arXiv:1611.09962v1 [math.PR] 30 Nov 2016
Large deviations for stochastic heat equations with memory driven by L´evy-type noise Markus Riedle Department of Mathematics King’s College London London WC2R 2LS United Kingdom
[email protected]
Jianliang Zhai˚ School of Mathematical Sciences University of Science and Technology of China Chinese Academy of Science Hefei, 230026, China
[email protected]
December 1, 2016
Abstract For a heat equation with memory driven by a L´evy-type noise we establish the existence of a unique solution. The main part of the article focuses on the FreidlinWentzell large deviation principle of the solutions of heat equation with memory driven by a L´evy-type noise. For this purpose, we exploit the recently introduced weak convergence approach.
AMS 2010 subject classification: 60H15, 35R60, 37L55, 60F10. Key words and phrases: large deviations, heat equation with memory, weak convergence.
1
Introduction
In this work we consider a non-linear heat equation with memory driven by a L´evytype noise. Heat equations with memory have been considered for a long time and their study has recently been published in the monograph [2]. In order to correct the non-physical property of instantaneous propagation for the heat equation, Gurtin and Pipkin introduced in [15] a modified Fourier’s law, which resulted in a heat equation with memory. More precisely, let upt, xq denote the temperature at time t at position ˚
The second author acknowledges funding by K. C. Wong Foundation for a 1-year fellowship at King’s College London
¯ By following the theory developed in [10], [15] and [18], x in a bounded domain O. the temperature upt, xq and the density ept, xq of the internal energy and the heat flux ϕpt, xq are related by ept, xq “ e0 ` b0 upt, xq ϕpt, xq “ ´c0 ∇upt, xq `
żt
¯ for t P R` , x P O,
´8
γprq∇upt ` r, xq dr
¯ for t P R` , x P O.
Here, the constant e0 denotes the internal energy at equilibrium and the constants b0 ą 0 and c0 ą 0 are the heat capacity and thermal conduction. The heat flux relaxation is described by the function γ : p´8, 0s Ñ R` . The energy balance for the system has the form Bt ept, xq “ ´divϕpt, xq ` Frpt, upt, xqq,
where Fr is the nonlinear heat supply which might describe temperature-dependent radiative phenomena. After rescaling the constants and generalizing Fr, we arrive at ż0 ` ˘ ` ˘ γprq∆upt ` r, xq dr. (1.1) Bt upt, xq “ ∆upt, xq ` F t, upt, xq ` div B t, upt, xq ` ´8
This equation models the heat flow in a rigid, isotropic, homogeneous heat conductor with linear memory and is considered for example in [12], [13] and [15]. It is well known that many physical phenomena are better described by taking into account some kind of uncertainty, for instance some randomness or random environment. For this purpose, we assume in this work that the heat supply F˜ in the derivation above contains a stochastic term representing an environmental noise. In order to accommodate a general non-Gaussian environmental noise with possibly discontinuous trajectories, we model the noise by a L´evy-tpe stochastic process. Repeating the above derivation with Fr containing such random noise, we arrive at a stochastically perturbed version of Equation (1.1); see Equation (2.1) in the next section. The main aim of our work is to establish the existence and uniqueness theorem and to investigate the Freidlin-Wentzell large deviation principle of the solutions of the stochastically perturbed equation. For the case of a Gaussian environmental noise, there exists a great amount of literature. For instance, results on the well-posedness of the resulting equation were obtained in [4], [8] and [17]. Long time behaviors of the stochatically perturbed equation for a Gaussian environmental noise were studied in [4], [5], [8] and [9]. The FreidlinWentzell large deviation principle in this situation had been studied in [17] under certain restriction on the the nonlinearity. In our case of a L´evy-type environmental noise, we will obtain the well-posedness of the stochastically perturbed equation by the classical cutting-off method. Due to the 2
appearance of jumps in our setting, the Freidlin-Wentzell large deviation principle are distinctively different to the Gaussian case in [17]. We will use the weak convergence approach introduced in [6] and [7] for the case of Poisson random measures. This approach is a powerful tool to establish the Freidlin-Wentzell large deviation principle for various finite and infinite dimensional stochastic dynamical systems with irregular coefficients driven by a non-Gaussian L´evy noise, see for example [3], [6], [11], [19], [20] and [21]. The main point of our approach is to prove the tightness of some controlled stochastic dynamical systems. For this purpose, we exploit estimates of the controlled stochastic dynamical systems which significantly differ from those in the Gaussian setting. The organization of this paper is as follows. In Section 2, we introduce the assumptions and establish the well-posedness of the stochastically perturbed equation. Section 3 is devoted to establishing the Freidlin-Wentzell’s large deviation principle.
2
Existence of a solution
Let pΩ, F, F :“ tFt utPr0,T s , P q be a filtered probability space with an adapted, stanr be a compensated dard cylindrical Brownian motion W on a Hilbert space H. Let N time homogeneous Poisson random measure on a Polish space X with intensity ν and r to be independent of W . If S is a separable metric space we denote the assume N space of equivalence classes of random variables V c : Ω Ñ S by L0 pΩ, Sq and the Borel σ-algebra by BpSq. Consider the following stochastic heat equation on L2 :“ L2 pOq for a bounded domain O Ď Rd for t P r0, T s: ˙ żtˆ ż0 ` ˘ ` ˘ U ptq “ u0 ` γprq∆U ps ` rq dr ds ∆U psq ` F s, U psq ` div B s, U psq ` 0 ´8 żtż żt ` ˘ ` ˘ r pds, dxq, (2.1) G2 s, U ps´q, x N ` G1 s, U psq dW psq ` 0
U psq “ ̺psq
0
X
for s ď 0,
where the initial condition is given by u0 P L2 and the square Bochner integrable function ̺ : p´8, 0s Ñ L2 . The past dependence is described by the function γ P L1 pR´ , R` q. The non-linear drift is described by the operator F : r0, T s ˆ L2 Ñ L2 . The diffusion coefficients are described by the operators G1 : r0, T s ˆ L2 Ñ L2 and G2 : r0, T s ˆ L2 ˆ X Ñ L2 , where L2 denotes the space of Hilbert-Schmidt operators from H to L2 . We begin with introducing some notations. For p ě 1, we denote by Lp pOq the usual Lp -space over O with the standard norm } ¨ }p . For m P N, let H0m pOq be the usual m-order Sobolev space over O with Dirichlet boundary conditions, and denote 3
its norm and the dual space by } ¨ }2,m and H ´m pOq, respectively. For simplicity, we will write Lp :“ Lp pOq, H0m :“ H0m pOq, H ´m :“ H ´m pOq.
Sobolev’s embedding theorem (see e.g. [1]) guarantees for any q ě 2 and q ˚ :“ q{pq ´ 1q ˚ that H0d ãÑ Lq and Lq ãÑ H ´d . It is well known that the Laplacian ∆ establishes an isomorphism between H01 and H ´1 . Since H01 coincides with the domain of the operator p´∆q1{2 , we will use the following equivalent norm in H01 : }u}2,1 :“ }∇u}2 “ }p´∆q1{2 u}2 . Notice that there exists a constant λ1 ą 0 such that λ1 }u}22 ď }∇u}22 for all u P H01 . ˚ For q ě 2 we define Vq :“ H01 X Lq and Vq˚ :“ H ´1 ` Lq . By identifying L2 with itself by the Riesz representation, we obtain an evolution triple Vq Ă L2 Ă Vq˚ . ˚
That is, for any v P Vq and w “ w1 ` w2 P H ´1 ` Lq we have xv, wyVq ,Vq˚ “ xv, w1 yH01 ,H ´1 ` xv, w2 yLq ,Lq˚ . For the simplicity of notation, when no confusion may arise, we will use the unified notation x¨, ¨y to denote the above dual relations between different spaces. Denote the Lebesgue measures on r0, T s and r0, 8q by LebT and Leb8 , respectively and define νT :“ LebT b ν. We introduce the function space ż ! H :“ h : r0, T s ˆ X Ñ R : exppδh2 pt, xqq νpdxq dt ă 8, Γ ) for all Γ P Bpr0, T s ˆ Xq with νT pΓq ă 8 and for some δ ą 0 . Throughout this work we will assume the following assumptions:
H1: there exist constants c1 , c2 ą 0 and h1 P L1 pr0, T s, R` q such that we have for all v1 , v2 P L2 and t P r0, T s: kBpt, v1 q ´ Bpt, v2 qk22 ď c1 kv1 ´ v2 k22 ,
kBpt, v1 qk22 ď c2 kv1 k22 ` h1 ptq.
