Acta Appl Math https://doi.org/10.1007/s10440-018-0174-1

Moderate Deviations for a Stochastic Wave Equation in Dimension Three Lingyan Cheng1 · Ruinan Li2 · Ran Wang3 · Nian Yao4

Received: 8 May 2016 / Accepted: 15 March 2018 © Springer Science+Business Media B.V., part of Springer Nature 2018

Abstract In this paper, we prove a central limit theorem and establish a moderate deviation principle for a perturbed stochastic wave equation defined on [0, T ] × R3 . This equation is driven by a Gaussian noise, white in time and correlated in space. The weak convergence approach plays an important role. Keywords Stochastic wave equation · Large deviations · Moderate deviations · Central limit theorem Mathematics Subject Classification 60H15 · 60F05 · 60F10

1 Introduction Since the pioneer work of Freidlin and Wentzell [16], the theory of small perturbation large deviations for stochastic dynamics has been extensively developed, see books [5, 11, 13]. The large deviation principle (LDP for short) for stochastic reaction-diffusion equations

B R. Li

[email protected] L. Cheng [email protected] R. Wang [email protected] N. Yao [email protected]

1

Center of Applied Mathematics, Tianjin University, Tianjin 300072, China

2

School of Statistics and Information, Shanghai University of International Business and Economics, Shanghai, 201620, China

3

School of Mathematics and Statistics, Wuhan University, Wuhan, 430072, China

4

College of Mathematics and Statistics, Shenzhen University, Shenzhen, 518060, China

L. Cheng et al.

driven by the space-time white noise was first obtained by Freidlin [15] and later by Sowers [26], Chenal and Millet [4], Cerrai and Röckner [3] and other authors. Also see [1, 25, 31] and references therein for further development. Like large deviations, the moderate deviation problems arise in the theory of statistical inference quite naturally. The moderate deviation principle (MDP for short) can provide us with the rate of convergence and a useful method for constructing asymptotic confidence intervals, see [14, 18] and references therein. Results on the 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 processes has been carried out as well, e.g., Gao [17] for martingales, Wu [30] for Markov processes, Guillin and Liptser [19] for diffusion processes. The problem of moderate deviations for stochastic partial differential equations (SPDEs for short) has been receiving much attention in very recently years, such as Wang and Zhang [28] for stochastic reaction-diffusion equations, Wang et al. [29] for stochastic Navier-Stokes equations, Budhiraja et al. [2] and Dong et al. [12] for stochastic systems with jumps. Those moderate deviation results are established for the stochastic parabolic equations. However, the hyperbolic case is much more complicated, one difficulty comes from the more complicated stochastic integral, another one comes from the lack of good regularity properties of the Green functions. See [7, 9] for the study of stochastic wave equations. Using the weak convergence approach in [1], Ortiz-López and Sanz-Solé [24] proved an LDP for a stochastic wave equation defined on [0, T ] × R3 , perturbed by a Gaussian noise which is white in time and correlated in space. In this paper, we shall study the central limit theorem and moderate deviation principle for the stochastic wave equation in dimension 3. The rest of this paper is organized as follows. In Sect. 2, we give the framework of the stochastic wave equation, and state the main results of this paper. In Sect. 3, we first prove some convergence results and then give the proof of the central limit theorem. In Sect. 4, we prove the moderate deviation principle by using the weak convergence method. Throughout the paper, C(p) is a positive constant depending on the parameter p, and C is a positive constant depending on no specific parameter (except T and the Lipschitz constants), whose value may be different from line to line by convention. We end this section with some notions. For any T > 0 and D ⊂ R3 , let C ([0, T ] × D) be the space of all continuous functions from [0, T ] × D to R, and let C α ([0, T ] × D) be the space of all Hölder continuous functions of degree α jointly in (t, x), with the Hölder norm gα :=

  |g(t, x) − g(s, y)| , ∀g ∈ C α [0, T ] × D α (t,x)=(s,y) (|t − s| + |x − y|) sup

and let 











C α,0 [0, T ] × D := g ∈ C α [0, T ] × D ; lim Og (δ) = 0 , δ→0

where Og (δ) := sup|t−s|+|x−y|<δ is denoted by Eα .

|g(t,x)−g(s,y)| (|t−s|+|x−y|)α

. Then C α,0 ([0, T ] × D) is a Polish space, which

Moderate Deviations for a Stochastic Wave Equation in Dimension Three

2 Framework and the Main Results 2.1 Framework Let us give the framework taken from Dalang and Sanz-Solé [9], Ortiz-López and Sanz-Solé [24]. Consider the following stochastic wave equation in spatial dimension d = 3: ⎧

    √  ∂2 ⎪ −  uε (t, x) = εσ uε (t, x) F˙ (t, x) + b uε (t, x) , ⎪ 2 ⎨ ∂t (1) uε (0, x) = ν0 (x), ⎪ ⎪ ⎩ ∂ uε (0, x) = ν˜ (x) 0

∂t

for all (t, x) ∈ [0, T ] × R3 (T > 0 is a fixed constant), where ε > 0, the coefficients σ, b : R → R are Lipschitz continuous functions, the term uε denotes the Laplacian of uε in the x-variable and the process F˙ is the formal derivative of a Gaussian random field, white in d+1 time and correlated in space. Precisely, for  any d ≥ 1, let D(R ) be the space of Schwartz d+1 test functions. F = F (ϕ), ϕ ∈ D(R ) is a Gaussian process defined on some probability space with zero mean and covariance functional     ˜ ds Γ (dx) ϕ(s) ∗ ψ(s) (x), (2) E F (ϕ)F (ψ) = R+

