Acta Appl Math DOI 10.1007/s10440-014-9969-x

Moderate Deviations for a Stochastic Heat Equation with Spatially Correlated Noise Yumeng Li · Ran Wang · Shuguang Zhang

Received: 19 March 2014 / Accepted: 12 September 2014 © Springer Science+Business Media Dordrecht 2014

Abstract In this paper, we proved a central limit theorem and established a moderate deviation principle for a perturbed stochastic heat equations defined on [0, T ] × [0, 1]d . This equation is driven by a Gaussian noise, white in time and correlated in space. Keywords Stochastic heat equation · Freidlin-Wentzell’s large deviation · Moderate deviations · Central limit theorem Mathematics Subject Classification 60H15 · 60F05 · 60F10

1 Introduction Since the work of Freidlin and Wentzell [17], the theory of small perturbation large deviations for stochastic (partial) differential equation has been extensively developed (see [10, 12, 13]). The large deviation principle (LDP) for stochastic reaction-diffusion equations driven by the space-time white noise was first obtained by Freidlin [16] and later by Sowers [31], Chenal and Millet [6], Carrai and Röckner [4] and other authors. An LDP for a stochastic heat equation driven by a Gaussian noise, white in time and correlated in space was proved by Márquez-Carreras and Sarrà [26].

Y. Li · S. Zhang Department of Statistics and Finance, University of Science and Technology of China, Hefei, Anhui Province 230026, China Y. Li e-mail: [email protected] S. Zhang e-mail: [email protected]

B

R. Wang ( ) School of Mathematical Sciences, University of Science and Technology of China, Hefei, Anhui Province 230026, China e-mail: [email protected]

Y. Li et al.

Like the large deviations, the moderate deviation problems arise in the theory of statistical inference quite naturally. The moderate deviation principle (MDP) can provide us with the rate of convergence and a useful method for constructing asymptotic confidence intervals, see [14, 20, 21, 23, 24] and references therein. Results on the MDP for processes with independent increments were obtained in De Acosta [1], Chen [5] and Ledoux [25]. The study of the MDP estimates for other processes has been carried out as well, e.g., Wu [34] for Markov processes, Guillin and Liptser [22] for diffusion processes, Budhiraja et al. [3] for stochastic differential equations with jumps, and references therein. Wang and Zhang [33] proved a central limit theorem and established a moderate deviation principle for stochastic reaction-diffusion equations on driven by the space-time white noise [0, T ] × [0, 1]. However the high dimensional case is much more complicated, one difficulty comes from the more complicated stochastic integral in the sense of Dalang [8], the other one comes from the Hölder continuity of the Green functions in high dimension. In this paper, we shall study it in this paper. More precisely, we shall prove a central limit theorem and a moderate deviation principle for a perturbed stochastic heat equations defined on [0, T ] × [0, 1]d . This equation is driven by a Gaussian noise, white in time and correlated in space. Let us give the framework taken from Márquez-Carreras and Sarrà [26]. Consider the following perturbed d-dimensional spatial stochastic heat equation on the compact set [0, 1]d , ⎧ √ ε ε ε d ⎪ ⎨Lu (t, x) = εα(u (t, x))F˙ (t, x) + β(u (t, x)), t ≥ 0, x ∈ [0, 1] , (1) uε (t, x) = 0, x ∈ ∂([0, 1]d ), ⎪ ⎩ ε d u (0, x) = 0, x ∈ [0, 1] , with ε > 0, L = ∂t∂ −  where  is the Laplacian on Rd and ∂([0, 1]d ) is the boundary of [0, 1]d . Assume that the coefficients satisfy the following conditions: (C)

the functions α and β are Lipschitz,

i.e., there exists some constant K such that |α(x) − α(y)| ≤ K|x − y|,

|β(x) − β(y)| ≤ K|x − y|

(2)

for all x, y ∈ R. The noise F = {F (ϕ), ϕ : Rd+1 → R} is an L2 (Ω, F , P)-valued Gaussian process with mean zero and covariance functional given by    J (ϕ, ψ) = ds dx dyϕ(s, x)f (x − y)ψ(s, y), (3) R+

Rd

Rd

and f : Rd → R+ is a continuous symmetric function on Rd − {0} such that there exists a non-negative tempered measure λ on Rd , whose Fourier transform is f . The function in (3) is said to be a covariance functional if all these assumptions are satisfied. Then, in addition,   J (ϕ, ψ) = ds λ(dξ )F ϕ(s, ·)(ξ )F ψ(s, ·)(ξ ), R+

Rd

where F is the Fourier transform and z is the conjugate complex of z. In (3), we could also work with a non-negative and non-negative definite tempered measure, therefore symmetric,

Moderate Deviations for a Stochastic Heat Equation

instead of the function f but, in this case, all the notation over the sets appearing in the integrals is becoming tedious. In this paper, moreover, we assume the following hypothesis on the measure λ:  λ(dξ ) < +∞, (4) (Hη ) 2 η Rd (1 + ξ  ) for some η ∈ (0, 1]. For instance, the function f (x) = x−κ , κ ∈ (0, d), satisfies (4). As in Dalang [8], the Gaussian process F can be extended to a worthy martingale measure, in the sense of Walsh [32], 

 M = Mt (A), t ∈ R+ , A ∈ Bb Rd , where Bb (Rd ) denotes the collection of all bounded Borel measurable sets in Rd . Under the assumptions (C) and (Hη ), following the approach of Walsh [32], one can prove that there is a unique solution to Eq. (1) given by the following mild form  t  √ Gt−s (x, y)α uε (s, y) F (dsdy) uε (t, x) = ε 

0

[0,1]d



t

+

ds 0

[0,1]d

 dyGt−s (x, y)β uε (s, y) ,

where Gt (x, y) is the Green kernel associated with the heat equation on [0, 1]d : ⎧ ∂ d ⎪ ⎨ ∂t Gt (x, y) = x Gt (x, y), t ≥ 0, x, y ∈ [0, 1] , d Gt (x, y) = 0, x ∈ ∂([0, 1] ), ⎪ ⎩ G0 (x, y) = δ(x − y).

