d, b, σ), see [9]. To start with, let us denote by H the Hilbertspace obtained by the completion of S(Rd) with the inner production
〈φ, ψ〉H :=∫
Rd
Γ(dx)(φ ∗ ψ)(x)
=∫
Rd
µ(dξ)Fφ(ξ)Fψ(ξ), φ, ψ ∈ S(Rd).
The norm induced by 〈·, ·〉H is denoted by ‖·‖H.By the Walsh’s theory of the martingale measures [26], for t ≥ 0 and h ∈ H the
stochastic integral
Bt(h) =∫ t
0
∫Rd
h(y)F (ds, dy),
is well-defined and the process Bt(h); t ≥ 0, h ∈ H is a cylindrical Wiener processon H, that is:
(a) for every h ∈ H with ‖h‖H = 1, Bt(h)t≥0 is a standard Wiener process,(b) for every t ≥ 0, a, b ∈ R and f, g ∈ H,
Bt(af + bg) = aBt(f) + bBt(g) almost surely.
Let ekk≥1 be a complete orthonormal system (CONS) of the Hilbert space H,then
Bkt :=
∫ t
0
∫Rd
ek(y)F (ds, dy); k ≥ 1
defines a sequence of independent standard Wiener processes and we have the fol-lowing representation
Bt :=∑k≥1
Bkt ek. (9)
Let Ftt∈[0,T ] be the σ-field generated by the random variables Bks ; s ∈ [0, t], k ≥
1. We define the predictable σ-field in Ω × [0, T ] generated by the sets (s, t] ×A;A ∈ Fs, 0 ≤ s ≤ t ≤ T . In the following, we can define the stochastic integralwith respect to cylindrical Wiener process (Bt(h))t≥0 (see e.g., [7] or [9]) of anypredictable square-integrable process with values in H as follows∫ t
0
∫Rd
g · dB :=∑k≥1
∫ t
0
〈g(s), ek〉HdBks .
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Note that the above series converges in L2(Ω,F ,P) and the sum does not depend onthe selected CONS. Moreover, each summand, in the above series, is a classical Itointegral with respect to a standard Brownian motion, and the resulting stochasticintegral is a real-valued random variable.
In the sequel, we shall consider the mild solution to equation Eqαδ,ε(d, b, σ)
given by
uε(t, x) =√ε∑k≥1
∫ t
0
〈Gα,δ(t− s, x− ·)σ(uε(s, )), ek〉HdBks
+∫ t
0
[Gα,δ(t− s) ∗ b(uε(s, ))](x)ds, (10)
for any t ∈ [0, T ], x ∈ Rd.
2.3. Existence, uniqueness and Holder regularity to equation
For a multi-index α = (α1, . . . , αd) such that αi ∈ ] 0, 2]\1, i = 1, . . . , d and anyξ ∈ R
d, let
Sα(ξ) =d∑
i=1
|ξi|αi .
Assume the following assumptions on the functions σ, b and the measure µ:
(C): The functions σ and b are Lipschitz, that is there exists some constant Lsuch that
‖σ(x) − σ(y)‖ ≤ L|x− y|, |b(x) − b(y)| ≤ L|x− y| (11)
for all x, y ∈ Rd.
(Hαη ): Let α as defined above and η ∈ ] 0, 1], it holds that∫
Rd
µ(dξ)(1 + Sα(ξ))η
< +∞.
The last assumption stands for an integrability condition w.r.t. the spectralmeasure µ. Indeed, the following stochastic integral∫ T
0
∫Rd
Gα,δ(T − s, x− y)F (ds, dy)
is well-defined if and only if∫ T
0
ds
∫Rd
µ(dξ)|FGα,δ(s, ∗)(ξ)|2 < +∞.
More precisely, by [2, Lemma 1.2], there exist two positive constants c1, c2 suchthat
c1
∫Rd
µ(dξ)1 + Sα(ξ)
≤∫ T
0
ds
∫Rd
µ(dξ)|FGα,δ(s, ∗)(ξ)|2 ≤ c2
∫Rd
µ(dξ)1 + Sα(ξ)
. (12)
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Under the assumptions (C) and (Hαη ), Boulanba et al. proved that Eq. (10)
admits a unique solution uε such that
supt∈[0,T ]
supx∈Rd
E|uε(t, x)|p < +∞, ∀T > 0, p ≥ 2. (13)
See [2, Theorem 2.1]. Moreover, Theorem 3.1 in [2] tells us that the trajectories ofthe solution uε(t, x) : (t, x) ∈ R+×R
d to Eq. (10) are β = (β1, β2)-Holder continuousin (t, x) ∈ [0, T ]×K for every K compact subset of R
d and every β1 ∈ (0, 1 − η/2),β2 ∈ (0,minα0(1 − η), 1/2), where α0 := min1≤i≤dαi.