Rd

˜ where Γ is a non-negative and non-negative definite tempered measure on Rd , ψ(s)(x) := ψ(s)(−x) and the notation “∗” means the convolution operator. According to [8], the process F can be extended to a martingale measure

  M = Mt (A); t ≥ 0, A ∈ Bb Rd , where Bb (Rd ) denotes the collection of all bounded Borel measurable sets in Rd . Using the tempered measure Γ above, we can define an inner product on D(Rd ):     Γ (dx) ϕ ∗ ψ˜ (x), ∀ϕ, ψ ∈ D Rd . ϕ, ψ H := Rd

Let H be the Hilbert space obtained by the completion of D(Rd ) with the inner product ·, · H , and denote by  · H the induced norm. By Walsh’s theory of stochastic integration with respect to (w.r.t. for short) martingale measures, for any t ≥ 0 and h ∈ H, the stochastic integral t Bt (h) :=

h(y)M(ds, dy) 0

is well defined, and



Rd



t Btk := 0

Rd

ek (y)M(ds, dy); k ≥ 1

defines a sequence of independent standard Wiener processes, here {ek }k≥1 is a complete  orthonormal system of the Hilbert space H. Thus, Bt := k≥1 Btk ek is a cylindrical Wiener process on H. See [5] or [27].

L. Cheng et al.

Hypothesis H (H.1) The coefficients σ and b are real Lipschitz continuous, i.e., there exists some constant K > 0 such that     σ (x) − σ (y) ≤ K|x − y|, b(x) − b(y) ≤ K|x − y|, ∀x, y ∈ R. (3) (H.2) The spatial covariance measure Γ is absolutely continuous with respect to the Lebesgue measure, and the density is f (x) = ϕ(x)|x|−β , x ∈ R3 \{0}. Here the function ϕ is bounded and positive, ϕ ∈ C 1 (R3 ), ∇ϕ ∈ Cbδ (R3 ) with δ ∈ ]0, 1] and β ∈ ]0, 2[. (H.3) The initial values ν0 , ν˜ 0 are bounded, ν0 ∈ C 2 (R3 ), ∇ν0 is bounded, ν0 and ν˜ 0 are Hölder continuous with degrees γ1 , γ2 ∈ ]0, 1], respectively. We remark that the hypothesis (H.2) on Γ implies that, for any T > 0,   F G(t)(ξ )2 μ(dξ ) < ∞, sup t∈[0,T ] R3

where F denotes the Fourier transform operator and μ = F −1 Γ ([6]). According to Dalang and Sanz-Solé [9], under Hypothesis H, Eq. (1) admits a unique solution uε as follows:    √  t uε (t, x) = w(t, x) + ε G(t − s, x − ·)σ uε (s, ·) , ek (·) H dBsk k≥1



t

+

0

  G(t − s) ∗ b uε (s, ·) (x) ds,

(4)

0

where

 w(t, x) :=

 d G(t) ∗ ν0 (x) + G(t) ∗ ν˜ 0 (x), dt

and G(t) = 4π1 t σt , σt is the uniform surface measure (with total mass 4πt 2 ) on the sphere of radius t . Furthermore, for any p ∈ [2, ∞[, p   sup E uε (t, x) < +∞, (5) sup ε∈]0,1] (t,x)∈[0,T ]×R3

and for any

  2−β 1+δ ∧ α ∈ I := 0, γ1 ∧ γ2 ∧ , 2 2

(6)

there exists C > 0 such that for any (t, x), (s, y) ∈ [0, T ] × D, it holds that p   αp  E uε (t, x) − uε (s, y) ≤ C |t − s| + |x − y| . Consequently, almost all the sample paths of the process {uε (t, x); (t, x) ∈ [0, T ] × D} are Hölder continuous of degree α jointly in (t, x). See Dalang and Sanz-Solé [9] or Hu et al. [20] for details.

Moderate Deviations for a Stochastic Wave Equation in Dimension Three

Intuitively, as the parameter ε tends to zero, the solution uε of (4) will tend to the solution of the deterministic equation t   0 G(t − s) ∗ b u0 (s, ·) (x) ds. (7) u (t, x) = w(t, x) + 0

In this paper, we shall investigate deviations of uε from u0 , as ε decreases to 0. That is, the asymptotic behavior of the trajectories, Z ε (t, x) := √

 1  ε u − u0 (t, x), εh(ε)

(t, x) ∈ [0, T ] × D.

(8)

√ (LDP) The case h(ε) = 1/ ε provides some large deviation estimates. Ortiz-López and Sanz-Solé [24] proved that the law of the solution uε satisfies an LDP, see Theorem 2.1 below. (CLT) If h(ε) ≡ 1, we are in the √domain of the central limit theorem (CLT for short). We will show that (uε −u0 )/ ε converges as ε → 0+ to a random field, see Theorem 2.2 below. (MDP) To fill in the gap between the central limit theorem scale and the large deviations scale, we will study moderate deviations, that is when the deviation scale satisfies √ εh(ε) → 0, as ε → 0. (9) h(ε) → +∞ and In this case, we will prove that {Z ε ; ε ∈ ]0, 1]} satisfies an LDP, see Theorem 2.3 below. This special type of LDP is called the MDP for {uε ; ε ∈ ]0, 1]}, see [11, Sect. 3.7]. Throughout this paper, we assume (9) is in place.