(5)

(6)

The stochastic integral in (5) is defined with respect to the Ft -martingale measure M. Denote DTd := [0, T ] × [0, 1]d . If d > 1, the evolution equation can not be driven by the Brownian sheet because G does not belong to L2 (DTd ) and we need to work with a smoother noise. The study of existence and uniqueness of solution to Eq. (5) on Rd has been analyzed by Dalang in [8]. Many other authors have also studied existence and uniqueness of solutions to d-dimensional spatial stochastic equations, in particular, wave and heat equations, see [7, 9, 19, 27, 30] and references therein. Nualart et al. studied the probability densities of the mild solutions of the SPDEs with such a noise, using tools of the Malliavin calculus in [28, 29]. Under the assumption (C) and (Hη ) for some η ∈ (0, 1), Márquez-Carreras and Sarrà [26] proved that the trajectories of the process uε (t, x) are (γ1 , γ2 )-Hölder continuous with respect to the parameters t and x, satisfying γ1 ∈ (0, (1 − η)/2) and γ2 ∈ (0, 1 − η). As the parameter ε tends to zero, the solutions uε of (1) will tend to the solution of the equation defined by ⎧ 0 0 d ⎪ ⎨Lu (t, x) = β(u (t, x)), t ≥ 0, x ∈ [0, 1] , (7) x ∈ ∂([0, 1]d ), u0 (t, x) = 0, ⎪ ⎩ 0 u (0, x) = 0, x ∈ [0, 1]d . In this paper we shall investigate deviations of uε from the deterministic solution u0 , as ε decreases to 0, that is, the asymptotic behavior of the trajectories, U ε (t, x) := √

1  ε u − u0 (t, x), εh(ε)

(t, x) ∈ [0, T ] × [0, 1]d ,

Y. Li et al.

where h(ε) is some deviation scale, which strongly influences the asymptotic behavior of U ε . √ The case h(ε) = 1/ ε provides some large deviations estimates. Under the assumption (C) and (Hη ) for some η ∈ (0, 1), Márquez-Carreras and Sarrà in [26] proved that the law of the solution uε satisfies a large deviation principle on the Hölder space, see Theorem 2.1 for details. If h(ε) is identically equal to 1, we are in the domain of the central limit theorem. We √ will show that (uε − u0 )/ ε converges as ε ↓ 0 to a random field. To fill in the gap between the central limit theorem scale [h(ε) = 1] and the large devi√ ations scale [h(ε) = 1/ ε], we will study moderate deviations, that is when the deviation scale satisfies √ εh(ε) → 0 as ε → 0. (8) h(ε) → +∞, The moderate deviation principle (MDP) enables us to refine the estimates obtained through √ the central limit theorem. It provides the asymptotic behavior for P(uε − u0  ≥ δ εh(ε)) √ while the central limit theorem gives asymptotic bounds for P(uε − u0  ≥ δ ε). Throughout this paper, we assume (8) is in place. To prove the central limit theorem and the moderate deviation principle for the SPDE (1), we shall establish some exponential estimates in the space of Hölder continuous functions. The Garsia-Rodemich-Rumsey’s lemma will play a very important role. The rest of this paper is organized as follows. In Sect. 2, the precise results are stated. In Sect. 3, we shall prove some convergence results of solutions. We establish the CLT and MDP in Sect. 4. We give some precise estimates of the fundamental solution G in the Appendix. We conclude this section by introducing some basic notations and conventions used in the paper. For any function φ : DTd → R, t ∈ [0, T ], let 

|φ|t,∞ = sup |φ(s, x)| : (s, x) ∈ [0, t] × [0, 1]d ,  |φ(s1 , x1 ) − φ(s2 , x2 )| d : (s , x ) =

(s , x ) ∈ [0, t] × [0, 1] , φt,γ1 ,γ2 = sup 1 1 2 2 |s1 − s2 |γ1 + |x1 − x2 |γ2 and let |φ|γ1 ,γ2 = |φ|T ,∞ + φT ,γ1 ,γ2 . Let C γ1 ,γ2 (DTd ; R) be the set of functions φ : DTd → R such that |φ|γ1 ,γ2 < +∞, endowed with the | · |γ1 ,γ2 -norm. Throughout the paper, C(p) is a positive constant depending on the parameter p, and C is a constant depending on no specific parameter (except T and the Lipschitz constants), whose value may be different from line to line by convention.

2 Main Results Let E be the space of all measurable functions ϕ : Rd → R such that 

 dy Rd

Rd

dz|ϕ(y)|f (y − z)|ϕ(z)| < +∞,

Moderate Deviations for a Stochastic Heat Equation

endowed with the inner product  ϕ, ψ E :=

 dy Rd

Rd

dzϕ(y)f (y − z)ψ(z).

Let H be the completion of (E , ·, · E ). For T > 0, let HT := L2 ([0, T ]; H), which is a real separable Hilbert space such that, if ϕ, ψ ∈ HT ,  E F (ϕ)F (ψ) =



T

 ϕ(s, ·), ψ(s, ·) H ds =: ϕ, ψ HT ,

0

(9)

where F is the noise introduced in Sect. 1. For any (t, x), (t , x ) ∈ DTd , let   Γ˜ (t, x), t , x :=





t∧t



ds 0

dy R(x)

R(x )

dzf (y − z),

where R(x) is the rectangle [0, x] (here [0, x] is a product of the rectangles). By Márquez-Carreras and Sarrà [26], for any γ ∈ [0, 12 ), there exists an abstract Wiener measure ν on C γ ,γ (DTd ; R) such that     w(t, x)w t , x dν(ω) = Γ˜ (t, x), t , x . C γ ,γ (DTd ;R)

For any (t, x) ∈ DTd , set

 t WF (t, x) =

F (ds, dy). 0

R(x)

Then WF is a Gaussian process with covariance function Γ˜ . Its trajectories belong to

C γ ,γ (DTd ; R), and the law of the process WF is ν. Let the space H be the set of the functions h˜ such that, for any h ∈ HT , (t, x) ∈ DTd ,

˜ x) = 1[0,t]×R(x) , h HT , h(t, with the scalar product ˜ k ˜ H = h, k HT . h, Classical results on Gaussian processes (see [11, Theorem 3.4.12]) show that, for any γ ∈ [0, 12 ), the family {εWF , ε > 0} satisfies a large deviation principle on C γ ,γ (DTd ; R) with the rate function  1 ˜ hH , if h˜ ∈ H ; ˜ = 2 I˜(h) (10) +∞, otherwise. For any h˜ ∈ H, (t, x) ∈ DTd , consider the solution to the deterministic evolution equation ˜





t

S h (t, x) =

ds [0,1]2d

0

 +



t

ds 0

[0,1]d

 ˜ dydzGt−s (x, y)α S h (s, y) f (y − z)h(s, z)  ˜ dyGt−s (x, y)β S h (s, y) .

