Large deviations for stochastic nonlinear beam equations

Journal of Functional Analysis 248 (2007) 175–201 www.elsevier.com/locate/jfa

Large deviations for stochastic nonlinear beam equations Tusheng Zhang School of Mathematics, University of Manchester, Oxford Road, Manchester M13 9PL, England, UK Received 20 August 2006; accepted 30 March 2007

Communicated by Paul Malliavin

Abstract We establish a large deviation principle for the solutions of stochastic partial differential equations for nonlinear vibration of elastic panels (also called stochastic nonlinear beam equations). © 2007 Elsevier Inc. All rights reserved. Keywords: Stochastic partial differential equations; Stochastic beam equations; Large deviations; Exponential martingales; Exponential integrability

1. Introduction Consider a bounded open interval on the real line, say, (0, 1). Let L2 = L2 (0, 1). Denote by H01 = H01 (0, 1) and H02 = H02 (0, 1) the Sobolev spaces of order one and two satisfying the homogeneous boundary conditions. Denote by H0−k the dual space of H0k . (·,·) will denote the L2 -inner product and ·,· denotes the dual pairing. The norms on L2 , H0k and H0−k will be denoted respectively by  · ,  · k and  · −k . Consider the linear operator Au = α∂x2 u − γ ∂x4 u,

E-mail address: [email protected] (T. Zhang). 0022-1236/$ – see front matter © 2007 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2007.03.029

176

T. Zhang / Journal of Functional Analysis 248 (2007) 175–201

and the nonlinear operator  1 B(u) = β

 |∂x u| dx ∂x2 u. 2

0

The mathematical model for the nonlinear penal vibration is governed by the following partial differential equation:  ∂t2 ut



1

= α+β

|∂y ut | dy ∂x2 ut − γ ∂x4 ut + F (u˙ t , ut ), 2

0

ut (0) = ut (1) = 0,

∂x ut (0) = ∂x ut (1) = 0,

u0 (x) = φ0 (x),

∂t u0 (x) = φ1 (x),

(1)

where u˙ t denotes the derivative of u with respect to the variable t. A detailed study of the model can be found in the book by Dowell [14]. The equation was also proposed by Woinowsky-Krieger in [22] as a model for the transversal deflection of an extensible beam of natural length 1. An equation in two space variables similar to (1) was suggested in [10] as a model of nonlinear oscillations of a plate in a supersonic flow of gas. It has also been studied by many other people, see [2,4,15,16] and references therein. Let Wt , t  0, be a Wiener process taking values in a Hilbert space. Without loss of generality, we may assume that Wt is l 2 -valued Wiener process which admits the following representation: Wt =

∞ 

λk βtk ek ,

k=1

 2 k where λk , k  1, is a sequence of non-negative numbers such that ∞ k=1 λk < ∞, βt , k  1, is a sequence of independent standard Brownian motions and {ek , k  1} is the canonical orthonormal basis of l 2 . Taking into account the random fluctuations, Chow and Menaldi [9] considered the stochastic nonlinear partial differential equation for vibration of elastic panels: 

1

∂t2 uεt = α + β

  ε 2   ∂y u  dy ∂ 2 uε − γ ∂ 4 uε + εσ uε W˙ t + F u˙ ε , uε , t x t x t t t t

0

uεt (0) = uεt (1) = 0, uε0 (x) = φ0 (x) ∈ H02 ,

∂x uεt (0) = ∂x uεt (1) = 0, ∂t uε0 (x) = φ1 (x) ∈ L2 ,

(2)

where for every u ∈ H02 , σ (u) stands for a map from l 2 into H02 which will be specified later, and F (·,·) denotes a map from L2 × H02 into L2 . It is proved in [9] that under reasonable conditions on σ , (2) has a unique solution with the property:   uε ∈ C [0, T ]; H02 and u˙ ε ∈ C [0, T ]; L2 .

T. Zhang / Journal of Functional Analysis 248 (2007) 175–201

177

A general formulation of the equation in an abstract Hilbert space was later studied by Brze´zniak, Maslowski and Seidler [4], where existence, uniqueness and asymptotic stability of the solution were discussed. The aim of this paper is to establish a large deviation principle (LDP) for the vector process vtε = (uεt , u˙ εt ) on the product space C([0, T ]; H02 ) × C([0, T ]; L2 ) as ε → 0. The large deviation problem for stochastic partial differential equations (SPDEs) has been studied by many people, but mainly for stochastic parabolic equations. For example, an LDP for stochastic reaction equations with nonlinear reaction term was established by Cerrai and Röckner [6]. An LDP for stochastic Burgers’-type SPDEs was considered by Cardon-Weber [5]. A uniform LDP for parabolic SPDEs was proved by Chenal and Millet [7]. In [19], Rovira and Sanz-Sole proved an LDP for a class of nonlinear hyperbolic SPDEs. An LDP was obtained by Chow [8] for some parabolic SPDEs. An LDP for stochastic reaction equations was established by Sowers [20]. A small time large deviation principle for stochastic parabolic equations was obtained by the author [23]. For the general theory of large deviations, readers are referred to the monograph [12]. For SPDEs in general, we refer readers to [18]. Because of the different nature of nonlinearity for different types of equations, the large deviations for SPDEs has to be dealt with on individual bases. There are two main issues which distinguish the current work from the previous ones. The first is the cubic nonlinear term B(u) in Eq. (2) and the second is the second-order differentiation in t (not like the parabolic cases). Note that even the existence and uniqueness of the solution of this kind of equation was newly established. Although the second-order (in t) Eq. (2) can also be written as a system of parabolic equations as it was done in [4], but by doing so the operator (differential) becomes degenerate. The properties of the corresponding semigroups are therefore not good enough for the large deviation estimates, not like the parabolic cases in the existing literature. To tackle the first issue, our idea is to prove that the probability that the energy of the solution is big is exponentially small. To this end, a remarkable result of Davis [11], Barlow and Yor in [3] on the moment estimates of martingales plays a key role. To treat the second-order differentiation in t, we fully exploit the energy equality proved by Chow, Menaldi [9] and Pardoux [17], and establish some exponential integrability of Hilbert space-valued martingales. To achieve this, some exponential martingales are specially constructed. The rest of the paper is organized as follows. In Section 2, the precise result is stated. In Section 3, the skeleton equation is studied. It is proved that the solution is a continuous map from the level set into the space C([0, T ]; H02 ) × C([0, T ]; L2 ). Section 4 is devoted to the proof of the large deviation principle. The long proof is split into several lemmas for clarity. We end this introduction with a remark. Remark 1.1. The main result in this paper is stated in the setting of one space dimension. This is just for simplicity. Our approach works equally well in high space dimensions and also in the general setting formulated in [4]. Throughout the paper, the generic constants may be different from line to line. If it is essential, the dependence of a constant on the parameters will be written explicitly. 2. Statement of the main result We now state the precise conditions on σ . Let σk (·), k  1, be a sequence of mappings from H02 into H02 and F (·,·) a mapping from L2 × H02 into L2 . Introduce

178

T. Zhang / Journal of Functional Analysis 248 (2007) 175–201 ∞   

2

 trace σ (u)2 = λ2 σk (u)  c 1 + u2 .