Consequently, the random field solution uε(t, x); (t, x) ∈ [0, T ]×K to Eq. (10)lives in the Holder space Cβ([0, T ]×K; Rd) equipped with the norm ‖f‖β,K givenin (4).
Particularly, taking ε = 0, the deterministic solution u0 to (2) has the followingestimates
supt∈[0,T ]
supx∈Rd
E|u0(t, x)|p < +∞, ∀T > 0, p ≥ 2 (14)
and ‖u0‖β,K <∞ for any compact set K ⊂ Rd.
3. Skeleton Equations
The purpose of this section is to study the Skeleton equation, which will be usedin the weak convergence approach.
From now on, we furthermore suppose that
(D): The function b is differentiable, and its derivative b′ is Lipschitz. Moreprecisely, there exists a positive constant L′ such that
|b′(y) − b′(z)| ≤ L′|y − z| for all y, z ∈ R. (15)
Combined with the Lipschitz continuity of b, we conclude that
|b′(z)| ≤ L, ∀ z ∈ R. (16)
For T > 0, let HT := L2([0, T ];H), which is a real separable Hilbert space suchthat, if ϕ, ψ ∈ HT ,
〈ϕ, ψ〉HT :=∫ T
0
〈ϕ(s, ·), ψ(s, ·)〉Hds.
Denote ‖·‖HT the norm induced by 〈·, ·〉HT . For any N > 0, define
HNT := h ∈ HT ; ‖h‖HT ≤ N,
and we consider that HNT is endowed with the weak topology of HT .
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For any h ∈ HT , consider the deterministic evolution equation (called Skeletonequation)
Zh(t, x) =∫ t
0
〈Gα,δ(t− s, x− )σ(u0(s, )), h(s, )〉Hds
+∫ t
0
[Gα,δ(t− s) ∗ (b′(u0(s, )Zh(s, ))](x)ds, (17)
where the first term on the right-hand side of the above equation can be written as∑k≥1
∫ t
0
〈Gα,δ(t− s, x− )σ(u0(s, ), ek〉Hhk(s)ds,
with hk(t) := 〈h(t), ek〉H, t ∈ [0, T ], k ≥ 1.Using the strategy in the proof of Proposition 2.7 in [13], one can obtain the
following result. Here we omit its proof.
Proposition 3.1. Assuming conditions (C), (Hαη ) and (D), there exists a unique
solution Zh to Eq. (17), which satisfies
suph∈HN
T
sup(t,x)∈[0,T ]×Rd
|Zh(t, x)| < +∞. (18)
Theorem 3.2. Assuming conditions (C), (Hαη ) and (D), the mapping h : HN
T →Zh ∈ Eβ([0, T ]×K; Rd) is continuous with respect to the weak topology, where K isa compact set in R
d, β = (β1, β2) satisfies that 0 < β1 < α0(1−η)/2, 0 < β2 < 1−ηand α0 = min1≤i≤dαi.
Proof. Let 0 < β1 < α0(1 − η)/2, 0 < β2 < 1 − η and h, (hn)n≥1 ⊂ HNT such
that for any g ∈ HT ,
limn→∞〈hn − h, g〉HT = 0.
We need to prove that
limn→∞ ‖Zhn − Zh‖β,K = 0. (19)
According to Lemma A.4 below (a particular case of Lemma A.1 in [1]), the proofof (19) can be divided into two steps:
(1) Pointwise convergence: for any (t, x) ∈ [0, T ]×K,
limn→∞ |Zhn(t, x) − Zh(t, x)| = 0. (20)
(2) Estimation of the increments: for any (t, x), (s, y) ∈ [0, T ]×K,
supn≥1
|(Zhn(t, x) − Zh(t, x)) − (Zhn(s, y) − Zh(s, y))| ≤ C(|t− s|β1 + |x− y|β2).
(21)
We will prove those two estimates in the following two steps.