(11)

Y. Li et al.

Recall the following theorem from Márquez-Carreras and Sarrà [26, Theorem 5.2]: Theorem 2.1 Assume (C) and (Hη ) for some η ∈ (0, 1). Then the law of the solution uε of Eq. (1) satisfies on C γ ,γ (DTd ; R), γ ∈ (0, (1 − η)/4), a large deviation principle with the good rate function  S(g) =

˜ S h = g}, inf{I˜(h); +∞, ˜

if g ∈ Im(S · ), otherwise.

(12)

More precisely, for any Borel measurable subset A of C γ ,γ (DTd ; R),  − info S(g) ≤ lim inf ε log P uε ∈ A g∈A

ε→0

 ≤ lim sup ε log P uε ∈ A ≤ − inf S(g), g∈A¯

ε→0

where Ao and A¯ denote the interior and the closure of A, respectively. We furthermore suppose that (D)

the function β is differentiable, and its derivative β is Lipschitz.

More precisely, there exists a positive constant K such that |β (y) − β (z)| ≤ K |y − z|

for all y, z ∈ R.

(13)

Combined with the Lipschitz continuity of β, we conclude that |β (z)| ≤ K,

∀z ∈ R.

(14)

Our first result is the following central limit theorem. Theorem 2.2 Assume (C), (D) and (Hη ) for some η ∈ (0, 1). Then for any γ ∈ (0, (1 − √ η)/2) and r ≥ 1, the random field (uε − u0 )/ ε converges in Lr to a random field Y on d C γ ,2γ (DT ; R), determined by ⎧ 0

0 ⎪ ⎨LY (t, x) = α(u (t, x))F˙ (t, x) + β (u (t, x))Y (t, x), d Y (t, x) = 0, x ∈ ∂([0, 1] ), ⎪ ⎩ Y (0, x) = 0, x ∈ [0, 1]d ,

(15)

for all (t, x) ∈ [0, T ] × [0, 1]d . Under conditions (C), (D) and (Hη ) for some η ∈ (0, 1), by Theorem 2.1 and in view of the assumption (8), we see that, for any γ ∈ (0, (1 − η)/4), Y / h(ε) obeys an LDP on C γ ,γ (DTd ; R) with the speed h2 (ε) and with the good rate function  I (g) =

˜

˜ SYh = g}, inf{I˜(h); +∞,

if g ∈ Im(SY· ), otherwise,

(16)

Moderate Deviations for a Stochastic Heat Equation ˜

where SYh is the solution to the deterministic evolution equation  t   ˜ ds dydzGt−s (x, y)α u0 (s, y) f (y − z)h(s, z) SYh (t, x) = 0

[0,1]2d





t

+

˜  dyGt−s (x, y)β u0 (s, y) SYh (s, y),

ds [0,1]d

0

(17)

for any fixed h˜ ∈ H, (t, x) ∈ DTd . Our second main result reads as follows: Theorem 2.3 Assume √ (C), (D) and (Hη ) for some η ∈ (0, 1). Then for any γ ∈ (0, (1 − η)/4), (uε − u0 )/( εh(ε)) obeys an LDP on the space C γ ,γ (DTd ; R) with speed h2 (ε) and with rate function I given by (16). 3 Convergence of Solutions The following estimate can be found in Márquez-Carreras and Sarrà [26, Proposition 3.1]. Lemma 3.1 Assume (C) and (H1 ). Then (1) has a unique solution. Moreover, for any T > 0, p ∈ [1, ∞), sup E|uε (t, x)|p < ∞.

sup

(18)

0<ε≤1 (t,x)∈D d T

The next result is concerned with the convergence of uε as ε → 0. Proposition 3.2 Assume (C) and (H1 ). Then for any p ≥ 2, there exists some constant C(p, K, T ) depending on p, K, T such that  p E |uε − u0 |T ,∞ ≤ ε p/2 C(p, K, T ) → 0 as ε → 0. (19) Proof Since 



t

uε (t, x) − u0 (t, x) =

ds [0,1]d

0

+

√

 t ε 0

we have 

|u − u |T ,∞ ε

0

p

p−1

(t,x)∈DT p 2



+ε 2

p−1

[0,1]d

0

 t   sup  d

(t,x)∈DT

Set

 M1ε (t, x) :=

[0,1]d

 t   sup  ds d

 ≤2

    dyGt−s (x, y) β uε (s, y) − β u0 (s, y)



t

ds 0

[0,1]d

0

 Gt−s (x, y)α uε (s, y) F (dsdy),

   ε  0  p dyGt−s (x, y) β u (s, y) − β u (s, y) 

[0,1]d

p   ε Gt−s (x, y)α u (s, y) F (dsdy) . (20)

    dyGt−s (x, y) β uε (s, y) − β u0 (s, y) ,

Y. Li et al.

 t M2ε (t, x) :=

 Gt−s (x, y)α uε (s, y) F (dsdy).

[0,1]d

0

By the Lipschitz condition (C), Hölder’s inequality and (58), for any p > (d + 2)/2, we obtain that E



p |M1ε |T ,∞

 ≤K

p



 pq



t

sup

ds

(t,x)∈DTd

[0,1]d

0



T

≤ C(p, K, T )E

dyGqs (x, y)

 ×E

T

p |uε − u0 |T ,∞ dt

0

p |uε − u0 |T ,∞ dt,

(21)

0

where 1/q + 1/p = 1. For any p > 2, x, y ∈ [0, 1]d , 0 ≤ t ≤ T , by Burkholder’s inequality and Hölder’s inequality and (64), there exists γ1 ∈ (0, 1 − η) such that   p EM2ε (t, x) − M2ε t, x   t  ≤ C(p)E ds

[0,1]2d

0

     dydz Gt−s (x, y) − Gt−s x , y α uε (s, y) f (y − z)

     × Gt−s (x, z) − Gt−s x , z α uε (s, z)   t  ≤ C(p) ds

[0,1]2d

0

 p2

   dydzGt−s (x, y) − Gt−s x , y f (y − z)

   × Gt−s (x, z) − Gt−s x , z 

 p−2  2



t

ds [0,1]2d

0

   dydz Gt−s (x, y) − Gt−s x , y 

     p    p   × f (y − z)Gt−s (x, z) − Gt−s x , z E α uε (s, y)  2 α uε (s, z)  2    ≤ C(p, K, T ) 1 + sup sup E |uε (t, x)|p 0<ε≤1 (t,x)∈D d

 ×

T



t

ds 0

[0,1]2d

   dydzGt−s (x, y) − Gt−s x , y f (y − z)

   × Gt−s (x, z) − Gt−s x , z   ≤ C(p, K, T ) 1 + sup

 p2

  sup E |uε (t, x)|p · |x − x |γ1 p .