H2: there exist constants c3 , c4 , c5 ą 0, q ě 2 and h2 , h3 P L1 pr0, T s, R` q such that we have for all v1 , v2 P L2 and t P r0, T s: xv1 ´ v2 , F pt, v1 q ´ F pt, v2 qy ď c3 kv1 ´ v2 k22 , 4
(2.2)
` ˘ xv1 , F pt, v1 qy ď ´c4 kv1 kqLq ` h2 ptq 1 ` kv1 k22 ,
for any x P
˚ kF pt, v1 qkqLq˚ Lq , y, z P H01 ,
ď
c5 kv1 kqLq
` h3 ptq,
(2.3) (2.4)
the mapping
η ÞÑ xx,F pt, y ` ηzqyLq ,Lq˚ is continuous on r0, 1s.
(2.5)
H3: there exist c6 ą 0, h4 P L1 pr0, T s, R` q such that we have for all v1 , v2 P L2 and t P r0, T s: kG1 pt, v1 q ´ G1 pt, v2 qk2L2 ď c6 kv1 ´ v2 k22 , ˘ ` kG1 pt, v1 qk2L2 ď h4 ptq 1 ` kv1 k22 .
(2.6)
kG2 pt, v1 , xq ´ G2 pt, v2 , xqk2 ď h5 pt, xq kv1 ´ v2 k2 , ` ˘ kG2 pt, v1 , xqk2 ď h6 pt, xq 1 ` kv1 k2 .
(2.8)
(2.7)
H4: there exist h5 , h6 P L2νT pr0, T s ˆ X, Rq X H such that we have for all v1 , v2 P L2 , x P X and t P r0, T s: (2.9)
Our main theorem in this section guarantees the existence of a unique solution of Equation (2.1). Theorem 2.1. Assume (H1)-(H4). Then for every` pu0 , ̺q P L2 ˆ L2 pR´ , H01 q, there˘ exists a unique F-adapted stochastic process U P L0 Ω, Dpr0, T s, L2 q X L2 pr0, T s, H01 q satisfying Equation (2.1) and ff « żT żT q 2 2 }U ptq}q dt ă 8. }U ptq}2,1 ds ` E sup }U ptq}2 ` tPr0,T s
0
0
Proof. As the proof is rather standard, see for example [17], we suppress some details. Step 1. As ν is σ-finite on the Polish space X, there exist measurable subsets Km Ò X satisfying νpKm q ă 8 for all m P N. For each m P N define the function Um,0 ” 0 and consider recursively the equations żtˆ ` ˘ Um,n ptq :“ u0 ` ∆Um,n psq ` F s, Um,n psq 0 ˙ ż0 ` ˘ γprq∆Um,n´1 ps ` rq dr ds (2.10) ` div B s, Um,n´1 psq ` `
żt 0
´8
`
˘ G1 s, Um,n psq dW psq ` 5
żtż 0
Km
` ˘ r pds, dxq. G2 s, Um,n ps´q, x N
As in the proof of [17, Th.3.2] it follows that there exists an F-adapted stochastic process ˘ ` Um,n P L0 Ω, Dpr0, T s, L2 q X L2 pr0, T s, H01 q satisfying (2.10) for each m, n P N. In a first step we show that there exists T0 ą 0 and a constant C ą 0 such that for all m, n P N we have: ff « ż T0 ż T0 q 2 2 (2.11) kUm,n psqkLq ds ď C. kUm,n psqk2,1 ds ` E sup kUm,n ptqk2 ` tPr0,T0 s
0
0
For this purpose, we apply Itˆo’s formula to obtain żt 2 2 kUm,n ptqk2 “ ku0 k2 ´ 2 kUm,n psqk22,1 ds 0 żtA Um,n psq, F ps, Um,n psqq `2 0 ż0 E ` ˘ γprq∆Um,n´1 ps ` rq dr ds ` div B s, Um,n´1 psq ` ´8
żt ˘ ` 2 xUm,n psq, G1 s, Um,n psq y dW psq ` kG1 ps, Um,n psqqk2L2 ds 0 0 żtż ` ˘ r pds, dxq `2 xUm,n ps´q, G2 s, Um,n ps´q, x y N żt
`
0
`
żtż 0
Km
Km
` ˘
G2 s, Um,n ps´q, x 2 N pds, dxq. 2
(2.12)
It follows from [17, Le.3.1] and Cauchy-Schwartz inequality that there exists a constant a ą 0 such that ż0 ż t A E ` ˘ γprq∆Um,n´1 ps ` rq dr ds 2 Um,n psq, div B s, Um,n´1 psq ` ´8 0 żt żt żt 2 2 2 kUm,n psqk2,1 ds ` 2δt ď kUm,n´1 psqk2,1 ds ` a ` a kUm,n´1 psqk22 ds, (2.13) 0
where δt :“
0
0
ş0
´t |γpsq|ds.
żt 0
Assumption (2.3) guarantees that
xUm,n psq, F ps, Um,n psqqy ds żt żt ` ˘ q ď ´c4 kUm,n psqkLq ds ` h2 psq 1 ` kUm,n psqk22 ds, 0
(2.14)
0
and Assumption (2.7) guarantees that żt żt ` ˘ 2 h4 psq 1 ` kUm,n psqk22 ds. kG1 ps, Um,n psqqkL2 ds ď 0
0
6
(2.15)
By applying (2.13) - (2.15) to equation (2.12) we obtain żt żt 2 2 kUm,n ptqk2 ` kUm,n psqk2,1 ds ` c4 kUm,n psqkqLq ds 0 0 żt ` ˘` ˘ ď ku0 k22 ` 2h2 psq ` h4 psq 1 ` kUm,n psqk22 ds 0 żt żt 2 2 ` 2δt kUm,n´1 psqk2,1 ds ` a ` a kUm,n´1 psqk22 ds 0 0 żt ` ˘ ` 2 xUm,n psq, G1 s, Um,n psq y dW psq 0 żtż ` ˘ r pds, dxq xUm,n ps´q, G2 s, Um,n ps´q, x y N `2 0
`
żtż 0
Km
Km
` ˘
G2 s, Um,n ps´q, x 2 N pds, dxq. 2
(2.16)
By applying Burkholder’s inequality and Young’s inequality, we obtain that for every r ą 0 and κ1 P p0, 1q there exists a constant cκ1 ą 0 such that « ż t ff ` ˘ E sup xUm,n psq, G1 s, Um,n psq y dW psq tPr0,rs 0 ff « „ż r 2 2 ď κ1 E sup kUm,n psqk2 ` cκ1 E kG1 ps, Um,n psqkL2 ds sPr0,rs
ď κ1 E
«
sup
sPr0,rs
kUm,n psqk22
ff
0
` cκ1 E
„ż r 0
` 2˘ h4 psq 1 ` kUm,n psqk2 ds ,
(2.17)
` ˘ h6 psq 1 ` kUm,n psqk22 ds ,
(2.18)
where we used Assumption (2.7) in the last equality. Similarly, we conclude from Burkholder’s inequality, Young’s inequality and Assumption (2.9), that for each r ą 0 and κ2 P p0, 1q there exists a constant cκ2 ą 0 such that « ż t ż ff ` ˘ r pds, dxq xUm,n ps´q, G2 s, Um,n ps´q, x y N E sup tPr0,rs
0
ď κ2 E
«
Km
sup
sPr0,rs
kUm,n psqk22
ff
` cκ2 E
„ż r 0
ş with h6 psq :“ X h26 ps, xq νpdxq for all s P r0, T s. Another application of Assumption (2.9) implies for each r P r0, T s that „ż r ż
` ˘
G2 s, Um,n ps´q, x 2 N pds, dxq E 2 0
Km
7
ďE
„ż r 0
` 2˘ h6 psq 1 ` kUm,n psqk2 ds .