2.2 Main Results Let HT := L2 ([0, T ]; H) and consider the usual L2 -norm  · HT on this space. For any h ∈ HT , we consider the deterministic evolution equation: t     h G(t − s, x − ·)σ V h (s, ·) , h(s, ·) H ds V (t, x) = w(t, x) + +

0

t

  G(t − s) ∗ b V h (s, ·) (x) ds.

(10)

0

By [24, Theorem 2.3], Eq. (10) admits a unique solution V h =: G1 (h) ∈ Eα , where G1 is the solution functional from HT to Eα . For any f ∈ Eα , define   1 2 I1 (f ) := hHT , inf (11) h∈HT ; G1 (h)=f 2 with the convention inf ∅ = +∞. Ortiz-López and Sanz-Solé [24] proved the following LDP result for uε . Theorem 2.1 (Ortiz-López and Sanz-Solé [24]) Under hypothesis H, the family {uε ; ε ∈ ]0, 1]} given by (4) satisfies a large deviation principle on Eα with the speed function ε −1 and with the good rate function I1 given by (11). More precisely,

L. Cheng et al.

(a) for any L > 0, the set {f ∈ Eα ; I1 (f ) ≤ L} is compact in Eα ; (b) for any closed subset F ⊂ Eα ,   lim sup ε log P uε ∈ F ≤ − inf I1 (f ); ε→0+

f ∈F

(c) for any open subset G ⊂ Eα ,   lim inf ε log P uε ∈ G ≥ − inf I1 (f ). ε→0+

f ∈G

In this paper, we further assume the condition (D): the function b is differentiable and its derivative b is also Lipschitz. More precisely, there exists a positive constant K  such that    b (y) − b (z) ≤ K  |y − z|, for all y, z ∈ R.

(12)

Since b is differentiable and Lipschitz continuous, we conclude that    b (z) ≤ K, for all z ∈ R.

(13)

Our first main result is the following central limit theorem. Theorem 2.2 √ Under conditions H and (D), for any α ∈ I and p ≥ 2, the random field (uε − u0 )/ ε converges in Lp to a random field Y on Eα as ε → 0, where Y is determined by ⎧

     ∂2 ⎪ −  Y (t, x) = σ u0 (t, x) F˙ (t, x) + b u0 (t, x) Y (t, x), ⎪ 2 ⎨ ∂t (14) Y (0, x) = 0, ⎪ ⎪ ⎩ ∂ Y (0, x) = 0, t ∈ [0, T ], x ∈ R3 . ∂t For any ε > 0, let qε := Y / h(ε). Then qε satisfies the following equation 

    1  0 ∂2 σ u (t, x) F˙ (t, x) + b u0 (t, x) qε (t, x), −  qε (t, x) = 2 ∂t h(ε)

(15)

with the same initial conditions as those of Y . Notice that Eq. (15) is a particular case of Eq. (1) if its coefficients σ and b were allowed to depend on (t, x). Now, assume that the coefficients σ and b in Eq. (1) depend on (t, x) and they are Lipschitz continuous in the third variable uniformly over (t, x) ∈ [0, T ] × R3 , that is, σ, b : [0, T ] × R3 × R → R satisfy that for all u, v ∈ R,     σ (t, x, u) − σ (t, x, v) + b(t, x, u) − b(t, x, v) ≤ K|u − v|. sup (t,x)∈[0,T ]×R3

By using the strategies in [9] and [24], we know that the wave equation under the above assumption admits a unique solution and the LDP result in Theorem 2.1 also holds. In fact, their proofs in this generalized case are the same as those in [9] and [24], only the notions need to be changed. For example, see [22, 31] for other types of SPDEs.

Moderate Deviations for a Stochastic Wave Equation in Dimension Three

Hence, {qε ; ε ∈ ]0, 1]} obeys an LDP on Eα with the speed h2 (ε) and with the good rate function   1 inf 2 h2HT ; Z h = g , if g ∈ Im(Z · ); (16) I (g) := +∞, otherwise, where Z h is the solution of the following deterministic evolution equation Z h (t, x) = 0

t

   G(t − s, x − ·)σ u0 (s, ·) , h(s, ·) H ds



+

t

    G(t − s) ∗ b u0 (s, ·) Z h (s, ·) (x) ds.

(17)

0

√ Our second main result is that {(uε − u0 )/[ εh(ε)]; ε ∈ ]0, 1]} satisfies the same LDP as qε , that is the following theorem. √ Theorem 2.3 Under conditions H and (D), the family {(uε − u0 )/[ εh(ε)]; ε ∈ ]0, 1]} satisfies a large deviation principle on Eα with the speed function h2 (ε) and with the good rate function I given by (16).

3 Central Limit Theorem 3.1 Convergence of Solutions For any function φ : [0, T ] × R3 → R, let 

  |φ|t,∞ := sup φ(s, x); (s, x) ∈ [0, t] × R3 . The following result is concerned with the convergence of uε as ε → 0. Proposition 3.1 Under Hypothesis H, for any p ≥ 2, there exists some positive constant C(p, K, T ) depending on p, K, T such that p   p E uε − u0 T ,∞ ≤ ε 2 C(p, K, T ) → 0,

as ε → 0.