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Step 1. Pointwise convergence. For any (t, x) ∈ [0, T ]×K,
Zhn(t, x) − Zh(t, x) =∫ t
0
〈Gα,δ(t− s, x− )σ(u0(s, )), hn(s, ) − h(s, )〉Hds
+∫ t
0
Gα,δ(t− s) ∗ [b′(u0(s, ))(Zhn(s, )−Zh(s, ))](x)ds
=: In1 (t, x) + In
2 (t, x). (22)
Since hn, h ∈ HNT and u0 is bounded in [0, T ]×R
d, by Cauchy–Schwarz’sinequality on the Hilbert space HT , (C) and (12), we have
|In1 (t, x)|2 ≤
∫ t
0
‖Gα,δ(t− s, x− )σ(u0(s, ))‖2Hds ·
∫ t
0
‖hn(s) − h(s)‖2Hds
≤ 4N2C
∫ t
0
‖Gα,δ(t− s, x− )‖2Hds
≤ C(N) < +∞,
here C(N) is independent of n, t, x. Since hn → h weakly in HNT , we know that
In1 → 0 in C([0, T ]× R
d; R) by Arzela–Ascoli theorem. This implies that
limn→∞ sup
t∈[0,T ],x∈Rd
|In1 (t, x)| = 0. (23)
Set ζn(t) := sup0≤s≤t,x∈Rd |Zhn(s, x)−Zh(s, x)|. By Lemma A.1 and (16), we have
|I2(t, x)| ≤∫ t
0
∫Rd
Gα,δ(t− s, x− y)|b′(u0(s, y))(Zhn(s, y) − Zh(s, y))|dyds
≤ L
∫ t
0
∫Rd
Gα,δ(t− s, x− y) sup0≤l≤s,z∈Rd
|Zhn(s, z) − Zh(s, z)|dyds
≤ L
∫ t
0
ζn(s)ds. (24)
By (22) and (24), we have
ζn(t) ≤ L
∫ t
0
ζn(s)ds+ supt∈[0,T ],x∈Rd
|In1 (t, x)|.
Hence, by the Gronwall’s lemma and (23), we obtain that
ζn(T ) ≤ eLT limn→∞ sup
t∈[0,T ],x∈Rd
|In1 (t, x)| → 0, as n→ ∞,
which is stronger than (20).
Step 2. Estimation of the increments. For any 0 ≤ t ≤ T, s > 0, x ∈ Rd, y ∈ K,
[Zhn(t+ s, x+ y) − Zh(t+ s, x+ y)] − [Zhn(t, x) − Zh(t, x)]
=∫ t+s
0
〈Gα,δ(t+ s− l, x+ y − )σ(u0(l, )), hn(l, ) − h(l, )〉Hdl
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Moderate deviations for a fractional stochastic heat equation
+∫ t+s
0
Gα,δ(t+ s− l) ∗ [b′(u0(l, ))(Zhn(l, ) − Zh(l, ))](x+ y)dl
−∫ t
0
〈Gα,δ(t− l, x− )σ(u0(l, )), hn(l, ) − h(l, )〉Hdl
−∫ t
0
Gα,δ(t− l) ∗ [b′(u0(l, ))(Zhn(l, ) − Zh(l, ))](x)dl
=∫ t
0
〈[Gα,δ(t+ s− l, x+ y − ) − Gα,δ(t+ s− l, x− )]σ(u0(l, )), hn(l, )
− h(l, )〉Hdl +∫ t
0
〈[Gα,δ(t+ s− l, x− )
−Gα,δ(t− l, x− )]σ(u0(l, )), hn(l, ) − h(l, )〉Hdl
+∫ t+s
t
〈Gα,δ(t+ s− l, x+ y − )σ(u0(l, )), hn(l, ) − h(l, )〉Hdl
+[∫ t+s
0
Gα,δ(t+ s− l) ∗ [b′(u0(l, ))(Zhn(l, ) − Zh(l, ))](x+ y)dl
−∫ t
0
Gα,δ(t− l) ∗ [b′(u0(l, ))(Zhn(l, ) − Zh(l, ))](x)dl]
=: An1 +An
2 +An3 +An
4 . (25)
Since u0 is bounded, hn, h ∈ HNT , we know that
supn≥1
∫ T
0
‖u0(s, )(hn − h)(s, )‖2Hds <∞. (26)
This, together with Cauchy–Schwarz’s inequality, (26) and Lemma A.2, implies thatfor each 0 < β1 < (1−η)/2, 0 < β2 < min(1−η)α0/2, 1/2, there exists a constantC independent of n such that
|An1 | ≤ C|y|β2 , |An
2 | ≤ Csβ1 , |An3 | ≤ Csβ1 . (27)
Let us now give the estimate of An4 . Denote
Vn(t, x) := b′(u0(t, x))(Zhn(t, x) − Zh(t, x)).
By the estimates in Step 1, (16) and (18), we know that
supn≥1
supt∈[0,T ],x∈Rd
|Vn(t, x)| <∞. (28)
After a change of variable, we have
An4 =
∫ s
0
∫Rd
Gα,δ(t+ s− l, x+ y − z)V (l, z)dldz
+∫ t
0
∫Rd
Gα,δ(t− l, x− z)[V (s+ l, y + z) − V (l, z)]dldz.
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By (i) of Lemma A.1 and (28), we know that
supn≥1
∫ s
0
∫Rd
Gα,δ(t+ s− l, x+ y − z)|Vn(l, z)|dldz ≤ c1s.