(2.19)
By choosing κ1 “ κ2 :“ 41 and applying (2.17) - (2.19) to inequality (2.16) we obtain for each r P r0, T s that « ff „ż r „ż r q 2 2 1 kUm,n psqkLq ds kUm,n psqk2,1 ds ` c4 E sup kUm,n ptqk2 ` E 2E tPr0,rs
ď
ku0 k22 `
`
żr 0
2δr2 E
„ż r 0
0
kUm,n´1 psqk22,1
0
ds ` a ` aE
ı¯ ” hpsq 1 ` E kUm,n psqk22 ds, ´
„ż r
kUm,n´1 psqk22 ds
0
(2.20)
where hpsq :“ 2h2 psq`p1`cκ1 qh4 psq`p1`cκ2 qh6 psq. Define the functions αm N : r0, T s Ñ R and βNm : r0, T s Ñ R by « ff „ż r 2 m m αN prq :“ sup E sup kUm,n ptqk2 , βN prq :“ sup E kUm,n ptqk22,1 dt . nďN
nďN
tPr0,rs
Inequality (2.20) implies for each r P r0, T s: 1 m 2 αN prq
`
m βN prq
2
ď ku0 k `
m 2δr2 βN prq
`a`
żr 0
hpsq ds `
0
żr 0
`
˘ a ` hpsq αm N psq ds.
By choosing T0 P r0, T s such that 2δT2 0 “ 1{2, we conclude from Gronwall’s inequality that ff « ż T0 kUm,n psqk22,1 ds ď C E sup kUm,n ptqk22 ` tPr0,T0 s
0
for a constant C ą 0. Applying this to inequality (2.20) completes the proof of (2.11). Step 2. For m P N and n1 , n2 P N define Γm n1 ,n2 ptq :“ Um,n1 ptq ´ Um,n2 ptq. By similar arguments as in Step 1 and by Gronwall’s lemma we obtain ff # « + „ż T0
m
2
m
2
Γn ,n ptq dt “ 0. lim E sup Γn ,n ptq ` E n1 ,n2 Ñ`8
1
tPr0,T0 s
2
2
1
0
2
2,1
˘ ` Hence, there exists a adapted process Um P L0 Ω, Dpr0, T s, L2 q X L2 pr0, T s, H01 q for each m P N such that « ff „ż T0 2 2 kUm,n psq ´ Um psqk2,1 ds “ 0, lim E sup kUm,n psq ´ Um psqk2 “ lim E nÑ8
sPr0,T0 s
nÑ8
8
0
and, due to (2.11), satisfying ff « „ż T0 „ż T0 q 2 2 kUm ptqkLq dt ď C. kUm ptqk2,1 dt ` E E sup kUm ptqk2 ` E tPr0,T0 s
(2.21)
0
0
Taking the limit in (2.10) as n Ñ 8 shows that Um is the unique solution of the equation żtˆ Um ptq “ u0 ` ∆Um psq ` F ps, Um psqq 0 ˙ ż0 ` ˘ γprq∆Um ps ` rq dr ds ` div B s, Um psq ` `
żt
G1 ps, Um psqq dW psq `
0
´8 żtż 0
Km
r pdx, dsq. G2 ps, Um ps´q, xq N
(2.22)
Step 3. For m, n P N with n ą m define Γn,m ptq :“ Un ptq ´ Um ptq. Applying Itˆo’s formula and similar arguments as in Step 1 result in żt 2 2 kΓn,m ptqk2 ` p1 ´ 2δt q kΓn,m psqk22,1 ds 0 żt żt 2 ď c kΓn,m psqk2 ds ` 2 xΓn,m psq, G1 ps, Un psqq ´ G1 ps, Um psqqy dW psq 0 0 żtż r pdx, dsq xΓn,m ps´q, G2 ps, Un ps´q, xq ´ G2 ps, Um ps´q, xqy N `2 0
`
0
`2 `
Km
żtż
Km
żtż
KnzKm
0
żtż 0
kG2 ps, Un ps´q, xq ´ G2 ps, Um ps´q, xqk22 N pdx, dsq
KnzKm
r pdx, dsq xΓn,mps´q, G2 ps, Un ps´q, xqy N
kG2 ps, Un ps´q, xqk22 N pdx, dsq.
(2.23)
It follows from (2.6) by Burkholder’s inequality and Young’s inequality that for each κ1 P p0, 1q there exists a constant cκ1 ą 0 such that ¸ ˜ ˇ ˇ żt ˇ ˇ E sup ˇ2 xΓn,m psq, G1 ps, Un psqq ´ G1 ps, Um psqqydW psqˇ tPr0,ls
ď cE
0
ˆż l 0
kΓn,mpsqk22 kG1 ps, Un psqq ´ G1 ps, Um psqqk2L2 pH,L2 q ds 9
˙1{2
ď κ1 E
˜
sup tPr0,ls
kΓn,m ptqk22
¸
` Cκ 1 E
ˆż l
kΓn,m psqk22 ds
0
˙
(2.24)
,
In the same way it follows from (2.8) that for each κ2 , κ3 P p0, 1q there exist constants cκ2 , cκ3 ą 0 such that ˜ ˇ żtż ˇ¸ ˇ ˇ r ˇ xΓn,m ps´q, G2 ps, Un ps´q, xq ´ G2 ps, Um ps´q, xqy N pdx, dsqˇˇ E sup ˇ2 0 Km tPr0,ls ˜ ¸ ˆż ż ˙ sup kΓn,m ptqk22
ď κ2 E
tPr0,ls
l
` cκ2 E
0
Km
h25 ps, xq kΓn,m psqk22 νpdxqds , (2.25)
and from (2.9) that ¸ ż ż ˜ t r pdx, dsq E sup 2 xΓn,mps´q, G2 ps, Un ps´q, xqyN tPr0,ls 0 KnzKm ˜ ¸ ¸ ˜ ´ ¯ żlż 2 2 ď κ3 E sup kΓn,m ptqk2 ` cκ3 E sup kUn psqk2 ` 1 tPr0,ls
sPr0,ls
0
(2.26)
KnzKm
h26 ps, xq νpdxqds.
Another application of (2.8) and (2.9) imply ˙ ˆż t ż 2 kG2 ps, Un ps´q, xq ´ G2 ps, Um ps´q, xqk2 N pdx, dsq E 0 Km ˆż t ż ˙ 2 2 h5 ps, xq kΓn,mpsqk2 νpdxqds , ďE 0
(2.27)
Km
and E
˜ż ż t 0
KnzKm
ďE
¸
kG2 ps, Un ps´q, xqk22 N pdx, dsq ˜
¸ ¯ żtż ´ sup kUn psqk22 ` 1 0
sPr0,ts
KnzKm
h26 ps, xq νpdxqds.
Define the functions αn,m : r0, T s Ñ R and βn,m : r0, T s Ñ R by ¸ ˜ ˆż αn,m plq :“ E
sup
tPr0,ls
kΓn,m ptqk22
l
βn,m plq :“ E
,
0
kΓn,mpsqk22,1 ds
(2.28)
˙
.