(18)

Proof Since for any 0 ≤ t ≤ T ,

t

uε (t, x) − u0 (t, x) =

     G(t − s) ∗ b uε (s, ·) − b u0 (s, ·) (x) ds

0

√  + ε k≥1



   G(t − s, x − ·)σ uε (s, ·) , ek (·) H dBsk

t 0

=: T1ε (t, x) + T2ε (t, x), we obtain that for any p ≥ 2,   ε u − u 0 p

t,∞

 p   p ≤ 2p−1 T1ε t,∞ + T2ε t,∞ .

(19)

L. Cheng et al.

By Hölder’s inequality w.r.t the measure on [0, T ] × R3 given by G(t − s, dy) ds and the Lipschitz continuity of b, we obtain that  p  E T1ε t,∞ ≤ C(K)

t

× 0

p−1

 t R3

0

G(t − s, dy) ds

 p   E uε − u0 s,∞

t

≤ C(p, K, T ) 0

R3

 G(t − s, dy) ds

p   E uε − u0 s,∞ ds.

(20)

By Burkholder’s inequality, Hölder’s inequality, the Lipschitz continuity of σ and (5), we have p   t       E  G(t − s, x − ·)σ uε (s, ·) , ek (·) H dBsk  k≥1

0

p   t 2    2 ≤ C(p)E  G(t − s, x − ·)σ uε (s, ·) H ds  0

 t  p2 −1  2   ≤ C(p, K) F G(t − s)(ξ ) μ(dξ ) ds ×

t

0

R3

0

1+

p   E uε (r, z)

sup (r,z)∈[0,s]×R3

≤ C(p, K, T )

t

1+

sup

R3

  F G(t − s)(ξ )2 μ(dξ ) ds

p   E uε (r, z) ds < ∞.

(r,z)∈[0,s]×R3

0

The above estimate yields that  p  p E T2ε T ,∞ ≤ ε 2 C(p, K, T ).

(21)

Putting (19)–(21) together, and using the Gronwall’s inequality, we obtain the desired inequality (18). The proof is complete. 

3.2 The Proof of Theorem 2.2 The following lemma is a consequence of the Garsia-Rodemich-Rumsey’s theorem, see Millet and Sanz-Solé [23, Lemma A2]. Lemma 3.2 ([23, Lemma A2]) Let {V ε (t, x); (t, x) ∈ [0, T ] × D, ε ∈ ]0, 1]} be a family of real-valued stochastic processes. Assume that there exists p ∈ ]1, ∞[ such that (A1) for any (t, x) ∈ [0, T ] × D, p   lim E V ε (t, x) = 0;

ε→0

Moderate Deviations for a Stochastic Wave Equation in Dimension Three

(A2) there exists γ0 > 0 such that for any (t, x), (s, y) ∈ [0, T ] × D, p   E V ε (t, x) − V ε (s, y) ≤ C(|t − s| + |x − y|)γ0 +4 , where C is a positive constant independent of ε. Then for any α ∈ ]0, γ0 /p[, r ∈ [1, p[, we have  r  lim E V ε  = 0. ε→0

α

√ Proof of Theorem 2.2 Denote Y ε := (uε − u0 )/ ε. We will prove that for any α ∈ I , p ≥ 2, p   lim E Y ε − Y α = 0. (22) ε→0

To this end, we only need to verify (A1) and (A2) in Lemma 3.2 for V ε := Y ε − Y . Notice that  t       Y ε (t, x) − Y (t, x) = G(t − s, x − ·) σ uε (s, ·) − σ u0 (s, ·) , ek (·) H dBsk 0

k≥1



t

+

G(t − s) ∗ 0

=: where I1ε (t, x) :=



t

:=

I3ε (t, x) :=

I1ε (t, x) + I2ε (t, x) + I3ε (t, x),

(23)

 t       G(t − s, x − ·) σ uε (s, ·) − σ u0 (s, ·) , ek (·) H dBsk , k≥1

I2ε (t, x)

     ε    b u (s, ·) − b u0 (s, ·) − b u0 (s, ·) Y (s, ·) (x) ds √ ε

0 t

0

     ε    b u (s, ·) − b u0 (s, ·) − b u0 (s, ·) Y ε (s, ·) (x) ds, G(t − s) ∗ √ ε     G(t − s) ∗ b u0 (s, ·) Y ε (s, ·) − Y (s, ·) (x) ds.

0

Next, we shall verify (A1) and (A2) for Iiε , i = 1, 2, 3. Step 1. Following the similar calculation to the proof of (21) and using the Lipschitz continuity of σ , we can deduce that for any p ≥ 2, p    p  p (24) E I1ε T ,∞ ≤ C(p, K, T )E uε − u0 T ,∞ ≤ ε 2 C(p, K, T ), where we have used Proposition 3.1 in the last inequality. √ Notice that uε = u0 + εY ε . By the mean theorem for derivatives, there exists a random ε field v (t, x) taking values in (0, 1) such that     √ 1    √ b uε − b u0 =b u0 + εv ε Y ε Y ε . ε By the Lipschitz continuity of b , we have  2          √ √ 1    √ b uε − b u0 − b u0 Y ε = b u0 + εv ε Y ε − b u0 Y ε ≤ εK  Y ε  . ε

(25)

L. Cheng et al.

Hence  ε  √ I (t, x) ≤ εK 



t

2

 2 G(t − s) ∗ Y ε (s, ·) (x) ds.

0

By Hölder’s inequality and Proposition 3.1, we obtain that for any p ≥ 2,  p  p E I2ε t,∞ ≤ ε 2 K  p

p−1

 t R3

0

G(t − s, dy) ds



t

×

  ε 2p    E Y

0

  ≤ ε C p, K, K  , T .

s,∞

R3

 G(t − s, dy) ds

p 2

(26)

By Hölder’s inequality and (13), we deduce that for any p ≥ 2,  p  E I3ε t,∞ ≤ K p

p−1

 t R3

0

G(t − s, dy) ds

≤ C(p, K, T ) 0



t

× 0

t

p   E Y ε − Y s,∞

 R3

 G(t − s, dy) ds

p   E Y ε − Y s,∞ ds.

