Asymptotics for stochastic reaction-diffusion equation driven by subordinate Brownian motions

0.26 MB·

by  Ran Wang

Asymptotics for stochastic reaction-diffusion
equation driven by subordinate Brownian motions
Ran Wang 1, Lihu Xu 2
∗ †
1 School of Mathematics and Statistics,
Wuhan University, Wuhan, P.R. China.
7 2 Department of Mathematics, Faculty of Science and Technology,
1
University of Macau, Taipa, Macau.
0
2
n January 6, 2017
a
J

5
Abstract

]
R
We study the ergodicity of stochastic reaction-diffusion equation driven by subordinate
P Brownian motions. After establishing the strong Feller property and irreducibility of the
.
h system,weprovethetightnessofthesolution’slaw. Thesepropertiesimplythatthisstochas-
t ticsystemadmitsauniqueinvariantmeasureaccordingtoDoob’sandKrylov-Bogolyubov’s
a
m theories. Furthermore, we establish a large deviation principle for the occupation measure
of this system by a hyper-exponential recurrence criterion. It is well known that S(P)DEs
[
driven by α-stable type noises do not satisfy Freidlin-Wentzell type large deviation, our

1 result gives an example that strong dissipation overcomes heavy tailed noises to produce a
v
Donsker-Varadhantype large deviation as time tends to infinity.
4
0
2 Keywords: Stochastic reaction-diffusion equation; Subordinate Brownian motions; Large deviation
1 principle (LDP); Occupation measure.
0
. Mathematics Subject Classification (2000): 60F10, 60H15, 60J75.
1
0
7
1 Introduction
1
:
v
i Consider a stochastic reaction-diffusion equation driven by subordinate Brownian motion on torus
X T:=R/Z as follows:
r dX ∂2Xdt (X X3)dt=Q dL , (1.1)
a − ξ − − β t
where X : [0,+ ) T Ω R and L is a subordinate Brownian motion. More details about
t
∞ × × →
this equation will be given in the next section. Sometimes the equation (1.1) is also called stochastic
Allen-Cahn equation or real Ginzburg-Landau equation. Recently, the study of invariant measures
and the long time behavior of stochastic partial differential equations (SPDEs) driven by α-stable
type noises has been extensively studied, we refer to [5, 6, 8, 12, 19] and the literatures therein.
In this paper, we firstly study the ergodicity of stochastic reaction-diffusion equation driven by
subordinate Brownian motions, showing that the system (1.1) admits a unique invariant probability
∗[email protected]
†[email protected], Corresponding author
1
measure π. To do this, we need to prove the system is strong Feller and irreducible. Those two
properties imply the uniqueness of the invariant measure according to Doob’s theory (see [10]). To
establish the strong Feller property, we truncate the nonlinearity and apply a gradient established in
[7]or[30]. Toestablishtheirreducibility,weneedtoprovetheirreducibilityofthestochasticevolution
and then apply a control problem result in [26]. Unlike the case of SPDEs driven by cylindrical α-
stable noises, the components of the noise are not independent, the approach in the proof of the
irreducibility is very different from that in our previous paper [26].
Another topic is the large deviation principle (LDP) about the occupation measure. Let be
t
L
the occupation measure of the system (1.1) given by
1 t
(A):= δ (A)ds for any measurable set A, (1.2)
Lt t Xs
Z0
whereδ istheDiracmeasureata. Bytheuniquenessofinvariantmeasure(see[2]),weknowthatthe
a
occupationmeasure convergestotheinvariantmeasureπ. Inthispaper,wealsostudytheLDPfor
t
L
the occupationmeasure . The LDP for empiricalmeasuresisone ofthe strongestergodicityresults
t
L
for the long time behavior of Markov processes. It has been one of the classical research topics in
probability since the pioneering work of Donsker and Varadhan [9]. Refer to the books [3, 4]. Based
on the hyper-exponential recurrence criterion developed by Wu [28], we prove that the occupation
measure obeys an LDP under τ-topology. As a consequence, we can obtain the exact rate of
t
L
exponential ergodicity.