(A.1)

k

2

2

k=1 ∞ 



σk (u) 2

(A.2)

is bounded on bounded subsets of H02 .

k=1 ∞  

2

   trace σ (u) − σ (v) 2 = λ2 σk (u) − σk (v)  c u − v2 .

(A.3)

k

k=1





F (v, u)  c 1 + v + u||2 .





F (v1 , u1 ) − F (v2 , u2 )  c v1 − v2  + u1 − u2 2 .

(A.4) (A.5)

Throughout this paper, we assume (A.1)–(A.5) are in place. The time interval we consider is fixed as [0, T ]. We notice that the Cameron–Martin space H corresponding to the Wiener process Wt is given by  H = ht =

∞ 

λk hkt ek ;

k=1

∞ T   k 2 ˙hs ds < ∞ . k=1 0

For h ∈ H, let uht denote the solution of the following deterministic PDE, the so called skeleton equation: ∞     λk σk uht h˙ kt dt, +F u˙ ht , uht dt, d u˙ ht = Auht dt + B uht dt + k=1

uh0 For ht =

∞

k k=1 λk ht ek

= φ0 ∈ H02 ,

u˙ h0 = φ1 ∈ L2 .

(3)

∈ H ⊂ C([0, T ]; l 2 ), define ∞

1 I (h) = 2

T

 k 2 h˙ t dt.

k=1 0

Set I (h) = ∞ if h ∈ C([0, T ]; l 2 ) \ H. Notice that I (·) is the rate function for the large deviations of the l 2 -valued Brownian motion Wt =

∞ 

λk βtk ek .

k=1

This is clear by considering the finite-dimensional version: Wtd = For f ∈ C([0, T ]; H02 ) × C([0, T ]; L2 ), introduce

  Lf = h ∈ H; f (t) = uht , u˙ ht .

d

k k=1 λk βt ek .

T. Zhang / Journal of Functional Analysis 248 (2007) 175–201

179

Define  R(f ) =

infh∈Lf I (h) +∞

if Lf = ∅, if Lf = ∅.

Theorem 1. Assume (A.1)–(A.5). Let με be the law of (uε , u˙ ε ) on the product space C([0, T ]; H02 ) × C([0, T ]; L2 ). Then {με , ε > 0} satisfies a large deviation principle with rate function R(f ), i.e., (1) for every closed subset C ⊂ C([0, T ]; H02 ) × C([0, T ]; L2 ), lim sup ε 2 log με (C)  − inf R(f ); f ∈C

ε→0

(4)

(2) for every open subset G ⊂ C([0, T ]; H02 ) × C([0, T ]; L2 ), lim inf ε 2 log με (G)  − inf R(f ). f ∈G

ε→0

(5)

3. The skeleton equation The purpose of this section is to study the skeleton equation. For h ∈ H, recall that uht denote the solution of the following deterministic PDE, the so called skeleton equation: ∞     d u˙ ht = Auht dt + B uht dt + λk σk uht h˙ kt dt + F u˙ ht , uht dt, k=1

uh0 = φ0 ∈ H02 ,

u˙ h0 = φ1 ∈ L2 .

(6)

For a > 0, we aim to show that the mapping v h = (uh , u˙ h ) from ({h; I (h)  a},  · ∞ ) into C([0, T ]; H02 ) × C([0, T ]; H ) is continuous, where  · ∞ denotes the uniform norm on C([0, T ]; l 2 ). Proposition 2. The map: v h = (uht , u˙ ht ) from ({h; I (h)  a},  · ∞ ) into C([0, T ]; H02 ) × C([0, T ]; H ) is continuous. Proof. Let hn ∈ {h; I (h)  a} with limn→∞ sup0tT hnt − ht l 2 = 0. Define  

2 2 1 β 2 2 4



. u˙ t  + α∂x ut  + ∂x ut  + γ ∂x ut e(t, u) = 2 2 By the energy equality proved in [9, Theorem 3.1] and (A.1), we have ∞    e t, uh = e 0, uh + λk k=1

t 0

 h  h k u˙ s , σk us h˙ s ds +

t 0

  h h h F u˙ s , us , u˙ s ds

(7)

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T. Zhang / Journal of Functional Analysis 248 (2007) 175–201

t

  e 0, uh +

 ∞ 1  ∞ 1 2  2 2

h 



2

u˙

h˙ ks λ2k σk uhs

ds s k=1

0

t

k=1

h 



u˙ 1 + u˙ h + uh ds s s s 2

+c 0

  e 0, uh + c

t

 ∞ 1

h 

h   k 2 2

u˙ 1 + u

h˙ s ds s s 2 k=1

0

t + cT + c

 e s, uh ds

0

  cT + e 0, uh + c

t

 ∞  1  2 2  h h˙ ks e s, u + 1 ds.

(8)

k=1

0

When h is fixed, it is easy to check that sup0tT e(t, uh ) < ∞. Thus, applying Gronwall’s inequality, we get sup



 sup e t, uh = M < ∞.

(9)

h∈ h;I (h)a 0tT

Observe that ||Aφ−2  (γ + α)||φ||2 , φ ∈ H02 ,



 |B(φ) − B(ψ)  β ∂ 2 φ ∂x φ + ∂x ψ∂x φ − ∂x ψ x



+ β∂x ψ2 ∂x2 φ − ∂x2 ψ , φ, ψ ∈ H02 . Thus, (9) implies that there exist constants C1 and C2 such that



sup  sup Auht −2  C1 ,

(10)

(11)

(12)

h∈ h;I (h)a 0tT

and

 h1





B ut − B uht 2  C2 uht 1 − uht 2 , 2

 h1 , h2 ∈ h; I (h)  a .