(27)

Putting (23), (24), (26) and (27) together, we have   t p    p    E Y ε − Y t,∞ ≤ C p, K, K  , T ε 2 + E |Y ε − Y |ps,∞ ds . 0

By Gronwall’s inequality, we have    p p  E |Y ε − Y |T ,∞ ≤ ε 2 C p, K, K  , T −→ 0,

as ε → 0,

(28)

which, in particular, implies (A1) in Lemma 3.2. Step 2. Notice that Y ε satisfies that      √ G(t − s, x − ·)σ u0 (s, ·) + εY ε (s, ·) , ek (·) H dBsk Y ε (t, x) = k≥1



t

+ 0

    √ b u0 (s, ·) + εY ε (s, ·) − b u0 (s, ·) (x) ds. G(t − s) ∗ √ ε

(29)

For any ε ∈ ]0, 1], set the mapping σ˜ ε , b˜ε : [0, T ] × R3 × R → R by  √  σ˜ ε (t, x, r) :=σ u0 (t, x) + εr ,   √  1   b˜ε (t, x, r) := √ b u0 (t, x) + εr − b u0 (t, x) . ε By the Lipschitz continuity of σ and b, we know that σ˜ ε and b˜ε are Lipschitz continuous in the third variable uniformly over (t, x) ∈ [0, T ] × R3 and ε ∈ ]0, 1]. Using the same strategy as that in the proof of [24, Theorem 2.3], one can obtain that for any α ∈ I , p > 4/α, p    αp (30) sup E Y ε (t, x) − Y ε (s, y) ≤ C |t − s| + |x − y| ε∈]0,1]

Moderate Deviations for a Stochastic Wave Equation in Dimension Three

and

p    αp E Y (t, x) − Y (s, y) ≤ C |t − s| + |x − y| .

(31)

Putting (30) and (31) together, we obtain the Hölder continuity of V , that is for any α ∈ I , p > 4/α, there exists a constant C > 0 such that    p   αp sup E  Y ε (t, x) − Y (t, x) − Y ε (s, y) − Y (s, y)  ≤ C |t − s| + |x − y| . (32) ε

ε∈]0,1]

By Lemma 3.2, (28) and (32), we obtain that for any α˜ ∈ ]0, α − 4/p[, r ∈ [1, p[, r   lim E Y ε − Y α˜ = 0. ε→0

By the arbitrariness of p > 4/α, r ∈ [1, p[, we get the desired result in Theorem 2.2 for any α ∈ I , p ≥ 2. The proof is complete. 

4 Moderate Deviation Principle Let P denote the set of all predictable processes belonging to L2 (Ω × [0, T ]; H). For any N > 0, T > 0, we define HTN := {h ∈ HT ; hHT ≤ N },



PTN := v ∈ P ; v ∈ HTN , a.s. ,

and we endow HTN with the √ weak topology of HT . Let Z ε := (uε − u0 )/( εh(ε)). Then    √ 1  t G(t − s, x − ·)σ u0 (s, ·) + εh(ε)Z ε (s, ·) , ek (·) H dBsk Z ε (t, x) = h(ε) k≥1 0    √   0 t b u (s, ·) + εh(ε)Z ε (s, ·) − b u0 (s, ·) + G(t − s) ∗ (x) ds. √ εh(ε) 0

(33)

For any ε ∈ ]0, 1] and v ∈ PTN , consider the controlled equation Z ε,v defined by Z ε,v (t, x) =

1  h(ε) k≥1

+

t

0

+

0

t



   √ G(t − s, x − ·)σ u0 (s, ·) + εh(ε)Z ε,v (s, ·) , ek (·) H dBsk

t 0

   √ G(t − s, x − ·)σ u0 (s, ·) + εh(ε)Z ε,v (s, ·) , v(s, ·) H ds

   √   0 b u (s, ·) + εh(ε)Z ε,v (s, ·) − b u0 (s, ·) G(t − s) ∗ (x) ds. (34) √ εh(ε)

Following the proof of [24, Theorem 2.3], similarly to (29), one can prove that Eq. (34) admits a unique solution {Z ε,v (t, x); (t, x) ∈ [0, T ] × R3 } satisfying that for any p ∈ [2, ∞[, p   sup E Z ε,v (t, x) < ∞, (35) sup sup ε∈]0,1] v∈P N (t,x)∈[0,T ]×R3 T

L. Cheng et al.

and there exists C > 0 such that for (t, x), (s, y) ∈ [0, T ] × D and α ∈ I , p    αp sup sup E Z ε,v (t, x) − Z ε,v (s, y) ≤ C |t − s| + |x − y| .