By choosing κ1 “ κ2 “ κ3 “ 1{6 and recalling 2δT2 0 “ 12 , we obtain by applying (2.24) – (2.28) to the inequality (2.23) that 1 2 αn,m pT0 q
` 12 βn,m pT0 q 10
ď
ż T0 ˆ 0
˙
ż
h25 ps, xq νpdxq
αn,m psq ds c ` cκ1 ` p1 ` cκ2 q Km ¸ ˜ ´ ¯ ż T0 ż 2 h26 ps, xqνpdxqds. ` p1 ` cκ3 qE sup kUn psqk2 ` 1 sPr0,T0 s
KnzKm
0
Applying Gronwall’s inequality and using together with (2.21) implies
şT0 ş 0
KnzKm
h26 ps, xqνpdxqds Ñ 0 as m, n Ñ 8
lim αn,m pT0 q ` βn,m pT0 q “ 0.
n,mÑ8
Hence, there exists an such that « lim E
nÑ8
sup kUm psq ´
sPr0,T0 s
U psqk22
which, due to (2.21), satisfies « ff „ż E
sup
tPr0,T0 s
kU ptqk22
`
˘
F-adapted process U P L0 Ω, Dpr0, T0 s, L2 q X L2 pr0, T0 s, H01 q
`E
T0 0
ff
“ lim E nÑ8
kU ptqk22,1 dt
„ż T0 0
`E
kUm psq ´
„ż T0 0
U psqk22,1 ds
kU ptqkqLq
dt ď C.
“ 0,
(2.29)
Taking limits in (2.22) shows that that U is the unique solution of 2.1 on the interval r0, T0 s. Step 4. By repeating the above arguments we obtain the existence of a unique solution of (2.1) on the interval rT0 , 2T0 s which finally leads to the completion of the proof by further iterations.
3
Large Deviation Principle
Recall that H is a separable Hilbert space with an orthonormal basis tui uiPN and assume that X is a locally compact Polish space with a σ-finite measure ν defined on BpXq. Let S be a locally compact Polish space. The space of all Borel measures on S is denoted by M pSq and the set of all µ P M pSq with µpKq ă 8 for each compact set K Ď S is denoted by MF C pSq. We endow MF C pSq şwith the weakest topology such that for each f P Cc pSq the mapping µ P MF C pSq Ñ S f psqµpdsq is continuous. This topology is metrizable such that MF C pSq is a Polish space, see [7] for more details.
In this section we specify the underlying probability space pΩ, F , F :“ tF t utPr0,T s , P q in the following way: ` ˘ ` ˘ F :“ BpΩq. Ω :“ C r0, T s; H ˆ MF C r0, T s ˆ X ˆ r0, 8q , 11
We introduce the functions ` ˘ W : Ω Ñ C r0, T s; H ,
W pα, βqptq “
` ˘ N : Ω Ñ MF C r0, T s ˆ X ˆ r0, 8q ,
8 ÿ
i“1
xαptq, ui yui ,
N pα, βq “ β.
Define for each t P r0, T s the σ-algebra ` ` ˘ ` ˘(˘ Gt :“ σ W psq, N pp0, ss ˆ Aq : 0 ď s ď t, A P B X ˆ r0, 8q .
For a given ν P MF C pXq, it follows from [16, Sec.I.8] that there exists a unique probability measure P on pΩ, BpΩqq such that: (a) W is a cylindrical Brownian motion in H; (b) N is a Poisson random measure on Ω with intensity measure LebT b ν b Leb8 ; (c) W and N are independent. We denote by F :“ tFt utPr0,T s the P -completion of tGt utPr0,T s and by P the F-predictable σ-field on r0, T s ˆ Ω. Define R :“ tϕ : r0, T s ˆ X ˆ Ω Ñ r0, 8q : pP b BpXqqzBr0, 8q-measurableu . For ϕ P R, define a counting process N ϕ on r0, T s ˆ X by ż ϕ 1r0,ϕps,x,¨qs prq N pds, dx, drq, N pp0, ts ˆ Aqp¨q “ p0,tsˆAˆp0,8q
for t P r0, T s and A P BpXq. For each f P L2 pr0, T s, Hq, we introduce the quantity ż 1 T Q1 pf q :“ kf psqk2H ds, 2 0
and we define for each m P N the space ! ) S1m :“ f P L2 pr0, T s, Hq : Q1 pf q ď m .
Equiped with the weak topology, S1m is a compact subset of L2 pr0, T s, Hq. We will throughout consider S1m endowed with this topology. By defining the function ℓ : r0, 8q Ñ r0, 8q,
ℓpxq “ x log x ´ x ` 1 12
we introduce for each measurable function g : r0, T s ˆ X Ñ r0, 8q the quantity ż ` ˘ ℓ gps, xq ds νpdxq. Q2 pgq :“ r0,T sˆX
Define for each m P N the space ! ) S2m :“ g : r0, T s ˆ X Ñ r0, 8q : Q2 pgq ď m .
A function g P S2m can be identified with a measure gˆ P MF C pr0, T s ˆ Xq, defined by ż gps, xq ds νpdxq for all A P Bpr0, T s ˆ Xq. (3.1) gˆpAq “ A
This identification induces a topology on S2m under which S2m is a compact space, see the Appendix of [6]. Throughout, we use this topology on S2m .
On the probability space pΩ, BpΩq, F , F, P q carrying the cylindrical Brownian mor ε´1 pds, dxq :“ N ε´1 pds, dxq ´ tion W and the compensated Poisson random measure N ε´1 νpdxq ds we consider for each ε ą 0 the following stochastic heat equation for t P r0, T s: ż0 żt´ ¯ ` ˘ ` ˘ γprq∆Uε ps ` rq dr ds ∆Uε psq ` div B s, Uε psq ` F s, Uε psq ` Uε ptq “ u0 ` 0
? ` ε
żt 0
`
˘ G1 s, Uε psq dW psq ` ε
Uε psq “ ̺psq for s ă 0.
żtż 0
´8
X
`
˘ ε´1 r pds, dxq, (3.2) G2 s, Uε ps´q, x N
The initial condition is given by pu0 , ̺q P L2 ˆ L2 pR´ , H01 q. Theorem 2.1 guarantees that for each ε ą 0 there exists a unique solution Uε :“ Uε pu0 , ̺q with trajectories in the space D :“ Dpr0, T s, L2 q X L2 pr0, T s, H01 q. Theorem 3.1. Under the assumption (H1)-(H4), the solutions tUε : ε ą 0u of (3.2) satisfy a large deviation principle on D with rate function I : D Ñ r0, 8s, where Ipξq :“ inf tQ1 pf q ` Q2 pgq : ξ “ upf, gq, f P S1m , g P S2m and m P Nu ,
and u “ upf, gq P D solves the following deterministic partial differential equation: ż0 żt´ ¯ ` ˘ ` ˘ γprq∆ups ` rq dr ds ∆upsq ` div B s, upsq ` F s, upsq ` uptq “ u0 ` 0
`
żt 0
` ˘ G1 s, upsq f psq ds `
upsq “ ̺psq for s ă 0.
żtż 0
´8
X
˘ G2 s, upsq, x gps, xq ´ 1 νpdxq ds,
13
`
˘`
(3.3)
Proof. Define the space C :“ Cpr0, T s; HqˆMF C pr0, T sˆXq. Using the correspondence (3.1), we define a function G 0 : C Ñ D such that ˙ ˆż ¨ G0 f psq ds, gˆ “ upf, gq for all f P S1m , g P S2m , m P N, 0
where upf, gq is the unique solution of (3.3). Theorem 2.1 implies that for each ε ą 0 there exists a mapping G ε : C Ñ D such that Gε
`?
ε W, εN ε D
´1
˘
D
“ Uε ,
where Uε is the solution of (3.2) (and “ denotes equality in distribution). Define for each m P N a space of stochastic processes on Ω by S1m :“ tϕ : r0, T s ˆ Ω Ñ H :
F-predictable and ϕp¨, ωq P S1m for P -a.a. ω P Ωu.
Let pKn qnPN be a sequence of compact sets Kn Ď X with Kn Õ X. For each n P let # ! ) r n1 , ns, if x P Kn , Rb,n “ ψ P R : ψpt, x, ωq P for all pt, ωq P r0, T s ˆ Ω , t1u, if x P Knc . and let Rb “
Ť8
n“1 Rb,n .