(13)

Regarded as an equation in H0−2 , one has t u˙ ht

= φ1 +

t Auhs ds

0

+ 0

By (9), (A.1) and (A.4), we have

∞   B uhs ds + λk k=1

t 0

 σk uhs h˙ ks ds +

t 0

 F u˙ hs , uhs ds.

(14)

T. Zhang / Journal of Functional Analysis 248 (2007) 175–201 ∞ 

t

 h  k 

σk u h˙  dl

λk

k=1



181

l

l

s

t   ∞

 h 2 ul

1 

λ2k σk

2

t c

1 2

dl

l

k=1

s

∞   k 2 h˙ k=1

 ∞  ∞ 1 1 t

h    k 2 2  k 2 2  h  1 + ul 2 h˙ l h˙ l dl  c e l, u dl k=1

s

s

k=1

1

 CM,a (t − s) 2 ,

(15)

and t

 h h

F u˙ , u dl  c l

t

l

s

  1 + e l, uh dl  CM,a (t − s),

(16)

s

for some constant CM,a . Combining this with (12) and (13), we see that there exists a constant C3 so that

h

u − uh  C3 |t − s| 12 . sup (17) t s −2

 h∈ h;Id (h)a

Introduce eL (t, v) =



2  1 v˙t 2 + α∂x vt 2 + γ ∂x2 vt . 2

Set vtn = uht n − uht . Write hnt =

∞ 

λk hk,n t ek .

k=1

Note that sup0tT e(t, v) dominates the norm of (vt , v˙t ) in the space of C([0, T ]; H02 ) × C([0, T ]; H ). Applying the energy inequality in [9, Lemma 3.1], we have  eL t, v n =

t

  h  B us n − B uhs , v˙sn ds +

0

+

t

  h h  F u˙ s n , us n − F u˙ hs , uhs , v˙sn ds

0 ∞  k=1

t λk

 n  h k,n  v˙s , σk us n h˙ s − σk uhs h˙ ks ds.

0

In virtue of (13),   h    



  B u n − B uh , v˙ n   B uhn − B uh

v˙ n  ceL s, v n , s s s s s s

(18)

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T. Zhang / Journal of Functional Analysis 248 (2007) 175–201

for some constant c. By the Lipschitz condition and the Sobolev imbedding,   h h 

    F u˙ n , u n − F u˙ h , uh , v˙ n   c v˙ n 2 + un − uh 2  ceL s, v n . s

s

s

s

s

s

s

s 2

(19)

Let sm = [ms]/m. Write ∞ 

t λk

k=1

 n  h k,n  v˙s , σk us n h˙ s − σk uhs h˙ ks ds = Ct1 + Ct2 + Ct3 + Ct4 ,

(20)

0

where ∞ 

=

Ct1

=

Ct2

=

Ct3

=

Ct4

t λk

k=1

0

∞ 

t λk

k=1

0

∞ 

t λk

k=1

0

∞ 

t

k=1

λk

 n   h  v˙s , σk us n − σk uhs h˙ k,n ds, s  n   ˙k v˙s − v˙snm , σk uhs h˙ k,n s − hs ds,  n    ˙k v˙sm , σk uhs − σk uhsm h˙ k,n s − hs ds,  n   ˙k v˙sm , σk uhsm h˙ k,n s − hs ds.

0

We now estimate each of the terms. Keeping (A.3) in mind, we have  1 C  

t

t

 ∞ 1  ∞ 1

n 

 h  h 2 2   k,n 2 2 2

v˙

h˙ s λk σk us n − σk us

ds s k=1

0

t c

k=1

n

h

v˙

u n − uh

s

s 2

s



∞   k,n 2 h˙ s



1 2

ds

k=1

0

t c

 ∞ 1  n   k,n 2 2 ˙ hs eL s, v ds.

(21)

k=1

0

In view of (17) and (A.1),  2 C   t

t

n

v˙ − v˙ n

s

0



∞ 

sm −2 k=1

 h 2 us 2

λ2k σk

1  2

∞   k,n 2 h˙ s − h˙ ks k=1

1 2

ds

T. Zhang / Journal of Functional Analysis 248 (2007) 175–201

1  c√ m

t

183

 ∞ 1 2

h    k,n 2 1 + us 2 h˙ s − h˙ ks ds k=1

0

1  ca,M √ , m

(22)

where M is defined as in (9). Since u˙ hs  is dominated by

 e(s, uh ), (9) implies that

h

u − uh  C|t − s|.

sup

h∈{h;I (h)a}

t

(23)

s

This together with (A.3) implies that t

 3 C   t

 ∞ 1  ∞ 1

n 

 h  h 2 2   k,n 2 2 2 k

v˙

h˙ s − h˙ s λk σk us − σk usm

ds sm k=1

0

t c

k=1

 ∞ 1

  h  k,n 2 2 h k

u − u

h˙ s − h˙ s ds s sm k=1

0

 ca,M

1 . m

(24)

Now  ∞  [mt]−1    4     h  k,n  k,n  n k k C    v˙ l , σk u l h (l+1) − h (l+1) − h l − h l  λk t   m m m m m m k=1 l=1  ∞      k,n    k + ht − hkt − hk,n λk v˙ n[mt] , σk uh[mt]  [mt] − h [mt]   m m m m k=1



 ∞ 1  ∞ 1

n 

 h 2 2   k,n  k,n 2 2 2 k k

v˙ l

σk u l

λk h (l+1) − h (l+1) − h l − h l

[mt]−1 

m

l=1

k=1

m

k=1

m

m

m

m

 ∞ 1  ∞ 1

n 

 h 2 2   k,n  k,n 2 2 2 k k

σk u [mt]

+ v˙ [mt]

λk ht − ht − h [mt] − h [mt] m

 cm,M

k=1

m



sup hnt − ht l 2 ,

m

k=1

m

(25)

0tT

where we have used (9) and the assumption (A.2). Putting together (18)–(25) we arrive at   t

n

 n  1



eL t, v  c √ + cm,M sup ht − ht l 2 + c eL s, v n ds m 0tT 0

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T. Zhang / Journal of Functional Analysis 248 (2007) 175–201

t +c

 ∞ 1  n   k,n 2 2 h˙ s eL s, v ds.