(36)

ε∈]0,1] v∈P N T

Particularly, taking v ≡ 0, we know that for any p ∈ [2, ∞[, p   sup sup E Z ε (t, x) < ∞,

(37)

ε∈]0,1] (t,x)∈[0,T ]×R3

and

p    αp sup E Z ε (t, x) − Z ε (s, y) ≤ C |t − s| + |x − y| .

(38)

ε∈]0,1]

Recall Z h defined in Eq. (17). Consider the following conditions: (a) For any family {v ε ; ε > 0} ⊂ PTN which converges in distribution as ε → 0 to v ∈ PTN , as HTN -valued random variables, ε

lim Z ε,v = Z v ,

ε→0

in distribution,

as Eα -valued random variables, where Z v denotes the solution of Eq. (17) corresponding to the HTN -valued random variable v (instead of a deterministic function h); (b) The set {Z h ; h ∈ HTN } is compact in Eα . Proof of Theorem [1, Theorem 6] to the solution functional G ε : C ([0, T ]; √ 2.3 Applying ∞ ε ε R ) → Eα , G ( εB) := Z , the solution of Eq. (33), and the solution functional G 0 : HT → E α , G 0 (h) := Z h , the solution of Eq. (17), conditions (a) and (b) above imply the MDP result in Theorem 2.3. The verification of condition (a) will be given in Proposition 4.1. Since HTN is compact in the weak topology of H, condition (b) follows from the continuity of the mapping HTN  h → Z h ∈ Eα , which will be proved in Proposition 4.2. The proof is complete.  Proposition 4.1 Under conditions H and (D), for any family {v ε ; ε > 0} ⊂ PTN which converges in distribution as ε → 0 to v ∈ PTN , as HTN -valued random variables, it holds that ε

lim Z ε,v = Z v ,

ε→0

in distribution,

as Eα -valued random variables. ¯ F¯ , Proof By the Skorokhod representation theorem, there exist a probability space (Ω, ¯ and, on this basis, a sequence of independent Brownian motions B¯ = (B¯ k )k≥1 and (F¯t ), P), ¯ such also a family of F¯t -predictable processes {v¯ ε ; ε > 0}, v¯ taking values on HTN , P-a.s., ¯ under P¯ and ¯ B) that the joint law of (v ε , v, B) under P coincides with that of (v¯ ε , v,   lim v¯ ε − v, ¯ g H = 0,

ε→0

T

¯ ∀g ∈ HT , P-a.s.

(39)

ε ¯ and let Let Z¯ ε,v¯ be the solution to a similar equation as (34) replacing v by v¯ ε and B by B, v ¯ Z¯ be the solution of Eq. (17) corresponding to the HTN -valued random variable v¯ (instead of a deterministic function h).

Moderate Deviations for a Stochastic Wave Equation in Dimension Three

Now, we shall prove that for any p ≥ 2, α ∈ I , p   ε lim E¯ Z¯ ε,v¯ − Z¯ v¯ α = 0,

(40)

ε→0

which implies the validity of Proposition 4.1. Here the expectation in (40) refers to the probability P¯ . From now on, we drop the bars in the notation for the sake of simplicity, and we denote X ε,v

ε ,v

ε

:= Z ε,v − Z v .

By (35) and (37), we know that for any p ≥ 2, sup

sup

sup

ε∈]0,1] v,v ε ∈HN (t,x)∈[0,T ]×R3 T

p   ε E X ε,v ,v (t, x) < ∞.

According to Lemma 3.2, to prove (40), it is sufficient to prove that for any (t, x), (s, y) ∈ [0, T ] × D, p ≥ 2 and α ∈ I , the following conditions hold: (1) Pointwise convergence:

p   ε lim E X ε,v ,v (t, x) = 0.

(41)

ε→0

(2) Estimation of the increments: there exists a positive constant C satisfying that p    αp ε ε sup E X ε,v ,v (t, x) − X ε,v ,v (s, y) ≤ C |t − s| + |x − y| .

(42)

ε∈]0,1]

By (36) and (38), it is easy to obtain (42). Now, it remains to prove (41). Notice that for any (t, x) ∈ [0, T ] × R3 , X

ε,v ε ,v

1  (t, x) = h(ε) k≥1  +

t

0

−

t

0



t 0

   √ ε G(t − s, x − ·)σ u0 (s, ·) + εh(ε)Z ε,v (s, ·) , ek (·) H dBsk

   √ ε G(t − s, x − ·)σ u0 (s, ·) + εh(ε)Z ε,v (s, ·) , v ε (s, ·) H ds

   G(t − s, x − ·)σ u0 (s, ·) , v(s, ·) H ds



   √   0 ε b u (s, ·) + εh(ε)Z ε,v (s, ·) − b u0 (s, ·) + (x) ds G(t − s) ∗ √ εh(ε) 0  t     G(t − s) ∗ b u0 (s, ·) Z v (s, ·) (x) ds − 

t

0

=: Aε1 (t, x) + Aε2 (t, x) + Aε3 (t, x).