For stochastic partial differential equations, the problems of LDP have been extensively studied
in recentyears. Mostofthem, however,areconcentratedonthe smallnoise LDP ofFreidlin-Wentzell
type, which provide estimates for the probability that stochastic systems converge to their determin-
istic partasnoisestendtozero. Butthereareonlyveryfew papersontheLDPofDonsker-Varadhan
tpye for large time, which estimate the probability of the occupation measures’ deviation from in-
variant measure. Gourcy [13, 14] established the LDP for occupation measures of stochastic Burgers
and Navier-Stokes equations by the means of the hyper-exponential recurrence. Jak˘si`c et al. [16]
established the LDP for occupation measures of SPDE with smooth randomperturbations by Kifer’s
LDP criterion [18]. Jak˘si`c et al. [17] also gave the large deviations estimates for dissipative PDEs
with rough noise by the hyper-exponential recurrence criterion. In [27], using the hyper-exponential
recurrence criterion, an LDP for the occupation measure is derived for a class of non-linear mono-
tone stochastic partialdifferential equations,suchas stochastic p-Laplaceequation,stochastic porous
medium equation and stochastic fast-diffusion equation.
Thepaperisorganizedasfollows. InSection2,wegiveabriefreviewofsomeknownresultsabout
the stochastic reaction-diffusion equations, and present the main result of this paper. In Sections 3
and 4, we prove the strong Feller property and the irreducibility of the system separately. In Section
5,wefirstrecallthe hyper-exponentialcriterionaboutthe LDPforMarkovprocesses,andthenverify
this condition by establishing some uniform estimates which also imply the tightness of the solution.
Throughout this paper, C is a positive constant depending on some parameter p, and C is a
p
constant depending on no specific parameter (except α,β), whose value may be different from line to
line by convention.
2 The model and the results
LetT=R/ZbeequippedwiththeusualRiemannianmetric,andletdξ denotetheLebesguemeasure
on T. For any p 1, let
≥
1
p
Lp(T;R):= x:T R; x := x(ξ)pdξ < .
Lp
( → k k (cid:18)ZT| | (cid:19) ∞)
2
Denote
H:= x L2(T;R); x(ξ)dξ =0 .
∈ T
(cid:26) Z (cid:27)
H is a real separable Hilbert space with inner product
x,y H := x(ξ)y(ξ)dξ, x,y H.
h i T ∀ ∈
Z
1
Write x H :=( x,x H)2 .
Lekt ∆k be thhe LapilaceoperatoronH. Then A:= ∆ is a positive self-adjointoperatoronH with
the discrete spectral. More precisely, there exist an o−rthogonal basis e ;e =ei2πkξ, k Z with
k k ∗
∈
Z :=Z 0 , and a sequence of real numbers λ =4π2 k 2; k Z such that Ae =λ e .
∗ \{ } k | | ∈ ∗ (cid:8) k k k (cid:9)
For any θ ≥0, let Hθ be the domain of the(cid:8)fractional operator Aθ(cid:9)2, i.e.,
Hθ :=(kX∈Z∗λk−θ2ak·ek :(ak)k∈Z∗ ⊂R,kX∈Z∗a2k <+∞),
with the inner product
hu,viθ :=hAθ2u,Aθ2uiH = λθkhu,ekiH·hv,ekiH,
kX∈Z∗
and with the norm kukθ :=hu,uiθ21 =kAθ2ukH. Clearly, Hθ is densely and compactly embedded in H.
Particularly, let
V:=H1 and x V := x 1.
k k k k
someLefitlt{eWretdk,ptr≥ob0a}bki∈liZt∗ybsepaacseeq(Ωue,nc,e(of)inde,pPe)n.deTnhtestcaynlidnadrrdicoanleB-driomwenniasinonmalotBioronwonniaHnmisodtieofinnoedn
t t≥0
F F
by
W := Wk e .
t t · k
kX∈Z∗
For α (0,2), let S be an independent α/2-stable subordinator, i.e., an increasing one dimensional
t
∈
L´evy process with Laplace transform
E e−ηSt =e−t|η|α/2, η >0.
Then L :=W defines a subordinate(cid:2)d cylin(cid:3)drical Brownian motion on H. Refer to [1, 24].
t St
For a sequence of bounded real numbers β =(βk)k∈N, let us define
Qβ :H H such that Qβu:= βk u,ek H ek for u H.
→ h i · ∈
kX∈Z∗
We shall rewrite the system (1.1) into the following abstract form:
dX +AX dt=N(X )dt+Q dL ,
t t t β t
(2.1)
(X0 =x,
where
(i) the nonlinear term N is defined by
N(u)=u u3, u H;
− ∈
3
(ii) L is a subordinated cylindrical Brownianmotion on H with α (1,2), and the intensity
t t≥0
{ } ∈
Q satisfies that for some δ >0 and 3 <θ′ θ <2,
β 2 ≤
′
δλ−θ2 β δ−1λ−θ2 , k Z .
k ≤| k|≤ k ∀ ∈ ∗
Definition 2.1 We say that a predictable H-valued stochastic process X = (Xx) is a mild solution
t
to Eq. (2.1), if for any t 0,x H, it holds (P-a.s.):
≥ ∈
t t
Xx(ω)=e−Atx+ e−A(t−s)N(Xx(ω))ds+ e−A(t−s)Q dL (ω). (2.2)
t s β s
Z0 Z0
ByLemma3.1inthenextsection,usingthesimilarapproachasintheproofof[29,Theorem2.2],
wecaneasilyobtainthatEq. (2.1)admitsauniquemildsolutionX(ω) D([0, );H) D((0, );V).