(26)

k=1

0

Applying the Gronwall’s inequality, we get 

sup eL t, v

n

0tT

1    T   ∞

n

 k,n 2 2 1  c √ + cm,M sup ht − ht l 2 exp cT + c h˙ s ds m 0tT 





0

k=1



1  ca √ + cm,M sup hnt − ht l 2 . m 0tT

(27)

Given ε > 0. We first choose m such that ca m1  2ε . Then for such a m, there exists N so that for n  N,



ε ca cm,M sup hnt − ht l 2  . 2 0tT

(28)

Therefore, for n  N ,  n sup eL t, uh − uh  ε, 0tT

which finishes the proof of the proposition.

2

Corollary 3. The rate function R(·) defined in Section 2 is a good rate function, i.e., for every a > 0, {g; R(g)  a} is compact. Proof. Notice that

   g; R(g)  a = uh , u˙ h ; I (h)  a .

So the corollary is a consequence of Proposition 2 and the fact that {h; I (h)  a} is compact in C([0, T ]; l 2 ). 2 4. Large deviations Consider  d u˙ εt

1

= α+β

 t ∞   ε 2  ε k  2 ε 4 ε ∂y u  dy ∂ u dt − γ ∂ u dt + ε λk σk ut dβt + F u˙ εs , uεs ds, t x t x t k=1

0

uεt (0) = uεt (1) = 0, uε0 (x) = φ0 (x) ∈ H02 ,

∂x uεt (0) = ∂x uεt (1) = 0, ∂t uε0 (x) = φ1 (x) ∈ L2 .

0

(29)

In this section, we will establish the large deviation principle. We first prepare a number of preliminary results. Let e(t, uε ) be defined as in (7) in Section 3.

T. Zhang / Journal of Functional Analysis 248 (2007) 175–201

185

Lemma 4. It holds that lim lim sup ε 2 log P

M→∞

  sup e t, uε > M = −∞.



ε→0

(30)

0tT

Proof. By the energy equality [9, (3.14)], ∞    e t, uε = e 0, uε + ε λk k=1

1 + ε2 2

t  ∞

t

 ε  u˙ s , σk uεs dβsk

0

 ε 2 us ds +

λ2k σk

0 k=1

t

 ε  ε ε u˙ s , F u˙ s , us ds.

(31)

0

Recall that it is proved in [3,11] that there exists a universal constant c such that, for any p  2 and any continuous martingale (Mt ) with M0 = 0, one has 1

∗



M  cp 12 Mt2 , t p p

(32)

where Mt∗ = sup0st |Ms |, and  · p stands for the Lp (Ω)-norm. Using (32) and (A.1), we have for p  2,   p  p1  E sup e t, uε 0tl

p  1    l ∞ p   

 1 2    ε 2

ε 2  e 0, u + ε E  λk σk us ds    2 0 k=1

p  1  ∞ p  1    l p p      t          u˙ εs , σk uεs dβsk  + ε E sup  λk + E   u˙ εs , F u˙ εs , uεs  ds     0tl   

k=1

0

0

p  1    l p   



 1 2   1 + uεs 2 ds   e 0, uε + c ε 2 E    2 0

p  1    l p  

ε 2 ε 2   1 + u˙ s + us 2 ds  +c E    0

 p  1    l ∞ p    2  1 2   + εcp 2 E  λ2k u˙ εs , σk uεs ds    0 k=1

1  cT + c ε 2 2

l 0

 p  1    l p  

ε 2  2    ε 2p  1 1 2  ε p







u˙ s 1 + us 2 ds  1 + E us 2 ds + εcp 2 E    0

186

T. Zhang / Journal of Functional Analysis 248 (2007) 175–201

l +c

   ε p  1 p ds E e s, u

0



1  cT + c ε 2 + c 2

 l

   ε p  1 1 p ds + εcp 2 E e s, u

0



1  cT + c ε 2 + c 2

 l

2 p  2   ε p  E u˙ s 1 + uεs 2 2 p ds

1 2

0

 l

   ε p  1 p ds E e s, u

0

 l    2  12

ε 2 p p

1

1 ε 2p



E u˙ + εcp + 1 + us 2 ds 2 s 2 1 2

0



1  cT + c ε 2 + c 2

 l

   ε p  1 1 p ds + εcp 2 E e s, u

0

 l

   ε p  2 p ds E e s, u

1 2

,

(33)

0

where we have used the inequality m  1  l    l m     m  1   m ds E  fs ds  E |fs |    0

0

in several places for an appropriate m. Therefore, l      ε p  2  ε p  p2  2 4 2 p ds. E sup e t, u E e s, u  cT + ε cp + cε + c 0tl

0

By Gronwall’s inequality, there exist constants c1 and c2 so that  E

p   2 2 p  c1 e c2 ε p . sup e t, uε

0tT

This implies, by Chebyshev inequality,  P

  2 2 p sup e t, uε > M  M −p c1 ec2 ε p .

0tT

Letting p =

1 , ε2

we get ε 2 log P



    1 sup e t, uε > M  log + log(c1 ) + c2 M 0tT

(34)

T. Zhang / Journal of Functional Analysis 248 (2007) 175–201

187

which yields lim lim sup ε 2 log P

M→∞

  sup e t, uε > M = −∞.



ε→0

2

(35)

0tT

We need a result of exponential integrability for a Hilbert space-valued martingale. Proposition 5. Let fk (s), k  1, be a sequence of adapted L2 -valued stochastic processes. Assume that there exists a constant K such that ∞ 



fk (s) 2  K

almost surely for all s  0.

k=1

Define

Mt =

∞  

t

fk (s) dβsk .

k=1 0

Then there exists a constant δ0 > 0 such that    Mt − Ms 2 < ∞. sup E exp δ0 |t − s| t =s

(36)

Proof. For simplicity, denote L2 by H . Without loss of generality, we may assume s = 0. Otherwise consider Yu = Ms+u − Ms . For g ∈ C 2 (H ), by Ito’s formula, ∞    exp g(Mt ) = exp g(M0 ) +

t

  exp g(Ms ) g (Ms ), fk (s) dβsk

k=1 0 ∞

1 + 2

t

  exp g(Ms ) g (Ms ) ⊗ g (Ms ) + g

(Ms ) fk (s), fk (s) ds.