(43)

Step 1. For the first term Aε1 (t, x), noticing that u0 is bounded, by Burkholder’s inequality, Hölder’s inequality and the linear growth property of σ , we have

L. Cheng et al.

sup (t,x)∈[0,T ]×R3

p   E Aε1 (t, x)

1 sup E ≤ p h (ε) (t,x)∈[0,T ]×R3 C(p, u0 , K) ≤ hp (ε) T

× 1+

R3

0

  √ G(t − s, x − ·)σ u0 (s, ·) + εh(ε)Z ε,vε (s, ·) 2 ds H

0



T

t

  F G(T − s)(ξ )2 μ(dξ ) ds

sup

 p2

 p2 −1

 p    ε,vε F G(T − s)(ξ )2 μ(dξ ) ds   E Z (r, z) R3

(r,z)∈[0,s]×R3

0

≤





C(p, u0 , K, T ) , hp (ε)

(44)

where (35) is used in the last inequality. Step 2. The second term is further divided into two terms: t     √ ε Aε2 (t, x) = G(t − s, x − ·) σ u0 (s, ·) + εh(ε)Z ε,v (s, ·) 0

   − σ u0 (s, ·) , v ε (s, ·) H ds t     + G(t − s, x − ·)σ u0 (s, ·) , v ε (s, ·) − v(s, ·) H ds 0

=:

Aε2,1 (t, x) + Aε2,2 (t, x).

(45)

By Cauchy-Schwarz inequality, Hölder’s inequality, the Lipschitz continuity of σ and the fact that v ε ∈ PTN , we obtain that p   sup E Aε2,1 (t, x) (t,x)∈[0,T ]×R3



p

≤ ε 2 hp (ε)K p  × 0

sup (t,x)∈[0,T ]×R3

t

 v ε (s, ·)2 ds H 

p

T

×

T

R3

0



 p2

  F G(T − s)(ξ )2 μ(dξ ) ds

 p2 −1

  p    ε,vε F G(T − s)(ξ )2 μ(dξ ) ds   E Z (r, z)

sup 0

0

 G(t − s, x − ·)Z ε,vε (s, ·)2 ds H

 p2 



≤ ε 2 hp (ε)N p K p

t

E

R3

(r,z)∈[0,s]×R3

p

≤ ε 2 hp (ε)C(N, p, K, T ),

(46)

where (35) is also used in the last inequality. Next, we will show that lim

ε→0

sup (t,x)∈[0,T ]×R3

p   E Aε2,2 (t, x) = 0.

(47)

Moderate Deviations for a Stochastic Wave Equation in Dimension Three

By Hölder’s inequality with respect to the measure on R3 given by |F G(t − s)(ξ )|2 μ(dξ ), we obtain that for any (t, x) ∈ [0, T ] × R3 ,

t 0

  G(t − s, x − ·)σ u0 (s, ·) 2 ds H t

≤ C(K)

R3

0

  F G(t − s)(ξ )2 μ(dξ ) ds × 1 +

sup (s,y)∈[0,T ]×R3

 0  u (s, y)2 < +∞.

  This implies that for any (t, x) ∈ [0, T ] × R3 , the function G(t − s, x − y)σ u0 (s, y) ;  (s, y) ∈ [0, T ] × R3 takes its values in HT . Since v ε → v weakly in HTN , we know that lim Aε2,2 (t, x) = 0,

ε→0

a.s.

(48)

By Cauchy-Schwarz inequality on the Hilbert space HT and the facts that v ε HT ≤ N , vHT ≤ N , we obtain that for any t ∈ [0, T ],  ε  A (t, x) ≤ 2,2



t 0

  G(t − s, x − ·)σ u0 (s, ·) 2 ds H

 12

 × 0

t

 v ε (s, ·) − v(s, ·)2 ds H

 12

 t  12    G(t − s, x − ·)σ u0 (s, ·) 2 ds ≤ C(N, T ) H 0

≤ C(N, K, T ) < +∞,

(49)

here C(N, K, T ) is independent of (ε, t, x). By the Hölder regularity of the path-wise integral t     G(t − s, x − ·)σ u0 (s, ·) , v ε (s, ·) − v(s, ·) H ds, 0



(see [24, Sect. 2]), we know that a.s., Aε2,2 (t, x), (t, x) ∈ [0, T ] × D has Hölder continuous sample paths of degree α ∈ I jointly in (t, x), and   sup Aε2,2 α < ∞, a.s. ε∈]0,1]

 This, in particular, implies that Aε2,2 (t, x); (t, x) ∈ [0, T ]×D is equicontinuous. By the ar  bitrariness of D ⊂ R3 , we known that Aε2,2 (t, x); (t, x) ∈ [0, T ] × R3 is equiv-continuous. Thus, by (48), (49) and Arzelà-Ascoli Theorem, we know that Aε2,2 converges to 0 in the space C ([0, T ] × R3 ; R), a.s. as ε → 0. This implies that  ε  A (t, x) = 0, a.s. sup (50) lim 2,2 ε→0

(t,x)∈[0,T ]×R3

By the dominated convergence theorem, (49) and (50) imply that   ε  A (t, x)p = 0, sup lim E ε→0

which is stronger than (47).

(t,x)∈[0,T ]×R3

2,2

L. Cheng et al.

Step 3. For the third term Aε3 , using the same argument as that in the proof of (25), we have    √    0 ε  ε   t b u (s, ·) + εh(ε)Z ε,v (s, ·) − b u0 (s, ·) A (t, x) ≤  G(t − s) ∗ √ 3  εh(ε) 0      ε − b u0 (s, ·) Z ε,v (s, ·) (x) ds    t     0  ε,vε  v  +  G(t − s) ∗ b u (s, ·) Z (s, ·) − Z (s, ·) (x) ds  0



≤ C K

√



t

εh(ε)

 2 ε G(t − s) ∗ Z ε,v (s, ·) (x) ds

0



t

+ C(K)

  ε G(t − s) ∗ Z ε,v (s, ·) − Z v (s, ·)(x) ds.