N,
Define for each m P N a space of stochastic process on Ω by
S2m :“ tψ P Rb : ψp¨, ¨, ωq P S2m for P -a.a. ω P Ωu.
According to Theorem 2.4 in [6] and Theorem 4.2 in [7], our claim is established once we have proved: (C1) if pfn qnPN Ď S1m converges to f P S1m and pgn qnPN Ď S2m converges to g P S2m for some m P N, then ´ż ¨ ´ż ¨ ¯ ¯ f psq ds, gˆ in D. G0 fn psq ds, gˆn Ñ G 0 0
0
(C2) if pϕε qεą0 Ď S1m converges weakly to ϕ P S1m and pψε qεą0 Ď S2m converges weakly to ψ P S2m for some m P N, then ż¨ ´ż ¨ ¯ ¯ ´ 0 ε´1 ψε ε ? ˆ converges weakly to G ϕpsq ds, ψ in D. G ε W ` ϕε psq ds, εN 0
0
In the sequel, we will prove Condition (C2). The proof of Condition (C1) follows analogously. 14
m Lemma 3.2. Let pϕε qεą0 Ď S1m and pψε qεą0 ˘ for each ` Ď S2 for2 some2 m P N. 1Then 0 ε ą 0 there exists a unique solution Vε P L Dpr0, T s, L q X L pr0, T s, H0 q of ż0 żt´ ¯ ` ˘ ` ˘ γprq∆Vε ps ` rq dr ds ∆Vε psq ` div B s, Vε psq ` F s, Vε psq ` Vε ptq “ u0 ` ´8
0
żt
żt ` ˘ ` ˘ ? ` G1 s, Vε psq ϕε psq ds ` ε G1 s, Vε psq dW psq 0 0 żtż ˘ ` ˘ ` ε´1 ψε pds, dxq ´ ε´1 ds νpdxq , G2 s, Vε ps´q, x N `ε 0
(3.4)
X
Vε psq “ ̺psq for s ă 0.
Moreover, the solution Vε has the same distribution as G ε
`?
ε W`
ş¨
0 ϕε psq ds, εN
ε´1 ψε
˘
.
Proof. The proof can be accomplished by following [6, p.543] and applying results in [7, Se.A.2]. Lemma 3.3. For each m P N there exists a constant cm ą 0 such that for any ε P p0, 1q: ¸ ˜ ż ż E
sup kVε ptqk22 `
tPr0,T s
T
0
kVε ptqk22,1 dt `
T
0
kVε ptqkqq dt
ď cm .
Proof. Lemma 3.4 in [6] guarantees that for each m P N there exists a constant cm ą 0 such that ż Tż ż Tż 2 hi ps, xq|gps, xq ´ 1| νpdxq ds ď cm , (3.5) hi ps, xqgps, xq νpdxq ds ` sup sup gPS2m 0
gPS2m 0
X
X
for i “ 5, 6. Using this inequality, the proof can be accomplished as the proof of (2.11). For each ε ě 0 let Yε be the unique solution of the SPDE żt żt ? Yε ptq “ ∆Yε psq ds ` ε G1 ps, Vε psqq dW psq 0 0 żtż r ε´1 ψε pdx, dsq for all t P r0, T s, `ε G2 ps, Vε ps´q, xq N 0
X
and let Zε be the unique solution of the random PDE żtż żt G2 ps, Vε psq, xqpψε ps, xq ´ 1q νpdxq ds ∆Zε psq ds ` Zε ptq “ 0
0
X
Furthermore, we define the difference
Jε ptq “ Vε ptq ´ Yε ptq ´ Zε ptq 15
for all t P r0, T s.
for all t P r0, T s.
Lemma 3.4. For each m P N there exists a constant cm independent of ε such that (a) Yε obeys E
´
sup tPr0,T s
kYε ptqk22
¯
`E
ˆż T 0
kYε ptqk22,1
dt
˙
ď ε cm ;
(b) tZε , ε P p0, 1qu is tight in Cpr0, T s, L2 q and obeys ˙ ˆż T ¯ ´ 2 2 kZε ptqk2,1 dt ď cm ; E sup kZε ptqk2 ` E tPr0,T s
0
(c) tJε , ε P p0, 1qu is tight in Cpr0, T s, H´d q X L2 pr0, T s, L2 q and obeys ˙ ˆż T ¯ ´ 2 2 kJε ptqk2,1 dt ď cm . E sup kJε ptqk2 ` E tPr0,T s
0
Proof. Part (a). Itˆo’s formula implies żt 2 kYε ptqk2 ` 2 kYε psqk22,1 ds 0 żt ? xG1 ps, Vε psqq, Y ε psqy dW psq “2 ε 0 ż tż r ε´1ψε pdx, dsq xG2 ps, Vε ps´q, xq, Yε ps´qy N ` 2ε 0 X żt żtż ´1 2 2 kG2 ps, Vε ps´q, xqk22 N ε ψε pdx, dsq ` ε kG1 ps, Vε psqqkL2 ds ` ε 0
0
X
“: I1 ptq ` I2 ptq ` I3 ptq ` I4 ptq.
It follows from (2.7) by Burkholder’s inequality and Young’s inequality that for each κ1 P p0, 1q there exists a constant cκ1 ą 0 such that ˇ¯ ˇ ¯ ¯ ´ ´ ´ ˇ ˇ E sup ˇI1 ptqˇ ď κ1 E sup kYε ptqk22 ` ε cκ1 E 1 ` sup kVε ptqk22 . tPr0,T s
tPr0,T s
tPr0,T s
Analogously, one obtains from (2.9) that for each κ2 P p0, 1q there exists a constant cκ2 ą 0 such that ˇ¯ ˇ ¯ ´ ´ ˇ ˇ E sup ˇI2 ptqˇ ď κ2 E sup kYε ptqk22 tPr0,T s
tPr0,T s
´
` ε cκ2 E 1 ` sup
tPr0,T s
kVε ptqk22 16
żTż ¯ ¯ ´ 2 h6 ps, xqgps, xq νpdxq ds . ¨ sup gPS2m 0
X
Inequality (2.7) yields that there exits a constant c3 ą 0 such that ¯ ´ ¯ ´ E sup |I3 ptq| ď ε c3 E 1 ` sup kVε ptqk22 . tPr0,T s
tPr0,T s
From inequality (2.9) we conclude that there exits a constant c4 ą 0 such that E
´
¯ ´ż T ż ¯ sup |I4 ptq| ď εE kG2 ps, Vε psq, xqk22 ψε ps, xq νpdxq ds
tPr0,T s
0
X
żTż ¯ ¯ ´ ´ 2 h26 ps, xqχps, xq νpdxq ds . ď εc4 E 1 ` sup kVε ptqk2 ¨ sup χPS2m 0
tPr0,T s
X
Combining all of the above estimates and applying Gronwall’s inequality together with Lemma 3.3 and (3.5) completes the proof of part (a). Part (b). By the chain rule we conclude from (2.9) and (3.5) that there exists a constant c ą 0 with żt 2 kZε ptqk2 ` 2 kZε psqk22,1 ds 0 żtż @ D “ G2 ps, Vε psq, xqpψε ps, xq ´ 1q, Zε psq νpdxq ds 0 X żtż ˇ ˇ` ˘ h6 ps, xqˇψε ps, xq ´ 1ˇ 1 ` kVε psqk2 kZε psqk2 νpdxq ds ď ¸˜ ¸˜ ˜0 X ż ż T
ď
ď
sup kZε psqk2
sPr0,T s
1 2
˜
sup kZ ε psqk22
sPr0,T s
¸
sup p1 ` kVε psqk2 q
sPr0,T s
˜
` c 1 ` sup kVε psqk22 sPr0,T s
sup
gPS2m 0