(37)

k=1 0

Put hs =

∞ 1  g (Ms ) ⊗ g (Ms ) + g

(Ms ) fk (s), fk (s) . 2 k=1

By integration by parts formula and (37), it is easy to verify that  g Nt



t

= exp g(Mt ) − g(0) −

hs ds 0

(38)

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T. Zhang / Journal of Functional Analysis 248 (2007) 175–201

is a non-negative local martingale. Now, for λ > 0 (which will be specified later), let gλ (x) = 1 (1 + λ|x|2H ) 2 . Then − 1  gλ (x) = λ 1 + λ|x|2H 2 x,  − 3 − 1  gλ

(x) = −λ2 1 + λ|x|2H 2 x ⊗ x + λ 1 + λ|x|2H 2 IH , where IH stands for the identity operator. It is easy to see that   1 supgλ (x)  λ 2 , x



1 sup gλ

(x)  λ 2 , x

where  ·  stands for the operator norm. Define hλs as in (38) replacing g by gλ . Then, ∞

 λ 1 

h   λ

fk (s) 2  1 λK. s 2 2

(39)

k=1

For any r > 0 and every λ > 0, we have  P

 Mt  √ >r t   1  2 2 = P gλ (Mt )  1 + λtr 

t

= P gλ (Mt ) − gλ (0) −

t hλs ds

0



1   1 + λtr 2 2



0

t

 P gλ (Mt ) − gλ (0) −

+ gλ (0) +

hλs ds

hλs ds

1  1  1 + λtr 2 2 − gλ (0) − λKt 2



0

   g   1 1  E Nt λ exp − 1 + λtr 2 2 + gλ (0) + λKt 2   1  1  exp − 1 + λtr 2 2 + gλ (0) + λKt , 2

(40)

where the fact E[Nt λ ]  1 has been used. Choosing λ = t −1 δr 2 , we get that g

 P

       1 1 Mt  1 1 √ > r  exp −δ 2 r 2 + 1 + Kδr 2 = exp −δ 1 − K r 2 . 2 2 t δ2

Take δ > 0 small enough so that δ ∗ := δ(1/δ 1/2 − 12 K) > 0. We arrive at 

  Mt  P √ > r  exp −δ ∗ r 2 + 1 , t

(41)

T. Zhang / Journal of Functional Analysis 248 (2007) 175–201

189

√ where δ ∗ is independent of t. We can now easily deduce (36). Fix δ0 < δ ∗ and let ξ = Mt / t. We have    ∞    2  Mt 2 = − exp δ0 r 2 dP (ξ > r) E exp δ0 = E exp δ0 ξ |t| 0

∞  1 + 2δ0

 exp δ0 r 2 P (ξ > r)r dr

0

∞  1 + 2δ0

  exp δ0 r 2 exp −δ ∗ r 2 + 1 r dr < +∞

(42)

0

2

which completes the proof of the proposition. Lemma 6. For every δ1 > 0, M > 0,  sup lim lim sup ε 2 log P m→∞

s

1 |s−t| m ,s,tT

ε→0

  ε  M = −∞.  δ , sup e t, u 1 −2

ε

u˙ − u˙ ε

t

(43)

0tT

Proof. In view of (10), (11) and (A.4), Auεs −2 , B(uεs ) and F (u˙ εs , uεs ) are uniformly bounded by a constant (depending only on M) on the set {ω, t  T ; sup0tT e(t, uε )  M}. Regarded as an equation in H0−2 , t u˙ εt = φ1 +

t Auεs ds +

0

∞   B uεs ds + ε λk k=1

0

t

 σk uεs dβsk +

0

t

 F u˙ εs , uεs ds.

(44)

0

Therefore, on {ω; sup0tT e(t, uε )  M}, sup 1 |s−t| m ,s,tT

ε

u˙ − u˙ ε

s −2

t

 cM

1 +ε sup Nt − Ns , m |s−t| 1 ,s,tT m

where Nt =

∞ 

t λk

k=1

 σk uεs dβsk .

0

Thus for sufficiently big m, 

ε

u˙ − u˙ ε

sup

P

1 |s−t| m ,s,tT

s

t −2

   δ1 , sup e t, uε  M 0tT

  ε 1 Nt − Ns   δ1 , sup e t, u  M . P ε sup 2 0tT |s−t| 1 ,s,tT 

m

(45)

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T. Zhang / Journal of Functional Analysis 248 (2007) 175–201

So it remains to show  lim lim sup ε 2 log P ε

m→∞

ε→0

  1 Nt − Ns   δ1 , sup e t, uε  M = −∞. 2 0tT |s−t| 1 ,s,tT sup

(46)

m

Notice that by (A.1) there exists a constant KM such that  ∞   

 ε 2  ε 2

λk σk ut  KM . ω; sup e t, u  M ⊂ ω; sup 0tT

(47)

0tT k=1

Define  τ = inf s  0;

∞ 



 ε 2 us > KM .

λ2k σk

k=1

It follows from (47) that

  ω; sup e t, uε  M ⊂ {τ  T }. 0tT

Therefore,  P ε

 1 Nt − Ns   δ1 , sup e t, uε  M 2 0tT |s−t| 1 ,s,tT



sup m



P ε

 1 Nt∧τ − Ns∧τ   δ1 . 2 |s−t| 1 ,s,tT sup m

So, to prove (46), we may drop the event {sup0tT e(t, uε )  M} and assume in the rest of the proof that sup

∞ 

0tT k=1

 2 λ2k σk uεt  KM .