By the Lipschitz continuity of b′, the Holder continuity of u0 and the bounded-ness of Zhn and Zh, we have
|V (s+ l, y + z) − V (l, z)|≤ |b′(u0(s+ l, y + z)) − b′(u0(l, z))| · |Zhn(s+ l, y + z) − Zh(s+ l, y + z)|
+ |b′(u0(l, z))| · |[Zhn(s+ l, y+ z)−Zh(s+ l, y+ z)]− [Zhn(l, z)−Zh(l, z)]|≤ C(L′)|u0(s+ l, y + z) − u0(l, z)| + |b′(u0(l, z))| · |[Zhn(s+ l, y + z)
−Zh(s+ l, y + z)] − [Zhn(l, z)− Zh(l, z)]|≤ c1(sβ1 + |y|β2) + c2 sup
z∈Rd
|[Zhn(s+ l, y + z)
−Zh(s+ l, y + z)] − [Zhn(l, z)− Zh(l, z)]|. (29)
Putting
φ(l, s, y) := supz∈Rd
[Zhn(s+ l, y + z) − Zh(s+ l, y + z)] − [Zhn(l, z) − Zh(l, z)]
.
Hence, by (25), (27) and (29), we have
φ(t, s, y) ≤ c3(s+ sβ1 + |y|β2) + c2
∫ t
0
φ(l, s, y)dl.
Therefore by the Gronwall’s inequality,
sup0≤t≤T
supx∈Rd
|[Zhn(t+ s, x+ y) − Zh(t+ s, x+ y)] − [Zhn(t, x) − Zh(t, x)]|
≤ c(s+ sβ1 + |y|β2).
The proof is complete.
4. Moderate Deviation Principle
The main aim of this paper is to prove that 1√ελ(ε)
(uε−u0) satisfies an LDP on theHolder space, where λ(ε) satisfies (3). This special type of LDP is usually calledthe moderate deviation principle of uε (cf. [11]).
4.1. Large deviation principle
First, recall the definition of large deviation principle. See [11].
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Moderate deviations for a fractional stochastic heat equation
Let (Ω,F ,P) be a probability space with an increasing family Ft0≤t≤T of thesub-σ-fields of F satisfying the usual conditions. Let E be a Polish space with theBorel σ-field B(E).
Definition 4.1. A function I : E → [0,∞] is called a rate function on E , if foreach M < ∞, the level set x ∈ E : I(x) ≤ M is a compact subset of E . Afamily of positive numbers λ(ε)ε>0 is called a speed function if λ(ε) → +∞ asε→ 0.
Definition 4.2. A family Xε of E-valued random elements is said to satisfythe large deviation principle on E with rate function I and with speed functionλ(ε)ε>0, if the following two conditions hold.
(a) (Large deviation upper bound) For each closed subset F of E ,
lim supε→0
1λ(ε)
log P(Xε ∈ F ) ≤ − infx∈F
I(x).
(b) (Large deviation lower bound) For each open subset G of E ,
lim infε→0
1λ(ε)
log P(Xε ∈ G) ≥ − infx∈G
I(x).
4.2. The main result
From now on, we always assume that K is a compact set in Rd, α =
(α1, . . . , αd), αi ∈ [0, 2]\1 for i = 1, . . . , d, β = (β1, β2) satisfies that 0 < β1 <
α0(1 − η)/2, 0 < β2 < 1 − η where α0 = min1≤i≤dαi, η ∈ ] 0, 1].The main result of this paper is the following theorem.
Theorem 4.1. Assuming conditions (C), (Hαη ) and (D) for η ∈ ]0, 1]. Let uε be
the solution of Eq. (10). Then the law of (uε − u0)/(√ελ(ε)) obeys an LDP on the
space Eβ([0, T ] ×K; Rd) with speed λ2(ε) and with rate function
I(f) = infh∈HT ;Zh=f
12‖h‖2
HT
, (30)
where Zh is defined by (17).
4.3. Proof of Theorem 4.1
We shall apply the weak convergence approach to establish moderate deviationprinciple.
Denote by AH the set of predictable process which belongs to L2(Ω× [0, T ];H).For any N > 0, let
ANH := h ∈ HT ; ‖h‖HT ≤ N.
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For any v ∈ ANH and ε ∈ (0, 1], define the controlled equation Zε,v by
Zε,v(t, x)
=1
λ(ε)
∑k≥1
∫ t
0
〈Gα,δ(t− s, x− )σ(u0(s, ) +√ελ(ε)Zε,v(s, )), ek〉HdBk
s
+∫ t
0
〈Gα,δ(t− s, x− )σ(u0(s, ) +√ελ(ε)Zε,v(s, )), v(s, )〉Hds
+∫ t
0
Gα,δ(t− s) ∗
[b(u0(s, ) +
√ελ(ε)Zε,v(s, )) − b(u0(s, ))√
ελ(ε)
]ds.