(22)

0<ε≤1 (t,x)∈D d T

Similarly, in view of (55) and (56), it follows that for any 0 ≤ t ≤ t ≤ T , x ∈ [0, 1]d and γ2 ∈ (0, 1−η ), 2  p  EM2ε (t, x) − M2ε t , x   ≤ C(p, K, T ) 1 + sup

  sup E |uε (t, x)|p

0<ε≤1 (t,x)∈D d T

Moderate Deviations for a Stochastic Heat Equation





t

×

ds [0,1]2d

0

dydz|Gt−s (x, y) − Gt −s (x, y)|

 p2 × f (y − z)|Gt−s (x, z) − Gt −s (x, z)|  +



t t

ds [0,1]2d

 p2 dydzGt−s (x, y)f (y − z)Gt−s (x, z)

 ≤ C(p, K, T ) 1 + sup

  sup E |uε (t, x)|p · |t − t |γ2 p .

(23)

0<ε≤1 (t,x)∈D d T

Putting together (22) and (23), we prove that for any (t, x), (t , x ) ∈ DTd , γ ∈ (0, 1 − η), p ≥ 2,   pγ  p (24) EM2ε (t, x) − M2ε t , x  ≤ C(p, K, T ) |x − x | + |t − t | 2 , where C(p, K, T ) is independent of ε. For p > 2(d + 1)/γ , applying Garsia-RodemichRumsey’s lemma (Corollary 1.2 in [32]), there exist a random variable Np,ε (ω) and a constant C such that,  ε   M (t, x) − M ε t , x p 2 2   2  pγ −(d+1) C ≤ Np,ε (ω) |x − x | + |t − t | 2 , (25) log |x − x | + |t − t | and sup E[Np,ε ] < +∞. ε∈(0,1]

Choosing t = 0 in (25), we obtain   t E sup (t,x)∈DTd

0

[0,1]d

p  Gt−s (x, y)α uε (s, y) F (dsdy)

≤ C(p, K, T ) sup E[Np,ε ] < +∞.

(26)

ε∈(0,1]

Putting (20), (21), (26) together, and using the Gronwall’s inequality, there exists a constant C(p, K, T ) such that p  E |uε − u0 |T ,∞ ≤ ε p/2 C(p, K, T ) → 0, as ε → 0. The proof is complete.



4 The Proofs of Main Results The following lemma is a consequence of the Garsia-Rodemich-Rumsey’s lemma, see [2, Lemma A1] for the proof. Lemma 4.1 Let {V ε (t, x) : (t, x) ∈ [0, T ] × [0, 1]d } be a family of real-valued stochastic processes and let p ∈ (1, ∞). Assume

Y. Li et al.

(A1) For any (t, x) ∈ [0, T ] × [0, 1]d , lim E|V ε (t, x)|p = 0.

ε→0

(A2) There exists γ0 > 0 such that for any (t, x), (t , x ) ∈ [0, T ] × [0, 1]d ,    p d+1+γ0 EV ε (t, x) − V ε t , x  ≤ C |t − t | + |x − x |2 , where C is a constant independent of ε. Then for any γ ∈ (0, γ0 /p), r ∈ [1, p),  r lim E |V ε |γ ,2γ = 0.

ε→0

4.1 Proof of Theorem 2.2 √ Proof Let Y ε = (uε − u0 )/ ε. We will prove that for any γ ∈ (0, (1 − η)/2), r ≥ 1,  r lim E |Y ε − Y |γ ,2γ = 0.

ε→0

(27)

To this end, we will verify (A1), (A2) for V ε = Y ε − Y . Write Y ε (t, x) − Y (t, x)  t    Gt−s (x, y) α uε (s, y) − α u0 (s, y) F (dsdy) = 0

[0,1]d



t

+ 0

=

   β(uε (s, y)) − β(u0 (s, y))

0 ds dyGt−s (x, y) − β u (s, y) Y (s, y) √ ε [0,1]d 

I1ε (t, x) + I2ε (t, x) + I3ε (t, x),

(28)

where  t I1ε (t, x) :=  I2ε (t, x)

[0,1]d

0



t

:=

   Gt−s (x, y) α uε (s, y) − α u0 (s, y) F (dsdy),

ds 0

[0,1]d

0

[0,1]d

   √ dyGt−s (x, y) β uε (s, y) − β u0 (s, y) / ε

  − β u0 (s, y) Y ε (s, y) ,  t    ds dyGt−s (x, y)β u0 (s, y) Y ε (s, y) − Y (s, y) . I3ε (t, x) := Now we shall divide the proof into the following two steps. Step 1. Following the similar calculations in the proof of (26), and using (2), we deduce that for p > 2, t ∈ [0, T ],   p p p E |I1ε |t,∞ ≤ C(p, K, T )E |uε − u0 |t,∞ ≤ ε 2 C(p, K, T ).

(29)

Moderate Deviations for a Stochastic Heat Equation

By Taylor’s formula, there exists a random field v ε (t, x) taking values in (0, 1) such that     β uε (t, x) − β u0 (t, x) = β u0 (t, x) + v ε (t, x) uε (t, x) − u0 (t, x)  × uε (t, x) − u0 (t, x) . Since β is also Lipschitz continuous and v ε (t, x) ∈ (0, 1), we have   0    β u (t, x) + v ε (t, x) uε (t, x) − u0 (t, x) − β u0 (t, x)    ≤ K v ε (t, x)uε (t, x) − u0 (t, x)   ≤ K uε (t, x) − u0 (t, x).