0

By Hölder’s inequality with respect to the Lebesgue measure on [0, t] × R3 and (35), we have p   E Aε3 (t, x) sup (t,x)∈[0,T ]×R3

  p ≤ ε 2 hp (ε)C p, K  t ×

R3

0

p−1

 t sup (t,x)∈[0,T ]×R3

G(t − s, x − y)

0

R3

G(t − s, x − y) dy ds

2p   ε E Z ε,v (r, z) dy ds

sup (r,z)∈[0,s]×R3

p−1

 t + C(p, K) t ×

R3

0

sup (t,x)∈[0,T ]×R3

G(t − s, x − y)

0

R3

G(t − s, x − y) dy ds p   ε E Z ε,v (r, z) − Z v (r, z) dy ds

sup (r,z)∈[0,s]×R3

  p ≤ ε 2 hp (ε)C p, K  , T + C(p, K, T )



p   ε E X ε,v ,v (r, z) ds.

t

sup 0 (r,z)∈[0,s]×R3

Putting (43), (44), (46), (50) and (51) together, we have p   ε E X ε,v ,v (s, x) sup (s,x)∈[0,t]×R3

   p ≤ C N, p, u0 , K, K  , T h−p (ε) + ε 2 hp (ε) +

t

+

sup

 p   ε E X ε,v ,v (r, z) ds .

0 (r,z)∈[0,s]×R3

By Gronwall’s inequality and (47), we obtain that p   ε E X ε,v ,v (s, x) sup (s,x)∈[0,T ]×R3

sup (s,x)∈[0,t]×R3

p   E Aε2,2 (s, x)

(51)

Moderate Deviations for a Stochastic Wave Equation in Dimension Three

 

p ≤ C N, p, u0 , K, K  , T h−p (ε) + ε 2 hp (ε) + −→ 0,

sup (s,x)∈[0,T ]×R3

p   E Aε2,2 (s, x)

as ε → 0. 

The proof is complete.

Proposition 4.2 Under conditions H and (D), for any α ∈ I , the mapping HTN  h → Z h ∈ Eα is continuous with respect to the weak topology. Proof Let {h, (hn )n≥1 } ⊂ HTN such that for any g ∈ HT ,   lim hn − h, g H = 0. T

n→∞

We need to prove that

  lim Z hn − Z h α = 0.

n→∞

(52)

Applying the deterministic version of Lemma 3.2 to Z hn and Z h , the proof of (52) can be divided into two steps: (1) Pointwise convergence: for any (t, x) ∈ [0, T ] × D,   lim Z hn (t, x) − Z h (t, x) = 0.

n→∞

(53)

(2) Estimation of the increments: for any (t, x), (s, y) ∈ [0, T ] × D, α ∈ I ,      α sup Z hn (t, x) − Z h (t, x) − Z hn (s, y) − Z h (s, y)  ≤ C |t − s| + |x − y| . (54) n≥1

By using the similar (but more easier) strategy to that in the proof of [24, Theorem 2.3], one can prove that the solution Z h of (17) satisfies that for any α ∈ I , there exists C > 0 such that for any (t, x), (s, y) ∈ [0, T ] × D,    α sup Z h (t, x) − Z h (s, y) ≤ C |t − s| + |x − y| .

(55)

h∈HN T

Thus, (54) holds. Next, it remains to prove (53). Notice that for any (t, x) ∈ [0, T ] × R3 ,

t

Z hn (t, x) − Z h (t, x) = 0

   G(t − s, x − ·)σ u0 (s, ·) , hn (s, ·) − h(s, ·) H ds



t

+

   G(t − s) ∗ b u0 (s, ·) Z hn (s, ·) − Z h (s, ·) (x) ds

0

=:

I1n (t, x) + I2n (t, x).

(56)

Using the similar arguments to that in the proof (50), we can obtain that lim

n→∞

sup (t,x)∈[0,T ]×R3

 n  I (t, x) = 0. 1

(57)

L. Cheng et al.

  Set ζ n (t) := sup(s,x)∈[0,t]×R3 Z hn (s, x) − Z h (s, x). By (13), we have   I2 (t, x) ≤

t 0

    G(t − s, x − y)b u0 (s, y) Z hn (s, y) − Z h (s, y)  dy ds

R3

t

≤K 0

R3

G(t − s, x − y)

 h  Z n (u, z) − Z h (u, z) dy ds

sup (u,z)∈[0,s]×R3



t

≤ C(K, T )

ζ n (s) ds.

(58)

0

By (56) and (58), we have

t

ζ n (t) ≤ C(K, T )

ζ n (s) ds + 0

sup (t,x)∈[0,T ]×R3

Hence, by Gronwall’s lemma and (57), we obtain that  n  I (t, x) −→ 0, sup ζ n (T ) ≤ eC(K,T )T 1 (t,x)∈[0,T ]×R3

The proof is complete.

 n  I (t, x). 1

as n → ∞. 

Acknowledgements The authors are grateful to the referee for his/her very numerous and conscientious comments and corrections. R. Wang was supported by Natural Science Foundation of China (11431014, 11671076). N. Yao was supported by Natural Science Foundation of China (11371283).

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