¸
X
h6 ps, xq|gps, xq ´ 1| νpdxq ds
,
which completes the proof of the second claim due to Lemma 3.3. Define for each M ě 1 and m P N the set of functions "ż G2 p¨, hp¨q, xqpgps, xq ´ 1q νpdxq : AM,m :“ X * 2 2 m h P Dpr0, T s, L q, sup }hpsq}2 ď M, g P S2 . sPr0,T s
Proposition 3.9 of [20] guarantees that the set * "ż ¨ p¨´sq∆ e f psq ds, f P AM,m BM,m “ 0
17
¸
is relatively compact in Cpr0, T s, L2 q. From Lemma 3.3 we conclude that ¯ ´ ¯ ´ ¯ 1 ´ cm P Z ε P BM,m ě P sup kVε ptqk22 ď M ě 1 ´ E sup kVε ptqk22 ě 1 ´ , M M tPr0,T s tPr0,T s which completes the proof of part (b). Part (c). The very definition of Jε yields ˙ żtˆ ż0 Jε ptq “ u0 ` γprq∆Vε ps ` rq dr ds ∆Jε psq ` divBps, Vε psqq ` F ps, Vε psqq ` 0 ´8 żt ` G1 ps, Vε psqqϕε psq ds. 0
By combining Part (a), Part (b) and Lemma 3.3 we conclude that for each m P N there exists a constant cm ą 0 such that ¯ ´ż T ¯ ´ kJε ptqk22,1 dt ď cm . (3.6) E sup kJε ptqk22 ` E tPr0,T s
0
Following the arguments in the proof of Lemma 4.3 in [17] by using (3.6) and Lemma 3.3 implies that there exist α P p0, 1q such that for all ε P p0, 1q we have „ kJε psq ´ Jε ptqk2,´d ă 8. (3.7) E sup |s ´ t|α s,tPr0,T s s‰t
According to Lemma 4.3 in [14], the estimates (3.6) and (3.7) imply that tJε , ε P p0, 1qu is tight in Cpr0, T s, H ´d q X L2 pr0, T s, L2 q, which completes the proof of part (c). Proof of Condition (C2). Step 1: We fix a sequence pϕε qεą0 Ď S1m converging weakly to ϕ P S1m and pψε qεą0 Ď S`2m converging weakly to˘ ψ P S2m for ( some m P N. We conclude from Lemma 3.4 that Yε , Zε , Jε , ϕε , ψε , W, N , ε P p0, 1q is tight in the space ´ ` ˘ ` ` ˘ ´ ` ˘¯ ˘ ` ˘¯ Π :“ D r0, T s, L2 X L2 r0, T s, H01 ˆ C r0, T s, L2 ˆ C r0, T s, H ´d X L2 r0, T s, L2 ` ˘ ` ˘ ˆ S1m ˆ S2m ˆ C r0, T s, H ˆ MF C r0, T s ˆ X ˆ r0, 8q .
Skorohod embedding theorem implies for any sequence pεk q`kPN Ď R` converging to 0, ˘ that there space pΩ1 , F 1˘, P 1 q, a sequence Yk1 , Zk1 , Jk1 , ϕ1k , ψk1 , Wk1 , Nk1 , ( exist a probability ` k P N as well as 0, Z 1 , J 1 , ϕ1 , ψ 1 , W 1 , N 1 defined on this probability space and taking values in Π such that: ˘D ` 1 1 1 1 1 Yk , Zk , Jk , ϕk , ψk , Wk1 , Nk1 “ pYεk , Zεk , Jεk , ϕk , ψk , W, N q for each k P N; (3.8) ˘ ` ˘ ` P 1 -a.s. in Π. (3.9) lim Yk1 , Zk1 , Jk1 , ϕ1k , ψk1 , Wk1 , Nk1 “ 0, Z 1 , J 1 , ϕ1 , ψ 1 , W 1 , N 1 kÑ8
18
(a) Lemma 3.4 together with (3.8) imply that there exists a constant c ą 0 such that E E E
1
1
1
´
´
´
sup tPr0,T s
sup tPr0,T s
sup tPr0,T s
}Yk1 ptq}22
¯
}Zk1 ptq}22 }Jk1 ptq}22
¯
¯
`E
1
¯ }Yk1 ptq}22,1 dt ď εk c;
0
`E `E
´ż T
1
1
´ż T
¯ }Zk1 ptq}22,1 dt ď c;
0
´ż T
¯ }Jk1 ptq}22,1 dt ď c.
0
(3.10) (3.11) (3.12)
Consequently, we conclude from (3.9) that E
1
E1
´
´
sup } Z
tPr0,T s
1
ptq}22
¯
`E
1
´ż T 0
¯ }Z 1 ptq}22,1 dt ď c,
´ż T ¯ ¯ }J 1 ptq}22,1 dt ď c; sup }J 1 ptq}22 ` E 1
tPr0,T s
(3.13) (3.14)
0
(b) In the following we establish that lim E
kÑ8
1
´ż T 0
}Jk1 ptq ´ J 1 ptq}22 dt
¯1{2
“ 0.
(3.15)
For this purpose, let δ ą 0 be arbitrary and define for each k P N and ω 1 P Ω1 the set ( Λk,δ pω 1 q :“ s P r0, T s : }Jk1 ps, ω 1 q ´ J 1 ps, ω 1 q}2 ě δ . Since (3.9) implies `
şT 0
}Jk1 ptq ´ J 1 ptq}22 dt Ñ 0 P 1 -a.s. as k Ñ 8 we obtain
˘ 1 Leb Λk,δ p¨q ď 2 δ
żT 0
}Jk1 ptq ´ J 1 ptq}22 dt Ñ 0 P 1 -a.s. as k Ñ 8.
By applying Fubini’s theorem and Cauchy inequality, we derive E1
ˆż T 0
}Jk1 ptq ´ J 1 ptq}22 dt ż ˜ż “
Ω1
˙1{2
r0,T sXpΛk,δ pω 1 qqc
`
ż
}Jk1 pt, ω 1 q ´ J 1 pt, ω 1 q}22 dt
r0,T sXΛk,δ pω 1 q
}Jk1 pt, ω 1 q ´ 19
1
J pt, ω
1
q}22 dt
¸1{2
P 1 pdω 1 q
(3.16)
ď δT 1{2 ` E 1 ď δT
1{2
`
˜
˜
E
1
˘` ` ˘˘1{2 ` sup }Jk1 ptq}2 ` }J 1 ptq}2 Leb Λk,δ p¨q
tPr0,T s
˜
` ˘ sup }Jk1 ptq}22 ` }J 1 ptq}22
tPr0,T s
¸¸1{2
¸
´ ` ` ˘˘ ¯1{2 E 1 Leb Λk,δ p¨q .
Since δ ą 0 is arbitrary, we complete the proof of (3.15) by applying Lebesgue’s dominated convergence theorem together with (3.16) and using (3.12) and (3.14). (c) Define for each k P N the stochastic processes Sk1 :“ Zk1 ` Jk1 ,
S 1 :“ Z 1 ` J 1 ,
Vk1 :“ Yk1 ` Zk1 ` Jk1 .
Note, that it follows from (3.8) and Vε “ Yε ` Zε ` Jε that Vk1 has the same distribution as Vεk . The convergence (3.9) implies Sk1 Ñ S 1 Vk1
ÑS
P 1 -a.s. in L2 pr0, T s, L2 q,
1
1
2
2
P -a.s. in L pr0, T s, L q.
(3.17) (3.18)
Moreover, since Yk1 Ñ 0 P 1 -a.s. in the Skorkhod space Dpr0, T s, L2 q according to (3.9), we have suptPr0,T s }Yk1 ptq}2 Ñ 0 P 1 -a.s. Consequently,we obtain lim sup }Sk1 ptq ´ S 1 ptq}2H ´d “ 0
P 1 -a.s.,
(3.19)
lim sup }Vk1 ptq ´ S 1 ptq}2H ´d “ 0
P 1 -a.s.