(30)

Hence |I2ε (t, x)| ≤ K



ds 0

=

√



t

εK



[0,1]d

  dyGt−s (x, y) uε (s, y) − u0 (s, y) Y ε (s, y)



t

ds [0,1]d

0

dyGt−s (x, y)|Y ε (s, y)|2 .

By Hölder’s inequality, for any p > (d + 2)/2,  p p E |I2ε |t,∞ ≤ ε 2 K p



 t   sup  ds d 0

(t,x)∈DT

[0,1]d

 p  t  q  2p q dyGt−s (x, y) × E |Y ε |s,∞ ds, 0

where 1/p +1/q = 1. Using (58) and Proposition 3.2, there exists a constant C(p, K, K , T ) depending on p, K, K , T such that  p  p E |I2ε |t,∞ ≤ ε 2 C p, K, K , T .

(31)

Noticing that |β | ≤ K, by Hölder’s inequality and (58), we deduce that for p > (d + 2)/2, E



p |I3ε |t,∞

 ≤K

p

 t   sup  ds d

(t,x)∈DT

0



t

≤ C(p, K, T )

 p   q

[0,1]d

q dyGt−s (x, y)

t

 p E |Y ε − Y |s,∞ ds,

0

 p E |Y ε − Y |s,∞ ds,

(32)

0

where 1/p + 1/q = 1. Putting (28), (29), (31), (32) together, we have    t  p   p p E |Y ε − Y |s,∞ ds . E |Y ε − Y |t,∞ ≤ C p, K, K , T ε 2 + 0

By Gronwall’s inequality, we have   p p E |Y ε − Y |T ,∞ ≤ ε 2 C p, K, K , T → 0, which in particular, implies (A1) in Lemma 4.1.

as ε → 0,

(33)

Y. Li et al.

Step 2. If the condition (A2) holds, that is there is a constant C independent of ε satisfying ), that for any γ0 ∈ (0, 1−η 2  ε     p pγ E Y (t, x) − Y (t, x) − Y ε t , x − Y t , x  ≤ C |t − t | + |x − x |2 0 . Then for any γ ∈ (0, γ0 −

and r ∈ [1, p), by Lemma 4.1, we have

d+1 ), p

 r lim E |Y ε − Y |γ ,2γ = 0.

ε→0

Since p ≥

d+2 ,0 2 1−η . 2

< γ < γ0 −

d+1 p

<

1−η 2

is arbitrary, we get the desired result (27) for any

0<γ < Next, we will verify the condition (A2) for Y ε − Y . It suffices to show that all the terms Iiε , i = 1, 2, 3 satisfy (A2). The term I1ε is the most complicated one of the three, and the other two terms can be treated easier. So we only consider the first term. For any p > 2, x, x ∈ [0, 1]d , 0 ≤ t ≤ T , γ0 ∈ (0, (1 − η)/2), by Burkholder’s and Hölder’s inequalities, we have   p EI1ε (t, x) − I1ε t, x   t  p2   0 2    ε   ≤ C(p)E Gt−s (x, ·) − Gt−s x , · α u (s, ·) − α u (s, ·) H ds 0

 t  p2    2 p   Gt−s (x, ·) − Gt−s x , · H ds ≤ C(p, K) × sup E |uε − u0 |T ,∞ ε∈(0,1]

0

≤ C(p, K)|x

− x |2pγ0 ,

(34)

where (2), (64) and Proposition 3.2 were used. Similarly, in view of (55) and (56), it follows ), that for 0 ≤ t ≤ t ≤ T , γ0 ∈ (0, 1−η 2  p  EI1ε (t, x) − I1ε t , x   t  p2    ε  0 2  

≤ C(p)E Gt−s (x, ·) − Gt −s (x, ·) α u (s, ·) − α u (s, ·) H ds 0

 + C(p)E

t t

    Gt−s (x, ·) α uε (s, ·) − α u0 (s, ·) 2 ds H

 p2

 t  p2  2  p   Gt−s (x, ·) − Gt −s (x, ·) H ds ≤ C(p, K) × sup E |uε − u0 |T ,∞ ε∈(0,1]

0

 t  p2  2  p   Gt−s (x, ·) H ds + C(p, K) × sup E |uε − u0 |T ,∞ t

pγ0

≤ C(p, K)|t − t |

ε∈(0,1]

,

where Proposition 3.2 were used. The proof is complete.

(35) 

4.2 Proof of Theorem 2.3 The following lemma is a consequence of Márquez-Carreras and Sarrà [26, Lemma 6.2.1].

Moderate Deviations for a Stochastic Heat Equation

Lemma 4.2 Let Z : ([0, T ] × [0, 1]d )2 → R, γ0 > 0 and CZ > 0 be such that for any (t, x) and (t , x ) ∈ [0, T ] × [0, 1]d ,     Z(t, x, ·, ·) − Z t , x , ·, · 2 ≤ CZ |t − t | + |x − x | 2γ0 . H

(36)

T

Let N : Ω × [0, T ] × [0, 1]d → R be an almost surely continuous, Ft -adapted process such that sup{|N (t, x)| : (t, x) ∈ [0, T ] × [0, 1]d } ≤ ρ, a.s., and for (t, x) ∈ [0, T ] × [0, 1]d , set 

T



F(t, x) =

Z(t, x, s, z)N (s, z)F (dsdz). [0,1]d

0

˜ , γ0 ) and C(γ , γ0 ) such Then for all 0 < γ < γ0 , there exist strictly positive constants L, C(γ 1 2 ˜ , γ0 ), that for all M ≥ ρCZ C(γ , γ0 )C(γ 

 M2 ≥ M) ≤ L exp − 2 . ρ CZ C 2 (γ , γ0 )

P(|F|γ ,γ

(37)

Proof of Theorem 2.3 By the statements after Theorem 2.2 in Sect. 2, Y / h(ε) obeys an LDP on C γ ,γ (DTd ; R), with the speed function h2 (ε) and the rate function I given by (16). Hence by [12, Theorem 4.2.13], to prove the LDP of Y ε / h(ε), it is enough to show that Y ε / h(ε) is h2 (ε)-exponentially equivalent to Y / h(ε), i.e., for any δ > 0,   ε |Y − Y |γ ,γ > δ = −∞. (38) lim sup h−2 (ε) log P h(ε) ε→0 Since there exists a constant C(T , d, γ ) satisfying that |Y ε − Y |γ ,γ ≤ C(T , d, γ )Y ε − Y T ,γ ,γ , to prove (38), it is enough to prove that   ε Y − Y T ,γ ,γ > δ = −∞, lim sup h−2 (ε) log P h(ε) ε→0

∀δ > 0.