(31)
Following the proof of Theorem 2.1 in [2] and Proposition 2.7 in [13], onecan show the existence and uniqueness of the stochastic controlled equation givenby (31).
Lemma 4.2. Assuming conditions (C), (Hαη ) and (D) for η ∈ ] 0, 1], there exists
a unique random field solution to Eq. (31), Zε,v(t, x); (t, x) ∈ [0, T ] × Rd, which
satisfies that for any p ≥ 1,
supε≤1
supv∈AN
H
sup(t,x)∈[0,T ]×Rd
E[|Zε,v(t, x)|p] <∞. (32)
Inspired by [3], let us consider the following two conditions which correspondon the weak convergence approach frame in our setting. Also refer to the weakconvergence approach used in [13].
(a) The set Zh;h ∈ HNT is a compact set of Eβ , where Zh is the solution of
Eq. (17).(b) For any family vε; ε > 0 ⊂ AN
H which converges in distribution as ε → 0 tov ∈ AN
H, as HNT -valued random variables, we have
limε→0
Zε,vε
= Zv
in distribution, as Eβ-valued random variables, where Zv denotes the solutionof Eq. (17) corresponding to the HN
T -valued random variable v (instead of adeterministic function h).
Proof of Theorem 4.1. Applying to Theorem 6 in [3], a verification of conditions(a) and (b) implies the validity of Theorem 4.1. Condition (a) follows from thecontinuity of the mapping h : HN
T → Zh ∈ Eβ([0, T ] × K; Rd), which has beenestablished in Theorem 3.2. Next, we verify Condition (b).
By the Skorokhod representation theorem, there exist a probability(Ω, F , (Ft), P), and, on this basis, a sequence of independent Brownian motionsB = (Bk)k≥1 and also a family of Ft-predictable processes vε; ε > 0, v belonging
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Moderate deviations for a fractional stochastic heat equation
to L2(Ω× [0, T ];H) taking values on HNT , P-a.s., such that the joint law of (vε, v, B)
under P coincides with that of (vε, v, B) under P and
limε→0
〈vε − v, g〉HT = 0, ∀ g ∈ HT , P-a.s.
Let Zε,vε
be the solution to a similar equation as (31) replacing v by vε and Bby B. Thus, to verify the condition (b), it is sufficient to prove that
limε→0
‖Zε,vε − Z v‖β,K = 0 in probability. (33)
From now on, we drop the bars in the notation for the sake of simplicity, andwe denote
Y ε,vε,v := Zε,vε − Zv.
According to Lemma A.4, the proof of (33) can be divided into two steps: forany (t, x), (s, y) ∈ [0, T ]×K with K being compact in R
d,
(1) Pointwise convergence:
limε→0
|Y ε,vε,v(t, x)| = 0 in probability. (34)
(2) Estimation of the increments: there exists a constant C satisfied that
supε≤1
E|Y ε,vε,v(t, x) − Y ε,vε,v(s, y)|2 ≤ C[|t− s|β1 + |x− y|β2 ]. (35)
Those two estimates will be established in the following steps.
Step 1. Pointwise convergence. For any (t, x) ∈ [0, T ]×K,
Zε,vε
(t, x) − Zv(t, x)
=1
λ(ε)
∑k≥1
∫ t
0
〈Gα,δ(t− s, x− )σ(u0(s, ) +√ελ(ε)Zε,vε
(s, )), ek〉HdBks
+∫ t
0
〈Gα,δ(t− s, x− )σ(u0(s, ) +√ελ(ε)Zε,vε
(s, )), vε(s, )〉Hds
−∫ t
0
〈Gα,δ(t− s, x− )σ(u0(s, )), v(s, )〉Hds
+∫ t
0
Gα,δ(t− s) ∗
[b(u0(s, )+
√ελ(ε)Zε,vε
(s, ))− b(u0(s, ))√ελ(ε)
](x)ds
−∫ t
0
Gα,δ(t− s) ∗ [b′(u0(s, ))Zv(s, )](x)ds
=: Aε1(t, x) +Aε
2(t, x) +Aε3(t, x). (36)
Let
J (t) :=∫
Rd
µ(dξ)|FGα,δ(t)(ξ)|2. (37)
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For the first term Aε1, by Burkholder’s inequality, the Lipschitz property of σ,
Eqs. (14) and (32), we have that
E[|Aε1(t, x)|2] = λ−2(ε)E
[∫ t
0
‖Gα,δ(t− s, x− )σ(u0(s, ) +√ελ(ε)Zε,v(s, ))‖2
Hds]
≤ λ−2(ε)c1∫ t
0
ds
(1 + sup
(r,y)∈[0,s]×Rd
E|u0(r, y) +√ελ(ε)Zε,vε
(s, y)|2)
×J (t− s)
≤ c2λ−2(ε). (38)
The second term is further divided into two terms:
|Aε2(t, x)| ≤
∫ t
0
|〈Gα,δ(t− s, x− )
× [σ(u0(s, ) +√ελ(ε)Zε,vε
(s, )) − σ(u0(s, ))], vε(s, )〉H|ds
+∫ t
0
|〈Gα,δ(t− s, x− )σ(u0(s, )), vε(s, ) − v(s, )〉H|ds
=: Aε2,1(t, x) +Aε
2,2(t, x).