·
∈ ∞ ∩ ∞
Moreover, X is a Markov process.
Our first main result is the following theorem about the ergodicity of solution.
Theorem 2.2 Assume that α (1,2). Then the Markov process X is strong Feller and irreducible
in H for any t>0, and X adm∈its a unique invariant measure.
Proof: We shall prove the the strong Feller property and irreducibility in Section 3 and Section 4.
By the well-known Doob’s Theorem (see [2]), we know that X admits at most one unique invariant
probabilitymeasure. AccordingtotheKrylov-Bogolyubov’stheorem(See[2]),ifthefamilyofthe law
X ;t 1 is tight, then there exists an invariant probability measure for (2.1). The tightness for
t
{ ≥ }
X ;t 1 follows from Theorem 5.4.
t
{ Th≥e p}roof is complete. (cid:4)
Recall that defined by
t
L
1 t
(A):= δ (A)ds for any measurable set A, (2.3)
Lt t Xs
Z0
whereδ istheDiracmeasureata H. Then isin (H),thespaceofprobabilitymeasuresonH.
a t 1
On (H), let σ( (H), (H)) ∈be the τ-toLpology oMf converence against measurable and bounded
1 1 b
funcMtions whichisMmuchstrBongerthanthe usualweakconvergencetopologyσ( (H),C (H)), where
1 b
C (H) is the space of all bounded continuous functions on H. See [9] or [3, SecMtion 6.2].
b
Our second main result is about the LDP for occupation time , whose proof will be given in
t
L
the last section.
Theorem 2.3 Assume that α (1,2). Then the family P ( ) as T + satisfies the LDP
ν T
∈ L ∈ · → ∞
with respect to the τ-topology, with speed T and rate function J defined by (5.1) below, uniformly for
any initial measure ν in (H). More precisely, the following three properties hold:
1
M
(a1) for any a 0, µ (H);J(µ) a is compact in ( (H),τ);
1 1
≥ { ∈M ≤ } M
(a2) (the lower bound) for any open set G in ( (H),τ),
1
M
1
liminf log inf P ( G) infJ;
ν T
T→∞ T ν∈M1(H) L ∈ ≥− G
(a3) (the upper bound) for any closed set F in ( (H),τ),
1
M
1
limsup log sup P ( F) infJ.
ν T
T→∞ T ν∈M1(H) L ∈ ≤− F
4
Remark 2.4 For every f :H R measurable and bounded, as ν fdν is continuous w.r.t. the
→ → H
τ-topology, then by the contraction principle ([3, Theorem 4.2.1]),
R
1 T
P f(X )ds
ν s
T Z0 ∈·!
satisfies the LDP on R uniformly for any initial measure ν in (H), with the rate function given by
1
M
Jf(r)=inf J(µ)<+ µ (H) and fdµ=r , r R.
1
∞| ∈M ∀ ∈
(cid:26) Z (cid:27)
3 Strong Feller property
3.1 Some useful estimates
We shall often use the following inequalities (see [29]):
kAσ1xkH ≤Cσ1,σ2kAσ2xkH, ∀ σ1 ≤σ2,x∈H; (3.1)
Aσe−At H Cσt−σ, σ >0,t>0; (3.2)
k k ≤ ∀
x 4 x 2 x 2, x V; (3.3)
k kL4 ≤k kVk kH ∀ ∈
1
x,N(x) H , x H; (3.4)
h i ≤ 4 ∀ ∈
N(x) V C( x V+ x 3V), x V; (3.5)
k k ≤ k k k k ∀ ∈
N(x) N(y) V C(1+ x 2V+ y 2V) x y V, x,y V; (3.6)
k − k ≤ k k k k ·k − k ∀ ∈
N(x) N(y) H C(1+ A41x 2H+ A41y 2H) x y H, x,y H. (3.7)
k − k ≤ k k k k ·k − k ∀ ∈
For all σ 1,
≥ 6
N(x) N(y) H C(1+ Aσx 2H+ Aσy 2H) Aσ(x y) H, x,y H; (3.8)
k − k ≤ k k k k ·k − k ∀ ∈
N(x) H C(1+ Aσx 3H), x H. (3.9)
k k ≤ k k ∀ ∈
Let us now consider the following stochastic convolution:
t t
Z := e−(t−s)AQ dL = e−(t−s)λkβ dWk e . (3.10)
t β s k Ss · k
Z0 kX∈Z∗Z0
The estimate about Z will play an important role in next sections (cf. [20, 22]).
t
Lemma 3.1 [7, Lemma 2.3] Suppose that for some γ R,
∈
K := λγ β 2 <+ .