(39)

Recall the decomposition in (28), Y ε (t, x) − Y (t, x) = I1ε (t, x) + I2ε (t, x) + I3ε (t, x). p , q = p−1 , γ0 ∈ (0, 12 − For any p > d+2 2 inequality, (14) and (65), we have

 ε   I (t, x) − I ε t, x  ≤ K 3



3



t

ds [0,1]d

0

 ×

x, x ∈ [0, 1]d , 0 ≤ t ≤ T , by Hölder’s

 q  dy Gt−s (x, y) − Gt−s x , y 



t

ds 0

d(q−1) ), 4

[0,1]d

dy|Y ε (s, y) − Y (s, y)|p

≤ C(p, γ0 , K, d, T )|x − x |

2γ0 q



t

×

 q1

 p1

p

|Y − Y |s,∞ ds

0

ε

 p1 .

(40)

Y. Li et al.

Similarly, in view of (57) and (58), it follows that for 0 ≤ t ≤ t ≤ T , p > (d + 2)/2, q =

p p−1

and γ0 ∈ (0, 12 − d(q−1) ), 4  ε   I (t, x) − I ε t , x  3 3 



t

≤K

ds [0,1]d

0





t

×

ds [0,1]d



p



t t

dy|Y (s, y) − Y (s, y)| ε

0

+K

dy|Gt−s (x, y) − Gt −s (x, y)|q

ds [0,1]d

dy|Gt−s (y, z)|q

≤ C(p, γ0 , K, d, T )|t − s|

2γ0 q



 q1

t

×

 q1

 p1

 ×



t

t

ds [0,1]d

p

|Y − Y |s,∞ ds ε

dy|Y ε (s, y) − Y (s, y)|p

 p1

 p1 (41)

.

0

Putting together (40) and (41), we obtain that for p > t ≤ t ≤ T , x, x ∈ [0, 1]d ,

d+2 , 2

q=

p , p−1

γ0 ∈ (0, 12 − d(q−1) ), 0 ≤ 4

 ε  2γ0   I (t, x) − I ε t , x  ≤ C(p, γ0 , K, d, T ) |t − t | + |x − x | q 3

3



t

×

p

|Y − Y |s,∞ ds ε

 p1 .

(42)

0

and γ0 ∈ (0, 12 − d(q−1) ) is arbitrary, we can choose p and γ0 satisfying that Since p > d+2 2 4 γ = 2γ0 /q for any γ ∈ (0, 1/4). Noticing that |Y ε − Y |s,∞ ≤ (1 + s)γ Y ε − Y s,γ ,γ , we obtain that  t  p1  p (1 + s)γ Y ε − Y s,γ ,γ ds . I3ε t,γ ,γ ≤ C 0

Thus, for t ∈ [0, T ], we have 

Y − Y t,γ ,γ ε

p

   t  ε p  ε p ε ≤ C(p, T , K) I1 t,γ ,γ + I2 t,γ ,γ + Y − Y s,γ ,γ ds . 0

Applying Gronwall’s lemma to h(t) = (Y − Y t,γ ,γ ) , we obtain that ε



Y ε − Y T ,γ ,γ

p

p

 p ≤C(p, T , K) I1ε T ,γ ,γ + I2ε T ,γ ,γ .

(43)

By (43), to prove (39), it is sufficient to prove that for any δ > 0   ε Ii T ,γ ,γ > δ = −∞, i = 1, 2. lim sup h−2 (ε) log P h(ε) ε→0 Step 1. For any ε > 0 and θ > 0,    P I1ε T ,γ ,γ > h(ε)δ ≤ P I1ε T ,γ ,γ > h(ε)δ, |uε − u0 |T ,∞ < θ + P |uε − u0 |T ,∞ ≥ θ . (44)

Moderate Deviations for a Stochastic Heat Equation

By Lemmas 5.1 and 5.2, Gt−s (x, y)1[s≤t] satisfies (36) for γ0 = (1 − η)/3. Applying Lemma 4.2 with Z(t, x, s, y) = Gt−s (x, y)1[s≤t] , γ0 = (1 − η)/3, CZ = C,    N (t, x) = α t, x, uε (t, x) − α t, x, u0 (t, x) 1[|uε −u0 |T ,∞ <θ] ,

M = h(ε)δ,

ρ = θ K,

˜ , γ0 ), for any γ ∈ (0, (1 − η)/4) and for small ε such that h(ε)δ ≥ ρCC(γ , γ0 )C(γ    h2 (ε)δ 2 P I1ε T ,γ ,γ > h(ε)δ, |uε − u0 |T ,∞ < θ ≤ L exp − 2 2 . θ K CC 2 (γ , γ0 )

(45)

Since uε satisfies the large deviation principle on C γ ,γ (DTd ; R) (see Theorem 2.1),   lim sup ε log P |uε − u0 |T ,∞ ≥ θ ≤ lim sup ε log P |uε − u0 |γ ,γ ≥ θ ε→0

ε→0



≤ − inf S(g) : |g − u0 |γ ,γ ≥ θ .

where S is given by (12). Since the good rate function S(g) has compact level sets, the inf{S(g) : |S(g) − u0 |γ ,γ ≥ θ } is obtained at some function g0 . Because S(g) = 0 if and only if g = u0 , we conclude that 

− inf S(g) : |g − u0 |γ ,γ ≥ θ < 0. By (8), we have

 lim sup h−2 (ε) log P |uε − u0 |T ,∞ ≥ θ = −∞.

(46)

ε→0

Since θ > 0 is arbitrary, putting together (44), (45) and (46), we obtain   ε I1 T ,γ ,γ −2 lim sup h (ε) log P > δ = −∞. h(ε) ε→0

(47)

Step 2. For the second term, 

ds 0

where



t

I2ε (t, x) =

[0,1]d

dyGt−s (x, y)Bε (s, y),

   √  Bε (s, y) := β uε (s, y) − β u0 (s, y) / ε − β u0 (s, y) Y ε (s, y),

by Lemmas 5.1 and 5.2, following the same method in the proof of (42), we obtain that for any γ ∈ (0, 1/4), I2ε T ,γ ,γ ≤ C(γ , T )|Bε |T ,∞ .