By the Lipschitz condition of σ, (32) and Cauchy–Schwarz’s inequality, we have
E|Aε2,1(t, x)|2 ≤ ελ2(ε)L2
(∫ t
0
sup(r,y)∈[0,s]×Rd
E|Zε,vε
(r, y)|2 · J (t− s)ds
)
×(∫ t
0
‖vε(s, )‖2Hds
) 12
≤ ελ2(ε)C(L,N). (39)
By the similar arguments as in the proof of (23), we can show that
limε→0
sup(t,x)∈[0,T ]×Rd
|Aε2,2(t, x)| → 0 in probability. (40)
For the third term Aε3, by the Taylor’s formula, the Lipschitz continuity of b′
and (16), we have
|Aε3(t, x)| ≤
∣∣∣∣∫ t
0
Gα,δ(t− s) ∗
[b(u0(s, ) +
√ελ(ε)Zε,vε
(s, )) − b(u0(s, ))√ελ(ε)
− b′(u0(s, ))Zε,vε
(s, )]
(x)ds
∣∣∣∣+∣∣∣∣∫ t
0
Gα,δ(t− s) ∗ [b′(u0(s, ))(Zε,vε
(s, ) − Zv(s, ))](x)ds∣∣∣∣
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Moderate deviations for a fractional stochastic heat equation
≤ L′√ελ(ε)∫ t
0
Gα,δ(t− s) ∗ |Zε,vε
(s, )|(x)ds
+L
∫ t
0
Gα,δ(t− s) ∗ |Zε,vε
(s, ) − Zv(s, )|(x)ds.
Then, by (32), we have
E|Aε3(t, x)|2 ≤ 2L′2ελ2(ε)
∫ t
0
∫Rd
Gα,δ(t− s, x− y) sup(r,z)∈[0,s]×Rd
E[|Zε,vε
(r, z)|2]dsdy
+ 2L2
∫ t
0
∫Rd
Gα,δ(t− s, y) sup(r,z)∈[0,s]×Rd
E[|Zε,vε
(r, z)−Zv(r, z)|2]dsdy
≤ C(L′)ελ2(ε) + 2L2
∫ t
0
sup(r,z)∈[0,s]×Rd
E[|Y ε,vε,v(r, z)|2]ds. (41)
Define the stopping time
τM,ε := inf
t ≤ T ; sup
(s,x)∈[0,t]×Rd
|Aε2,2(s, x)| ≥M
,
where M is some constant large enough.Putting (36), (38)–(41) together, we have
sup(s,x)∈[0,t]×Rd
E[|Y ε,vε,v(s ∧ τM,ε, x)|2]
≤ C
(sup
(s,x)∈[0,t]×Rd
E[|Aε1(s ∧ τM,ε, x)|2] + sup
(s,x)∈[0,t]×Rd
E[|Aε2,1(s ∧ τM,ε, x)|2]
+ sup(s,x)∈[0,t]×Rd
E[|Aε2,2(s ∧ τM,ε, x)|2] + E[|Aε
3(t ∧ τM,ε, x)|2])
≤ C
(λ−2(ε) + 2ελ2(ε) + sup
(s,x)∈[0,t]×Rd
E[|Aε2,2(s ∧ τM,ε, x)|2]
+∫ t
0
sup(r,z)∈[0,s]×Rd
E[|Y ε,vεv,(r ∧ τM,ε, z)|2ds]).
By Gronwall’s lemma, we obtain that
sup(s,x)∈[0,T ]×Rd
E[|Y ε,vε,v(s ∧ τM,ε, x)|2]
≤ C(T, L, L′)
(λ−2(ε) + 2ελ2(ε) + sup
(s,x)∈[0,T ]×Rd
E[|Aε2,2(s ∧ τM,ε, x)|2]
).
(42)
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Since |Aε2,2(s∧ τM,ε, x)| ≤M for any s ∈ [0, T ], by (40) and the dominated conver-
gence theorem, we know that
sup(s,x)∈[0,T ]×Rd
E[|Aε2,2(s ∧ τM,ε, x)|2] → 0, as ε→ 0.
This inequality, together with (42), implies that
sup(s,x)∈[0,T ]×Rd
E[|Y ε,vε,v(s ∧ τM,ε, x)|2] → 0, as ε→ 0. (43)
Letting M → ∞, we obtain that for any (t, x) ∈ [0, T ]× Rd,
limε→0
|Y ε,vε,v(t, x)| = 0 in probability.