(48)

Applying Proposition 5, there exists a constant λM > 0 such that    Nt − Ns 2 < ∞. sup E exp λM |t − s| t =s,s,tT Introduce T T D= 0 0

  Nt − Ns 2 ds dt. exp λM |t − s|

(49)

T. Zhang / Journal of Functional Analysis 248 (2007) 175–201

191

Then we have E[D] < ∞. Now by Garsia lemma (see [21]) we have 8 Nt − Ns   √ λM

|t−s|  

D log 2 u

1 2

dp(u),

(50)

0

1

where p(u) = u 2 . For any δ < 12 , say δ = 14 , (50) implies that there exists a constant c such that 1 8c  Nt − Ns   √ ( log D + 1)|t − s| 4 . λM

Consequently,  1 8c  1 4 Nt − Ns   √ ( log D + 1) . m λM |s−t| 1 ,s,tT sup m

Therefore, √     1 1 δ1 λM 1 P ε sup Nt − Ns   δ1  P log D > −1 2 2 ε 8c ( 1 ) 14 1 |s−t| m ,s,tT m √   2  1 δ1 λM 1  P D > exp − 1 2 ε 8c ( 1 ) 14 m √ 2    1 δ1 λM 1 . − 1  E[D] exp − 2 ε 8c ( 1 ) 14 m 

This yields  1 lim lim sup ε P ε sup Nt − Ns   δ1 = −∞, m→∞ ε→0 2 |s−t| 1 ,s,tT 

2

m

which completes the proof.

2

Lemma 7. Let fs , s  0 be a H0−2 -valued adapted stochastic process such that sup fs −2  δ1 .

0sT

Set Mt =

∞  k=1

t λk 0

 ! fs , σk uεs dβsk .

(51)

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T. Zhang / Journal of Functional Analysis 248 (2007) 175–201

Then there exist positive constants c1 > 0, c2 > 0 and cM such that for η1 > 0,      c2 η12 P sup |Mt | > η1 , sup e t, uε  M  c1 exp − . cM T δ12 0tT 0tT

(52)

Proof. Notice that Mt , t  0 is a martingale whose bracket satisfies Mt =

∞ 

t λ2k

k=1

 !2 fs , σk uεs ds

0

t 

fs 2−2

∞ 

 2 λ2k σk uεs 2 ds

k=1

0

t

2  fs 2−2 1 + uεs 2 ds

c 0

t  cδ12

 e s, uε ds.

(53)

0

Thus, on {ω, sup0tT e(t, uε )  M}, Mt  cM δ12 =: b. By the martingale representation theorem, there exists a standard Brownian motion Bs , s  0 such that Mt = BMt . We have    ω, sup |Mt | > η1 , sup e t, uε  M 0tT

0tT

 

= ω, sup |BMt | > η1 , sup e t, uε  M 0tT

0tT

   ⊂ ω; sup |Bu | > η1 , sup e t, uε  M 0ub

0tT

  √  = ω; b sup |B˜ u | > η1 , sup e t, uε  M , 0u1

0tT

where B˜ is another Brownian motion by the scaling invariance property. It is well known that there exists a constant c2 > 0 such that c1 := E[exp(c2 sup0u1 |B˜ u |2 )] < ∞. Thus,  P

 η1 ˜ sup |Bu | > √ b 0u1      c2 η12  E exp c2 sup |B˜ u |2 exp − b 0u1   c2 η12 = c1 exp − , cM δ12

  sup |Mt | > η1 , sup e t, uε  M  P 0tT

proving the lemma.

0tT

2



T. Zhang / Journal of Functional Analysis 248 (2007) 175–201

193

Recall eL (t, v) =



2  1 v˙t 2 + α∂x vt 2 + γ ∂x2 vt . 2

Theorem 8. For every η > 0, R > 0, h ∈ H, there exists δ > 0 such that    lim sup ε 2 log P sup eL t, uε − uh  η, sup εW − h∞ < δ  −R. ε→0

0tT

(54)

(55)

0tT

Proof. As an equation in H0−2 , we have t u˙ εt

− u˙ ht

=

 A uεs − uhs ds +

0

t

  ε  B us − B uhs ds

0

+ε

∞  k=1

t +

t λk

∞   σk uεs dβsk − λk

0

k=1

t

 σk uhs h˙ ks ds

0

  ε ε  F u˙ s , us − F u˙ hs , uhs ds.

(56)

0

For simplicity, denote vt := uεt − uht . By the energy inequality (3.10) in [9], we have t eL (t, v) =

  ε  1 B us − B uhs , v˙s ds + ε 2 2

0

+

∞ 

t λk

k=1

t +

t  ∞ 0 k=1

∞    v˙s , εσk uεs dβsk − λk

0

 2 λ2k σk uεs ds

k=1

t

  v˙s , σk uhs h˙ ks ds

0

  ε ε  F u˙ s , us − F u˙ hs , uhs , v˙s ds.

(57)

0

Let M > sup0tT e(t, uh ). In view of (11) and (A.1), on the event {ω, sup0tT e(t, uε )  M},



 ε 



B u − B uh  CM uε − uh , s s s s 2 ∞ 

 2 λ2k σk uεs  CM .

k=1

By the Lipschitz condition, (A.5) and the Sobolev imbedding,   ε ε    F u˙ , u − F u˙ h , uh , v˙s   ceL (s, v). s s s s

194

T. Zhang / Journal of Functional Analysis 248 (2007) 175–201

So it follows from (57) that on {ω, sup0tT e(t, uε )  M}, t eL (t, v)  CM

  1 eL (s, v) ds + ε 2 cM T + Mtε , 2

(58)

0

where Mtε =

t

∞ 

λk

k=1

∞    v˙s , εσk uεs dβsk − λk k=1

0

t

  v˙s , σk uhs h˙ ks ds.

(59)

0

Let sm = [ms]/m and write Mtε = Nt1,m + Nt2,m + Nt3,m + Nt4,m , where

Nt1,m

Nt2,m

Nt3,m

Nt4,m

=

=

=

=

∞ 

t λk

k=1

0

∞ 

t λk

k=1

0

∞ 

t λk

k=1

0

∞ 

t λk

k=1

   v˙s , σk uεs − σk uhs h˙ ks ds,    v˙s − v˙sm , σk uεs ε dβsk − h˙ ks ds ,     v˙sm , σk uεs − σk uεsm ε dβsk − h˙ ks ds ,    v˙sm , σk uεsm ε dβsk − h˙ ks ds .

0

Now,  1,m  N t  



t v˙s 

∞ 

 ε  2 us − σk uhs

λ2k σk

1  2

k=1

0

t c

c 0

1 2

ds

k=1

 ∞ 1  2 2

 ε h˙ ks v˙s  us − uhs 2 ds k=1

0

t

∞   k 2 h˙ s



∞   k 2 h˙ s eL (s, v) k=1

1 2

ds.