(48)

 2 √ |Bε |T ,∞ ≤ K |uε − u0 |T ,∞ / ε.

(49)

Similarly with (30), we have

By (2), (20) and (58) (taking p = 1), for any t ∈ [0, T ], we have   l    |uε − u0 |t,∞ ≤ sup  ds dyGl−s (x, y)K|uε (s, y) − u0 (s, y)| (l,x)∈Dtd

0

[0,1]d

Y. Li et al.

  l √ + sup  ε d (l,x)∈Dt

0



t

≤ C(K, T ) 0

where I˜2ε (t, x) =

√

|uε − u0 |s,∞ ds + |I˜2ε |t,∞ ,

 t ε 0

[0,1]d

   Gl−s (x, y)α uε (s, y) F (dsdy)

[0,1]d

 Gt−s (x, y)α uε (s, y) F (dsdy).

By the Gronwall’s inequality, we have |uε − u0 |T ,∞ ≤ C(K, T )|I˜2ε |T ,∞ ≤ C(K, T )|I˜2ε |γ ,γ .

(50)

Applying Lemma 4.2 with Z(t, x, s, y) = Gt−s (x, y)1[s≤t] , γ0 = (1 − η)/3, CZ = C, ρ = √  N (t, x) = εα uε (t, x) 1[|uε |T ,∞ <|u0 |T ,∞ +θ] ,

√

 εK 1 + |u0 |T ,∞ + θ ,

for any fixed θ > 0, we obtain that for any γ ∈ (0, (1 − η)/4), M ≥ 1 ˜ , γ0 ), θ )C 2 C(γ , γ0 )C(γ

√

εK(1 + |u0 |T ,∞ +

 P |I˜2ε |γ ,γ ≥ M, |uε |T ,∞ < |u0 |T ,∞ + θ   M2 ≤ L exp − 2 . εK CC 2 (γ , γ0 )(1 + |u0 |T ,∞ + θ )2

(51)

Similarly with (46), we have  lim sup h−2 (ε) log P |uε |T ,∞ ≥ |u0 |T ,∞ + θ ε→0

 ≤ lim sup h−2 (ε) log P |uε − u0 |T ,∞ ≥ θ ε→0

= −∞.

(52)

Putting (49)–(52) together, we have   ε I2 T ,γ ,γ −2 >δ lim sup h (ε) log P h(ε) ε→0 √    2 εh(ε)δ ≤ lim sup h−2 (ε) log P |I˜2ε |γ ,γ > C(γ , T , K) ε→0 √     2 εh(ε)δ , |uε |T ,∞ < |u0 |T ,∞ + θ ≤ lim sup h−2 (ε) log P |I˜2ε |γ ,γ > C(γ , T , K) ε→0   + P |uε |T ,∞ ≥ |u0 |T ,∞ + θ  ≤ lim sup √ ε→0

−δ 2 εh(ε)C(γ , T , K)K CC 2 (γ , γ0 )(1 + |u0 |T ,∞ + θ )2



Moderate Deviations for a Stochastic Heat Equation

   ∨ lim sup h−2 (ε) log P |uε |T ,∞ ≥ |u0 |T ,∞ + θ ε→0

= −∞. 

The proof is complete.

Acknowledgements The authors are grateful to the anonymous referees for conscientious comments and corrections. Y. Li and S. Zhang were supported by Natural Science Foundation of China (11471304, 11401556). R. Wang was supported by Natural Science Foundation of China (11301498, 11431014).

Appendix To make reading easier, we present here some results on the kernel G partially from Márquez-Carreras and Sarrà [26]. Recall that G denotes the fundamental solution of the heat equation ⎧ ∂ d ⎪ ⎨ ∂t Gt (x, y) = x Gt (x, y), t ≥ 0, x, y ∈ [0, 1] , d (53) Gt (x, y) = 0, x ∈ ∂([0, 1] ), ⎪ ⎩ G0 (x, y) = δ(x − y). The details about the construction of the fundamental solution G can be found in [18, Chap. 1]. This fundamental solution G is non-negative and can be decomposed into different terms as follows (see [15]):  Hi (t − s, x − y), (54) Gt−s (x, y) = H (t − s, x − y) + R(t − s, x, y) + i∈Id

where H is the heat kernel on Rd ,  H (t, x) =

1 4πt

 d2

  |x − y|2 , exp − 4t

R is a Lipschitz function, Id = {i = (i1 , . . . , id ) ∈ {−1, 0, 1}d − {(0, . . . , 0)}} and  Hi (t − s, x − y) = (−1)k H t − s, x − y i , with k =

d

j =1 |ij |

and ⎧ i ⎪ ⎨yj = yj , yji = −yj , ⎪ ⎩ i y j = 2 − yj ,

if ij = 0, if ij = 1, if ij = −1.

Lemma 5.1 Assume (Hη ) for some η ∈ (0, 1). There exists a positive constant C indepen), 1 ≤ p < 1 + d2 dent of t and x such that, for any 0 ≤ t < t ≤ T , x ∈ [0, 1]d , γ1 ∈ (0, 1−η 2 d(p−1) and γ2 ∈ [0, 12 − 4 ), 

t  0

 Gt−s (x, ·) − Gt −s (x, ·)2 ds ≤ C|t − t |2γ1 , H

(55)

Y. Li et al.

 

t

 Gt−s (x, ·)2 ds ≤ C|t − t |2γ1 , H

t



t

ds 

0



t

[0,1]d

ds

t

[0,1]d

(56)

dy|Gt−s (x, y) − Gt −s (x, y)|p ≤ C|t − t |2γ2 ,

(57)

dy|Gt−s (x, y)|p ≤ C|t − t |2γ2 .

(58)

Proof The proof of (55) and (56) can be found in Lemma 6.1.3 in [26]. The proofs of (57) and (58) are similar. For the convenience of reading, we shall give the proofs of (57) and (58). For any p ∈ [1, 1 + d2 ), (54) implies that    p |Gt−s (x, y) − Gt −s (x, y)|p ≤ C(d, p) H (t − s, x − y) − H t − s, x − y    p + R(t − s, x − y) − R t − s, x − y  +

    Hi (t − s, x − y) − Hi t − s, x − y p . (59) i∈Id

In order to check (57), we only need to bound the right-hand side of (59). Let L be the Lipschitz constant of the function R. Clearly 



t

ds [0,1]d

0

  p dy R(t − s, x, y) − R t − s, x, y  ≤ Lp T |t − t |p .