(3.20)
kÑ8 tPr0,T s kÑ8 tPr0,T s
(d) It follows from Lemma 3.3 that there exists a stochastic process V 1 on Ω1 such that ´ ¯ Vk1 Ñ V 1 weakly star in L2 Ω1 , L8 pr0, T s, L2 q , ´ ¯ Vk1 Ñ V 1 weakly in L2 Ω1 , L2 pr0, T s, H01 q , ´ ¯ Vk1 Ñ V 1 weakly in L2 Ω1 , Lq pr0, T s, Lq q , V 1 “ S1
P 1 bLeb-a.s.
In particular, it follows by (3.13), (3.14) and Lemma 3.3 that ¸ ˜ ˙ ˆż T ˙ ˆż T 1 1 2 1 1 2 1 q 1 }S ptq}2,1 dt ` E sup }S ptq}2 ` E }S ptq}q dt ď C. E tPr0,T s
0
0
20
(3.21)
Step 2: We will prove that S 1 solves equation (3.3) for f “ ϕ1 and g “ ψ 1 , that is xw, S 1 ptqy “ xw, u0 y ż0 żtA E ` ˘ ` ˘ γprq∆S 1 ps ` rq dr ds w, ∆S 1 psq ` div B s, S 1 psq ` F s, S 1 psq ` ` 0
żtA E ` ˘ w, G1 s, S 1 psq ϕ1 psq ds ` 0 żtż A ` ˘` ˘E w, G2 s, S 1 psq, z ψ 1 ps, zq ´ 1 νpdzq ds, ` 0
1
´8
(3.22)
X
S psq “ ̺psq for s ă 0,
for each w P H0d . For this purpose note that the very definition of Sk1 yields xw, Sk1 ptqy ´ xw, u0 y ż0 żtA E γprq∆Vk1 ps ` rq dr ds w, ∆Sk1 psq ` div Bps, Vk1 psqq ` F ps, Vk1 psqq ´ ´ ´8 0 żt @ D ´ w, G1 ps, Vk1 psqqϕ1k psq ds 0 żtż D @ “ (3.23) w, G2 ps, Vk1 psq, zqpψk1 ps, zq ´ 1q νpdzq ds. 0
X
It follows from (3.9), (3.11), (3.12), (3.17)–(3.20) by the same method as applied in [17, p.5234], that there is a subsequence such that the left hand side of (3.23) converges P 1 -a.s. to xw, S 1 ptqy ´ xw, u0 y żt ż0 D @ 1 1 1 γprq∆S 1 ps ` rq dr ds ` w, ∆S psq ` div Bps, S psqq ` F ps, S psqq ` 0 ´8 żt ` xw, G1 ps, S 1 psqqϕ1 psqy ds. 0
For taking the limit on the right hand side in (3.23), we define(for each δ ą 0 and ω 1 P Ω1 the set Λk,δ pω 1 q :“ s P r0, T s, }Vk1 ps, ω 1 q ´ S 1 ps, ω 1 q}2 ě δ and conclude from (3.18) as in (3.16) that LebpΛk,δ p¨qq Ñ 0 P 1 -a.s. Consequently, we obtain ˜ˇ ż ż ˇ t ` ˘ D @ 1 ˇ w, G2 s, Vk1 psq, x pψk1 ps, xq ´ 1q νpdxq ds E ˇ ˇ 0 X ˇ¸ żtż ˇ ` ˘ D @ ˇ ´ w, G2 s, S 1 psq, x pψk1 ps, xq ´ 1q νpdxq dsˇ ˇ 0 X 21
ď }w}2 E 1 ˜
˜ż ż t 0
ď }w}2 δ sup
}Vk1 psq ´ S 1 psq}2 h5 ps, xq|ψk1 ps, xq ´ 1| νpdxq ds
X
żTż
gPS2m 0
X
¸
h5 ps, xq|gps, xq ´ 1| νpdxq ds
¸ ¯¯2 ¯1{2 ´ ¯1{2 ´ ´ ˘ 2 , E 1 sup }Vk1 psq}2 ` }S 1 psq}2 ` E Rδ,k ´
` 1
(3.24)
sPr0,T s
where Rδ,k :“ sup
ż
gPS m r0,T sXΛk,δ
ż
X
h5 ps, xq|gps, xq ´ 1| νpdxq ds.
Since LebpΛk,δ p¨qq Ñ 0 P 1 -a.s. as k Ñ 8, we conclude from [6, Le.3.4] or [20, Eq.(3.19)] that Rδ,k Ñ 0 in P 1 -probability as k Ñ 8. As δ ą 0 is arbitrary, we obtain from (3.24) together with [6, Le.3.11] that the right hand side in (3.23) converges in P 1 -probability to żtż @ D w, G2 ps, S 1 psq, xqpψ 1 ps, xq ´ 1q νpdxq ds, 0
X
which varifies S 1 as a solution of (3.22). Step 3. In the last step we establish that Lk :“ Vk1 ´ S 1 obeys sup tPr0,T s
}Lk ptq}22
`
żT 0
}Lk ptq}22,1 dt Ñ 0 in P 1 -probability for k Ñ 8.
(3.25)
Recall that Vk1 has the same distribution as Vεk and S 1 is a solution of (3.3) according to Step 2. Thus, once we will have established (3.25), it shows that Condition (C2) is satisfied. Itˆo’s formula implies for each t P r0, T s that }Lk ptq}22
“ ´2
żt 0
}Lk psq}22,1 ds
`
9 ÿ
i“1
Ik,i ptq,
(3.26)
where ż0 żtA E γprq∆Lk ps ` rq dr ds, Lk psq, Ik,1 ptq “ 2
(3.27)
Ik,2 ptq “ 2
(3.28)
0 żtA 0
´s
˘E ` Lk psq, div Bps, Vk1 psqq ´ Bps, S 1 psqq ds, 22
Ik,3 ptq “ 2 Ik,4 ptq “ 2
żt
@
0 żt @
0 żtż
D Lk psq, F ps, Vk1 psqq ´ F ps, S 1 psqq ds,
(3.29)
D Lk psq, G1 ps, Vk1 psqqϕ1k psq ´ G1 ps, S 1 psqqϕ1 psq ds, @
Lk psq, G2 ps, Vk1 psq, xqpψk1 ps, xq ´ 1q ´ G2 ps, S 1 psq, xq 0 X D pψ 1 ps, xq ´ 1q νpdxq ds, żt D @ ? Lk psq, G1 ps, Vk1 psqq dWk1 psq, Ik,6 ptq “ 2 εk 0 żt Ik,7 ptq “ εk }G1 ps, Vk1 psqq}22 ds, 0 żtż ´1 1 r 1ε ψk pdx, dsq xLk psq, G2 ps, Vk1 ps´q, xqy N Ik,8 ptq “ 2εk k 0 X żtż 1 ε´1 ψk Ik,9 ptq “ ε2k pdx, dsq. }G2 ps, Vk1 psq, xq}22 Nk1 Ik,5 ptq “ 2
0
(3.30)
(3.31)
(3.32) (3.33) (3.34) (3.35)
X
By exploiting similar arguments as in [17, p.5235], we derive that P 1 -a.s. we have ˜ ˙2 ¸ ż t ˆż 0 }Lk psq}22,1 ds, (3.36) |γprq| dr Ik,1 ptq ď 1 ` Ik,2 ptq ď
1 2
żt
0 żt
´t
0
}Lk psq}22,1 ds ` c
żt 0
}Lk psq}22 ds,
}Lk psq}22 ds, 0 żt ` 1 ˘ }Vk psq}22 ` 1 ds Ik,7 ptq ď cεk
Ik,3 ptq ď c
(3.37) (3.38) (3.39)
0