(60)

T. Zhang / Journal of Functional Analysis 248 (2007) 175–201

195

So we deduce from (58) that on {ω, sup0tT e(t, uε )  M}, t eL (t, v)  CM





eL (s, v) 1 +

∞   k 2 h˙ s

1  2

k=1

0

 i,m  1 Nt . ds + ε 2 cM T + 2 4

(61)

i=2

By Gronwall’s inequality we obtain that on {ω, sup0tT e(t, uε )  M},  eL (t, v)  e

CM T

4    1 2 ε cM T + sup Nti,m  2 0tT



i=2

which implies that there exists a constant ε1 > 0 such that for ε  ε1 ,  P

   sup eL t, uε − uh  η, sup e t, uε  M, εW − h∞ < δ

0tT

 P

0tT

4  i=2

  i,m  1 −C T  ε sup Nt  > e M η, sup e t, u  M, εW − h∞ < δ . 2 0tT 0tT

(62)

Now, 

  i,m  1 −C T  ε M P sup Nt  > e η, sup e t, u  M, sup εW − h∞ < δ 2 0tT 0tT i=2 0tT     1  P sup Nt4,m  > e−CM T η, sup e t, uε  M, sup εW − h∞ < δ 6 0tT 0tT 0tT     1  +P sup Nt2,m  > e−CM T η, sup e t, uε  M 6 0tT 0tT    3,m  1 −C T  ε M   > e +P sup Nt η, sup e t, u  M . 6 0tT 0tT 4 

(63)

(64)

Furthermore, for δ1 > 0, 

  2,m  1 −C T  ε M   > e P sup Nt η, sup e t, u  M 6 0tT 0tT     1  P sup Nt2,m  > e−CM T η, sup e t, uε  M, sup v˙t − v˙s −2  δ1 6 0tT 0tT |t−s| 1 

+P



 sup v˙t − v˙s −2 > δ1 , sup e t, uε  M .

1 |t−s| m

0tT

m

(65)

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T. Zhang / Journal of Functional Analysis 248 (2007) 175–201

By the Girsanov theorem, we know that Wt − 1ε ht , t  0 is a Wiener process under the probability measure P ∗ given by  t   t ∞ 1 1 1  4  ˙ k 2 dP ∗  ˙ hs ds . = exp hs , dWs  − λk dP Ft ε 2 ε2 k=1

0

0

Through a change of measure and applying Lemma 7, we can show that there exist constants c1 , c2 such that    2,m  1 −C T  ε M   > e η, sup e t, u  M, sup v˙t − v˙s −2  δ1 P sup Nt 4 0tT 0tT |t−s| 1   c1 exp −

η2

m

 . 2

c2 cMT ε 2 δ1

(66)

Notice that on {ω; sup0tT e(t, uε )  M}, ∞ 

∞ 

    2  λ2k v˙sm , σk uεs − σk uεsm  v˙sm 2 λ2k σk uεs − σk uεsm 2

k=1

k=1

2 

 cv˙sm 2 uεs − uεsm

2   s  2



1 2

ε

 v˙sm  u˙ l dl  CM .



m

(67)

sm

Using this and following the same proof of Lemma 7, we can show that  P

  1  sup Nt3,m  > e−CM T η, sup e t, uε  M 6 0tT 0tT   2 c2 η .  c1 exp − cMT ε 2 ( m1 )2



(68)

Now given R > 0, η > 0. According to Lemma 4, we can choose M large enough and ε2 > 0 such that for ε  ε2 ,      R (69) P sup e t, uε > M  exp − 2 . ε 0tT Next, we choose δ1 , according to (66), so that for ε  ε3  P

  1  sup Nt2,m  > e−CM T η, sup e t, uε  M, sup v˙t − v˙s −2  δ1 4 1 0tT 0tT |t−s| m   R  exp − 2 , ε



(70)

T. Zhang / Journal of Functional Analysis 248 (2007) 175–201

197

where ε3 is a positive number . For such a δ1 > 0, by Lemma 6 and (68) there exist an integer m and ε4 > 0 so that for ε  ε4 ,     ε R P sup v˙t − v˙s −2 > δ1 , sup e t, u  M  exp − 2 , ε 0tT |t−s| 1 

 P

(71)

m

     1  R sup Nt3,m  > e−CM T η, sup e t, uε  M  exp − 2 . 6 ε 0tT 0tT

(72)

When such an m is fixed,

Nt4,m =

∞ 

λk

[mt]−1 

   k   v˙ l , σk uεl εβ l+1 − hkl+1 − εβ kl − hkl

k=1

+

m

l=0

∞ 

m

m

m

m

m

   k   εβt − hkt − εβ k[mt] − hk[mt] . λk v˙ [mt] , σk uε[mt]

k=1

m

m

m

(73)

m

Therefore,  ∞ 1/2  ∞ 1/2    2    4,m  [mt]−1  k 2 ε

2 k k k N t  

σk u l v˙ l  λk εβ l+1 − h l+1 − εβ l − h l m

l=0

k=1

m

m

k=1

m

m

m

 ∞ 1/2  ∞ 1/2  



 ε 2  k 2 2 k k k





σk u [mt] + v˙ [mt] λk εβt − ht − εβ [mt] − h [mt] m



m

k=1

m

k=1

m

 ∞ 1/2



 ε 2

v˙ l

σk u l

εW − h∞

[mt]−1 

m

l=0

k=1

m

 ∞ 1/2



 ε 2

σk u [mt]

+ v˙ [mt]

εW − h∞ . m

(74)

m

k=1

By the assumption (A.2), we know that on the event {ω, sup0tT e(t, uε )  M},

sup

∞ 

 ε 2

σk u

0sT k=1

s

and

sup v˙s  0sT

are uniformly bounded by some constant cM . Thus, we see from (74) that there exists δ > 0 such that the event 

  1  sup Nt4,m  > e−CM T η, sup e t, uε  M, sup εW − h∞ < δ 6 0tT 0tT 0tT



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T. Zhang / Journal of Functional Analysis 248 (2007) 175–201

is empty. Combining this with (62) and (69)–(72) we obtain that for sufficiently small ε,  P

    R sup eL t, uε − uh  η, sup εW − h < δ  5 exp − 2 . ε 0tT 0tT

This completes the proof.