Step 2. Estimation of the increments. For any (t, x), (s, y) ∈ [0, T ] × K with K
being compact in Rd and t ≥ s,
Y ε,vε,v(t, x) − Y ε,vε,v(s, y)
=1
λ(ε)
∑k≥1
∫ t
0
〈Gα,δ(t− r, x− )σ(u0(r, ) +√ελ(ε)Zε,vε
(r, )), ek〉HdBkr
−∫ s
0
〈Gα,δ(s− r, y − )σ(u0(r, ) +√ελ(ε)Zε,vε
(r, )), ek〉HdBkr
+∫ t
0
〈Gα,δ(t− r, x− )σ(u0(r, ) +√ελ(ε)Zε,vε
(r, )), vε(r, )〉Hdr
−∫ s
0
〈Gα,δ(s− r, y − )σ(u0(r, ) +√ελ(ε)Zε,vε
(r, )), vε(r, )〉Hdr
−∫ t
0
〈Gα,δ(t− r, x− )σ(u0(r, )), v(r, )〉Hdr
−∫ s
0
〈Gα,δ(s− r, y − )σ(u0(r, )), v(r, )〉Hds
+∫ t
0
Gα,δ(t− r) ∗[b(u0(r, ) +
√ελ(ε)Zε,vε
(r, )) − b(u0(r, ))√ελ(ε)
](x)dr
−∫ s
0
Gα,δ(s− r) ∗[b(u0(r, ) +
√ελ(ε)Zε,vε
(r, )) − b(u0(r, ))√ελ(ε)
](y)dr
−∫ t
0
Gα,δ(t− r) ∗ [b′(u0(r, ))Zv(r, )](x)dr
−∫ s
0
Gα,δ(s− r) ∗ [b′(u0(r, ))Zv(r, )](y)dr
=:1
λ(ε)Bε
1(t, s, x, y) +Bε2(t, s, x, y)
+Bε3(t, s, x, y) +Bε
4(t, s, x, y) +Bε5(t, s, x, y).
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For the first term, by the Lipschitz continuity of σ, (32) and Lemma A.2, weobtain that for any β1 ∈ ] 0, (1 − η)/2[, 0 < β2 < min(1 − η)α0/2, 1/2
E[|Bε1(t, s, x, y)|2]
≤ C1E
[∫ s
0
‖(Gα,δ(t− r, x− ) − Gα,δ(s− r, x− ))σ(u0(r, )
+√ελ(ε)Zε,vε
(r, ))‖2Hdr +
∫ t
s
‖Gα,δ(t− r, x− )σ(u0(r, )
+√ελ(ε)Zε,vε
(r, ))‖2Hdr +
∫ s
0
‖(Gα,δ(s− r, x− )
−Gα,δ(s− r, y − ))σ(u0(r, ) +√ελ(ε)Zε,vε
(r, ))‖2Hdr
]
≤ C2 sup(r,z)∈[0,T ]×Rd
E[|σ(u0(r, ) +√ελ(ε)Zε,vε
(r, ))|2]
×[∫ s
0
‖(Gα,δ(t− r, x− ) − Gα,δ(s− r, x− ))‖2Hdr
+∫ t
s
‖Gα,δ(t− r, x− )‖2Hdr +
∫ s
0
‖(Gα,δ(s− r, x− )
−Gα,δ(s− r, y − ))‖2Hdr
]
≤ C3(|t− s|2β1 + |x− y|2β2), (44)
where C3 is independent of ε > 0.Using the similar argument in (44) and noticing the fact vε, v ∈ AN
H , we givethe following estimates without the proof
E[|Bεi (t, s, x, y)|2] ≤ C(|t− s|2β1 + |x− y|2β2), i = 2, 3, (45)
where C is independent of ε > 0.We can also deal with the last two terms Bε
4 , Bε5 by the same argument. Next
we only estimate Bε4 , and the same result holds for Bε
5 .By Taylor’s formula, (16) and using the same approach in estimation of Λ3 in
the proof of [13, Proposition 2.10], we have for any t ∈ [0, T ],
E[|Bε4(t, s, x, y)|2]
≤ LE
∣∣∣∣∫ t
0
Gα,δ(t− r) ∗ |Zε,vε
(r, )|(x)dr−∫ s
0
Gα,δ(s− r) ∗ |Zε,vε
(r, )|(y)dr∣∣∣∣2
≤ C
[|t− s| +
∫ t
0
supz∈Rd
E|Zε,vε
(t− s+ r, x− y + z) − Zε,vε
(r, z)|2dr]
≤ C[|t− s| + T (|t− s|2β1 + |x− y|2β2)], (46)
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where in the last inequality we have used the estimate in the proof of Theorem 3.1in [2], C is independent of ε.