(60)

The terms Hi can be studied using the same arguments as H . Now it remains to analyze the term with H . First, for any p ∈ [1, 1 + d2 ), 



t

ds [0,1]d

0

  p dy H (t − s, x − y) − H t − s, x − y  ≤ C(A1 + A2 ),

with  pd2    2 2 p  1 exp − |x − y| − exp − |x − y|  , ds dy A1 =  4π(t − s) 4(t − s) 4(t − s)  0 [0,1]d    d2   d2 p  t  2   1 1  exp − p|x − y| . ds dy  − A2 = 4π(t − s) 4π(t − s)  4(t − s) 0 [0,1]d 

t





Since t < t , we have     |x − y|2 |x − y|2 exp − ≥ exp − . 4(t − s) 4(t − s) Using the inequality (a − b)p ≤ a p − bp

for all a > b > 0, p ≥ 1,

(61)

Moderate Deviations for a Stochastic Heat Equation

we obtain that 

 pd2      1 p|x − y|2 p|x − y|2 A1 ≤ ds dy exp − − exp − 4π(t − s) 4(t − s) 4(t − s) 0 Rd d    d(p−1)   t  2 1 t − s 2 d p− 2 1 − . = 4π(t − s) t −s 0 



t

Since for any γ2 ∈ [0, 12 − t − t < t −s



d(p−1) ) 4

t − t t −s

and s < t ,

2γ2

 ≤ 1,

t − s t −s

n ≤

t − s < 1, t −s

n ≥ 1,

we have 

t − s 1− t −s

 d2

      d  t −s 2 t − s t − s 2 t − s + ≤ 1− + ··· + × 1+ t −s t −s t −s t −s    

d t −t ≤ +1 2 t −s     d t − t 2γ2 , ≤ +1 2 t −s 



where  d2  is the largest integer number less or equal than d2 . Thus, for any γ2 ∈ [0, 12 − d(p−1) ), we have 4 A1 ≤ (4π)−

d(p−1) 2

d

p− 2

    t  − d(p−1) d −2γ2 2 + 1 · |t − t |2γ2 · ds t − s 2 0

≤ C(p, γ2 , d, T )|t − t |2γ2 .

(62)

Similar computations to the study of A2 , we have    d2   d2 p 2   1 1  exp − p|x − y|  ds dy  − A2 ≤ 4π(t − s) 4π(t − s)  4(t − s) 0 Rd d d d          t 2 2 p  1 4π(t − s) 2 1   ds  − · = 4π(t − s) 4π(t − s)  p 0   d   t  − d2 p−1 t −s 2 − d(p−1) − d2 − d2 2 = (4π) p ds|(t − s) − t − s | · 1− t −s 0 d     t  − d(p−1) d(p−1) t − s 2 d 2 ds t − s · 1− ≤ (4π)− 2 p − 2 t −s 0     t  − d(p−1) d(p−1) d d −2γ2 2 ≤ (4π)− 2 p − 2 ds t − s + 1 · |t − t |2γ2 · 2 0 

t



≤ C(p, d, γ2 , T )|t − t |2γ2 . Then (57) follows form (59)–(63).

(63)

Y. Li et al.

Next we shall prove (58). As before, we only need to check (58) replacing G by H . For any 0 ≤ t ≤ t ≤ T , 1 ≤ p < 1 + d2 and γ2 ∈ [0, 12 − d(p−1) ), we have 4  t  ds dyH (t − s, x − y)p t

[0,1]d

 pd2   1 p|x − y|2 ≤ ds dy exp − 4π(t − s) 4(t − s) t Rd  t (p−1)d (p−1)d d = (4π)− 2 p − 2 ds(t − s)− 2 



t



t

− (p−1)d 2

= (4π)

p

− d2



(p − 1)d 1− 2

−1



t − t

1− (p−1)d 2

≤ C(p, γ2 , d, T )|t − t |2γ2 . 

The proof is complete.

Lemma 5.2 Assume (Hη ) for some η ∈ (0, 1). There exists a positive constant C independent of t and x such that, for any 0 ≤ t ≤ T , x, x ∈ [0, 1]d , p ∈ [1, 1 + d2 ), γ1 ∈ (0, 1 − η) ), and γ2 ∈ [0, 12 − d(p−1) 4  t    Gt−s (x, ·) − Gt−s x , · 2 ds ≤ C|x − x |2γ1 , (64) H 0  t   p  ds dy Gt−s (x, y) − Gt−s x , y  ≤ C|x − x |2γ2 . (65) [0,1]d

0

Proof The proof of (64) can be found in Lemma 6.1.4 in [26]. Now we shall give the proof of (65). Using the similar arguments as in the proof of (57), we only need to check (65) replacing G by H . For any x, x ∈ [0, 1]d , p ∈ [1, 1 + d2 ) and γ2 ∈ [0, 12 − d(p−1) ), 4  t    p B := ds dy H (t − s, x − y) − H t − s, x − y  0

[0,1]d

      − pd  |x − y|2 2γ2 |x − y|2 dy 4π(t − s) 2 exp − − exp − 4(t − s) 4(t − s)  Rd 0      (p − 2γ2 )|x − y|2 (p − 2γ2 )|x − y|2 × exp − + exp − . 4(t − s) 4(t − s)

≤ C





t

ds

By the mean-value theorem, and the fact e−x < 0 for any x > 0, we have    t   − pd |x − x | 2γ2 B≤C ds dy 4π(t − s) 2 t −s Rd 0     2 (p − 2γ2 )|x − y| (p − 2γ2 )|x − y|2 × exp − + exp − 4(t − s) 4(t − s)  t d(p−1) d(p−1) d ds(t − s)− 2 −2γ2 ≤ 2C(4π)− 2 (p − 2γ2 )− 2 · |x − x |2γ2 · 0

2γ2

≤ C(p, γ2 , d, T )|x − x | The proof is complete.

. 

Moderate Deviations for a Stochastic Heat Equation

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