2

After we established the key results, Proposition 2 and Theorem 8, there exists now a wellknown method (see [1,13]) to deduce the large deviation principle. For completeness, we include a proof. Theorem 9. {με , ε > 0} satisfies a large deviation principle with rate function R(f ). Proof. Denote by d(·,·) the distance in the metric space C([0, T ]; H02 ) × C([0, T ]; L2 ). First, fix any closed subset C ⊂ C([0, T ]; H02 ) × C([0, T ]; L2 ) and choose a < infg∈C R(g). Define   

 Ka = g ∈ C [0, T ]; H02 × C [0, T ]; L2  R(g)  a ,  

Ca = f ∈ H  I (f )  a . Then Ka =



 f f u , u˙ ; f ∈ Ca ,

Ka ∩ C = ∅,

where (uf , u˙ f ) denotes the solution of the skeleton equation in Section 2 with h replaced by f . For any g ∈ Ka , there exists an open neighborhood of g, Vg such that Vg ∩ C = ∅. One can choose ρg > 0, such that  

 Gg = h ∈ C [0, T ]; H02 × C [0, T ]; L2 , d(h, g)  ρg ⊆ Vg . For any fg ∈ Ca such that g = (ufg , u˙ fg ) and R > a, by Theorem 8 one can find two constants εg > 0, αg > 0 such that for any ε < εg     R ,  exp − P εW ∈ Fg , uε , u˙ ε ∈ GC g ε2 where Fg = {f ∈ C([0, T ], l 2 ); f − fg ∞ < αg }. Therefore,      ε ε  ε ε R C C P εW ∈ Fg , u , u˙ ∈ Vg  P εW ∈ Fg , u , u˙ ∈ Gg  exp − 2 . ε

(75)

T. Zhang / Journal of Functional Analysis 248 (2007) 175–201

199

Since (Fg )g∈Ka forms a cover for the compact set Ca of C([0, T ], l 2 ), there exist g1 , . . . , gl ∈ Ka such that F=

l "

F g i ⊃ Ca .

i=1

Therefore, we have  l  "   ε ε 

 ε ε P εW ∈ F, u , u˙ ∈ C = P {εW ∈ Fgi } ∪ u , u˙ ∈ C 

i=1

 l "  

 ε ε C P {εW ∈ Fgi } ∪ u , u˙ ∈ Vgi i=1



l    P εW ∈ Fgi , uε , u˙ ε ∈ VgCi i=1

  R  l exp − 2 . ε It follows that      lim sup ε 2 log με (C)  lim sup ε 2 log P εW ∈ F C + P εW ∈ F, uε , u˙ ε ∈ C ε→0

ε→0

   − inf c I (f ) ∨ (−R) f ∈F

= −a, where we have used the fact that εW satisfies a large deviation principle with rate function I (·). Let a → infg∈C R(g) to get lim sup ε 2 log με (C)  − inf R(g). ε→0

g∈C

Let G ⊂ C([0, T ]; H02 ) × C([0, T ]; L2 ) be an open set, take g ∈ G with R(g) < ∞. Then there exists f ∈ H such that  g = uf , u˙ f ,

R(g) = I (f ).

Choose ρ > 0, such that   

 h ∈ C [0, T ]; H02 × C [0, T ]; L2  d(h, g)  ρ ⊂ G. For any R > R(g), by Theorem 8, ∃α > 0, ε0 > 0, such that for any ε ∈ (0, ε0 ),     ε ε  f f R > ρ, εW − f ∞ < α  exp − 2 . P d u , u˙ , u , u˙ ε

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T. Zhang / Journal of Functional Analysis 248 (2007) 175–201

Therefore,    P uε , u˙ ε ∈ G  P d uε , u˙ ε , g  ρ     P d uε , u˙ ε , uf , u˙ f  ρ, εW − f ∞ < α      P εW − f ∞ < α − P d uε , u˙ ε , uf , u˙ f > ρ, εW − f ∞ < α    R  P εW − f ∞ < α − exp − 2 . ε But 

 lim ε 2 log P εW − f ∞ < α  − inf I (ϕ), ϕ − f ∞  α

ε→0

 −I (f ) and since R > R(g),  lim ε 2 log P uε , u˙ ε ∈ G  −I (f ) = −R(g).

ε→0

Since g is the arbitrary,  lim ε 2 log P uε , u˙ ε ∈ G  − inf I (g). g∈G

ε→0

This completes the proof of the theorem.

2

Acknowledgment This work was initiated when the author visited the University of Oslo. He would like to thank Bernt Øksendal for his hospitality and for the stimulating discussions. References [1] R. Azencott, Grandes déviations et applications, in: Ecole d’été de Probabilité de Saint-Flour VIII, 1978, in: Lecture Notes in Math., vol. 774, Springer-Verlag, Berlin, 1980, pp. 1–176. [2] J.M. Ball, Initial-boundary value problems for an extensible beam, J. Math. Anal. Appl. 42 (1973) 61–90. [3] M.T. Barlow, M. Yor, Semi-martingale inequalities via the Garsia–Rodemich–Rumsey lemma, and applications to local times, J. Funct. Anal. 49 (1982) 198–229. [4] Z. Brze´zniak, B. Maslowski, J. Seidler, Stochastic beam equations, Probab. Theory Related Fields 132 (2005) 119– 149. [5] C. Cardon-Weber, Large deviations for a Burgers’-type SPDE, Stoch. Process. Appl. 84 (1999) 53–70. [6] S. Cerrai, M. Röckner, Large deviations for stochastic reaction–diffusion systems with multiplicative noise and non-Lipschitz reaction term, Ann. Probab. 32 (1) (2004) 1100–1139. [7] F. Chenal, A. Millet, Uniform large deviations for parabolic SPDEs and applications, Stoch. Process. Appl. 72 (1997) 161–186. [8] P.L. Chow, Large deviation problem for some parabolic Ito equations, Comm. Pure. Appl. Math 45 (1) (1992) 97–120. [9] P.L. Chow, J.L. Menaldi, Stochastic PDE for nonlinear vibration of elastic panels, Differential Integral Equations 12 (3) (1999) 419–432. [10] I.D. Chueshov, Introduction to the Theory of Infinite-dimensional Dissipative Systems, Acta, Kharkiv, 2002.

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