Putting together all the estimates, we get (35). The proof is complete.
Appendix
To make reading easier, we present here some results on the kernelG from Boulanbaet al. [2].
Lemma A.1. ([2, Lemma 1.1]) For α ∈ ]0, 2]\1 such that |δ| ≤ minα, 2 − α,the following statements hold.
(i) The function Gα,δ(t, x) is the density of a Levy α-stable process in time t.(ii) Semigroup property: Gα,δ(t, x) satisfies the Chapman–Kolmogorov equation,
i.e. for 0 < s < t,
Gα,δ(t+ s, x) =∫
R
Gα,δ(t, y)Gα,δ(s, y − x)dy.
(iii) Scaling property: Gα,δ(t, x) = t−1/αGα,δ(1, t−1/αx).(iv) There exists a constant cα such that 0 ≤ Gα,δ(1, x) ≤ cα/(1 + |x|1+α), for all
x ∈ R.
The next proposition studies the Holder regularity of the Green function, whoseproof is contained in the proof of Proposition 3.2 in [2].
Lemma A.2. Under (Hαη ), it holds that
(i) For each 0 ≤ s < t ≤ T, β1 ∈ ] 0, (1 − η)/2[, there exists a constants C > 0such that ∫ s
0
‖Gα,δ(t− r, ) − Gα,δ(s− r, )‖2Hdr ≤ C|t− s|2β1
and ∫ t
s
‖Gα,δ(t− r, )‖2Hdr ≤ C|t− s|2β1 .
(ii) For each 0 < β2 < min(1 − η)α0/2, 1/2, there exists a constant C > 0 forany x, y ∈ R
d,∫ T
0
‖Gα,δ(T − s, x− ) − Gα,δ(T − s, y − )‖2Hds ≤ C|x− y|2β2 .
The next lemma is about the Holder regularity of the stochastic integral. See[2, Proposition 3.2]. For a given predictable random field V , we set
U(t, x) :=∑k≥1
∫ t
0
〈Gα,δ(t− s, x− )V (s, ), ek〉HdBks .
Lemma A.3. Assume that sup0≤t≤T supx∈Rd E(|V (t, x)|p) is finite for some p largeenough. Then under (Hα
η ), we have
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(i) For each x ∈ Rd a.s. the mapping t → U(t, x) is β1-Holder continuous for
0 < β1 < (1 − η)/2.(ii) For each t ∈ [0, T ] a.s. the mapping x → U(t, x) is β2-Holder continuous for
0 < β2 < minα0(1 − η)/2, 1/2.The following result is a consequence of Lemma A.1 in [1].
Lemma A.4. Let K be a compact set in Rd and let V ε(t, x) : (t, x) ∈ [0, T ]×K
be a family of real-valued functions. Assume
(A1) For any (t, x) ∈ [0, T ]×K,
limε→0
|V ε(t, x)| = 0.
(A2) There exist β1, β2 > 0 satisfied that for any (t, x), (t′, x′) ∈ [0, T ]×K,
|V ε(t, x) − V ε(t′, x′)| ≤ C(|t− t′|β1 + |x− x′|β2),
where C is a constant independent of ε.
Then for any θ ∈ (0, 1), we have
limε→0
‖V ε‖θβ1,θβ2 = 0.
Proof. The stochastic version of this lemma in [1, Lemma A.1] is proved by theGarsia–Rodemich–Rumsey’s lemma. Here we give a direct proof for the determin-istic case.
In view of Arzela–Ascoli theorem, a sequence in C([0, T ] × K; R) convergesuniformly if and only if it is equicontinuous and converges pointwise. Thence, underconditions (A1) and (A2), we know that
limε→0
sup(t,x)∈[0,T ]×K
|V ε(t, x)| → 0.
Thus, for any θ ∈ (0, 1), (t, x) = (t′, x′) ∈ [0, T ]×K, we have
|V ε(t, x) − V ε(t′, x′)|(|t− t′|β1 + |x− x′|β2)θ
=|V ε(t, x) − V ε(t′, x′)|θ|V ε(t, x) − V ε(t′, x′)|1−θ
(|t− t′|β1 + |x− x′|β2)θ
≤ C(|V ε(t, x)| + |V ε(t′, x′)|)1−θ
→ 0 uniformly on ([0, T ]×K)2.
The proof is complete.
Acknowledgments
Y.L. and S.Z. were supported by Natural Science Foundation of China (Grant Nos.11471304 and 11401556). R.W. was supported by Natural Science Foundation ofChina (Grant Nos. 11301498 and 11431014) and the Fundamental Research Fundsfor the Central Universities (Grant No. WK 0010000048). N.Y. was supported byNatural Science Foundation of China (Grant Nos. 11101313 and 11371283).
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