where c ą 0 denotes a generic constant. Our Assumption (H3) yields Ik,4 ptq ż t ˇA Eˇ ˘ ` ˇ ˇ ď 2 ˇ Lk psq, G1 ps, Vk1 psqq ´ G1 ps, S 1 psqq ϕ1k psq ˇ ds 0 ż t ˇA ˘ Eˇˇ ` ˇ ` 2 ˇ Lk psq, G1 ps, S 1 psqq ϕ1k psq ´ ϕ1 psq ˇ ds 0 ¸ż ˜ ż t
ďc
0
}Lk psq}22 }ϕ1k psq}H ds `
1 ` sup }S 1 psq}2 sPr0,ts
23
t
0
˘ ` }Lk psq}2 }ϕ1k psq}H ` }ϕ1 psq}H ds
ďc
żt 0
}Lk psq}22 }ϕ1k psq}H
˜
1
ds ` c m 1 ` sup }S psq}2 sPr0,ts
¸ ˆż
t
0
}Lk psq}22 ds
˙1{2
. (3.40)
By applying our assumption (H4) we obtain P 1 -a.s. that |Ik,5 ptq| żtż }Lk psq}2 }G2 ps, Vk1 psq, xq ´ G2 ps, S 1 psq, xq}2 |ψk1 ps, xq ´ 1| νpdxq ds ď2 0 X żtż ` ˘ }Lk psq}2 }G2 ps, S 1 psq, xq}2 |ψk1 ps, xq ´ 1| ` |ψ 1 ps, xq ´ 1| νpdxq ds `2 0 X żtż ¯ ´ ď2 }Lk psq}22 h5 ps, xq|ψk1 ps, xq ´ 1| νpdxq ds ` 2 1 ` sup }S 1 psq}2 Kptq, (3.41) 0
X
sPr0,ts
where we use the notation żtż ˘ ` }Lk psq}2 h6 ps, xq |ψk1 ps, xq ´ 1| ` |ψ 1 ps, xq ´ 1| νpdxq ds. Kptq :“ 0
X
By applying the inequalities (3.36)–(3.41) to the equality (3.26) we obtain P 1 -a.s. for each t P r0, T s that ˜ ˙2 ¸ ż t ˆż 0 1 }Lk ptq}22 ` }Lk psq}22,1 ds |γprq| dr ´ 2 0 ´t ˙ ˆ żt ż h5 ps, xq|ψk1 ps, xq ´ 1| νpdxq ds ď c }Lk psq}22 2 ` }ϕ1k psq}H ` 2 0 ¸ ˆż X ˜ ˙1{2 żt t ˘ ` 1 1 2 ` cεk }Vk psq}22 ` 1 ds ` cm 1 ` sup }S psq}2 }Lk psq}2 ds sPr0,ts
0
0
¯ ´ ` 2 1 ` sup }S 1 psq}2 Kptq ` |Ik,6 ptq| ` |Ik,8 ptq| ` |Ik,9 ptq|.
(3.42)
sPr0,ts
From (3.18) and (3.21) we conclude that ¸ ˆż ˜ ˙1{2 t 1 2 Ñ0 Jk,1 ptq :“ cm 1 ` sup }S psq}2 }Lk psq}2 ds sPr0,ts
P 1 -a.s.
(3.43)
in L1 pΩ1 , Rq as k Ñ 8.
(3.44)
0
as k Ñ 8. The estimates (3.10) to (3.12) results in Jk,2 ptq :“ cεk
żT 0
`
˘ }Vk1 psq}22 ` 1 ds Ñ 0 24
In order to estimate the term ¯ ´ Jk,3 ptq :“ 2 1 ` sup }S 1 psq}2 Kptq, sPr0,T s
( we define for fixed δ ą 0 and ω 1 P Ω1 the set Λk,δ pω 1 q :“ s P r0, T s : }Lk ps, ω 1 q}2 ě δ . It follows from (3.18) that 1 LebpΛk,δ p¨qq ď 2 δ
żT 0
}Lk psq}22 ds Ñ 0 P 1 -a.s. as k Ñ 8.
By splitting the integration domain we obtain 1
E rKk ptqs ď 2δ sup
żtż
gPS2m 0
h6 ps, xq|gps, xq ´ 1| νpdxq ds ` 2E
X
1
˜
¸
sup }Lk psq}2 Rδ,k ptq ,
sPr0,ts
where we use the notation Rδ,k ptq :“ sup
ż
gPS2m r0,tsXΛk,δ
ż
X
h6 ps, xq|gps, xq ´ 1| νpdxq ds.
Since LebpΛk,δ p¨qq Ñ 0 P 1 -a.s. as k Ñ 8, we conclude from [6, Le.3.4] or [20, Eq.(3.19)] that Rδ,k ptq Ñ 0 in P 1 -probability as k Ñ 8. Thus, we obtain E 1 rKk ptqs Ñ 0 which results in in P 1 -probability.
lim Jk,3 ptq “ 0
kÑ8
Choose T1 ą 0 such that (3.42) we obtain sup tPr0,T1 s
}Lk ptq}22
`
1 4
ş0
´T1
ż T1 0
|γprq| dr “ 12 . Then by applying Gronwall’s inequality to
}Lk psq}22,1 ds
´ ď Jk,1 pT1 q ` Jk,2 pT1 q ` Jk,3 pT1 q
¯ ` sup |Ik,6 ptq| ` sup |Ik,8 ptq| ` sup |Ik,9 ptq| Kk pT1 q, tPr0,T1 s
(3.45)
tPr0,T1 s
(3.46)
tPr0,T1 s
where we define ˙ ˙ ˆ ż T1 ˆ ż h5 ps, xq|ψk1 ps, xq ´ 1| νpdxq ds . Kk pT1 q :“ exp c 2 ` }ϕ1k psq}H ` 2 X
0
25
By the definition of S1m and (3.5) we have Kk pT1 q ď c P 1 -a.s. for all k P N for a generic constant c ą 0. By applying Burkholder’s inequality to Ik,6 and Ik,8 we obtain ´ ¯ E 1 sup |Ik,6 ptq| tPr0,T1 s
´ ? ď 2 εk E 1
ż T1
¯1{2 }Lk psq}22 }G1 ps, Vk1 psqq}22 ds 0 ´ ż T1 ¯¯ ¯ ´ ´ ? 1 2 1 p}Vk1 psq}22 ` 1q ds , sup }Lk psq}2 ` cE ď 2 εk E sPr0,T1 s
and ´ ¯ E 1 sup |Ik,8 psq| tPr0,T1 s
ď 2E 1
´ ż T1 ż
ď 2E 1
1{4 εk
˜0
1{2
ď εk E 1 ď
1{2 εk E 1
´
´
X
1 ε´1 ψk k
ε2k }Lk psq}22 }G2 ps, Vk1 psq, xq}22 N 1
sup }Lk ptq}2
tPr0,T1 s
(3.47)
0
´ ż T1 ż 0
X
¯1{2 pdx, dsq
3{2 εk }G2 ps, Vk1 psq, xq}22
N
ε´1 ψ 1 1 k k
¯1{2 pdx, dsq
¸
´ ż T1 ż ¯ ¯ 1{2 p}Vk1 psq}22 ` 1qh26 ps, xqψk1 ps, xq νpdxq ds sup }Lk ptq}22 ` εk cE 1
tPr0,T1 s
0
¯
sup }Lk ptq}22 `
tPr0,T1 s
1{2 εk c hE 1
´
X
¯ sup }Vk1 ptq}22 ` 1 ,
(3.48)
tPr0,T1 s
where we define h :“ sup
ż T1 ż
gPS2m 0
X
h26 ps, xqgps, xq νpdxq ds.
Similarly, it follows from (2.9) that ¸ ˜ ż T1 ż ´ ¯ h26 ps, xqgps, xq νpdxq ds E 1 sup }Vk1 ptq}22 ` 1 . E 1 pIk,9 pT1 qq ď εk c sup gPS2N
0
tPr0,T1 s
X
(3.49)
By applying (3.43) – (3.45) and (3.47) –(3.49) to inequality (3.46) we conclude ż T1 2 }Lk ptq}22,1 dt Ñ 0 in P 1 - probability. sup }Lk ptq}2 ` tPr0,T1 s
0
As the definition of T1 only depends on γ, we can repeat the above procedure for the interval rT1 , 2T1 s, which after further iterations finally leads to the proof of (3.25). 26
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