γ k| k| ∞
kX∈Z∗
Then for any p (0,α) and T >0,
∈
sup E kZtkpγ+1 ≤Cα,pKγp2Tαp−p2; (3.11)
t∈[0,T]
(cid:2) (cid:3)
5
for any θ <γ,
E"t∈s[u0p,T]kZtkpθ#≤Cα,pKγp2Tαp 1+Tγ−2θ ; (3.12)
(cid:16) (cid:17)
for any ε>0,
P sup Z ε >0. (3.13)
t θ
t∈[0,T]k k ≤ !
Moreover, t Z is almost surely c`adla`g in H .
t θ
7→
3.2 Strong Feller property
For any f (H), t 0 and x H, define
b
∈B ≥ ∈
P f(x):=E[f(Xx)].
t t
The main result of this section is
Theorem 3.2 (P ) , as a semigroup on (H), is strong Feller.
t t≥0 b
B
To proveTheorem3.2, thanks to a standardargument(see [29, p. 943]for example), we only need to
prove that the following lemma.
Lemma 3.3 (P ) , as a semigroup on (V), is strong Feller.
t t≥0 b
B
Proof: The proof is inspired by the proof of Theorem [29, Theorem 6.2]. Let T >0 be arbitrary, it
0
suffices to show that for all t (0,T ], x V and f (V),
0 b
∈ ∈ ∈B
lim P f(y)=P f(x). (3.14)
t t
ky−xkV→0
Without loss of generality, we assume f := sup f(x) = 1. We divide the proof into three
k k∞ x∈V| |
steps.
Step 1. Since the nonlinearity N is not bounded and Lipschitz continuous, we need to use a
truncation technique. Consider the equation with truncated nonlinearity as follows:
dXρ+AXρdt=Nρ(Xρ)dt+Q dL , Xρ =x V, (3.15)
t t t β t 0 ∈
where ρ>0, Nρ(x):=N(x)χ( x V/ρ)for all x V andχ:R [0,1]is asmoothfunction suchthat
k k ∈ →
χ(z)=1 for z 1, χ(z)=0 for z 2.
| |≤ | |≥
By (3.5), for all x V,
∈
Nρ(x) V C( x 3V+ x V)χ( x V/ρ) C(ρ3+ρ). (3.16)
k k ≤ k k k k k k ≤
It follows from (3.6) that
Nρ(x) Nρ(y) V C(1+ρ2) x y V. (3.17)
k − k ≤ ·k − k
Hence, Eq. (3.15) admits a unique Markov solution Xρ D([0, );V).
. ∈ ∞
By Theorem 3.1 in [7] (choosing σ = γ = 1 and γ′ = 0 there), we have for any 0 < t T , and
0
x,y V, ≤
∈ |E[f(Xtρ,x)]−E[f(Xtρ,y)]|≤Ct−α1−θ−21kfk∞·kx−ykV. (3.18)
6
Step 2. Define
KT0(ω):= sup kZt(ω)kV, ω ∈Ω.
0≤t≤T0
By Lemma 3.1 and Markov inequality, we have
P(K >ρ/2) C(α,T )/ρ, (3.19)
T0 ≤ 0
where C(α,T ) is some constant depending on α and T .
0 0
Choose ρ so large that x V √ρ<ρ/2 1 and define
k k ≤ −
G:= K ρ/2 .
{ T0 ≤ }
For all ω Ω, define Y (ω):=X (ω) Z (ω), then
t t t
∈ −
dY +AY dt=N(Y +Z )dt, Y =x V.
t t t t 0
∈
By (ii) of Lemma 4.1 in [29], there exists some 0<t T depending on ρ such that for all ω G,
0 0
≤ ∈
sup kYtx(ω)kV ≤1+kxkV ≤1+√ρ<ρ/2.
0≤t≤t0
Then
P sup kXtxkV ≥ρ ≤P sup kYtxkV+ sup kZtkV ≥ρ
(cid:18)0≤t≤t0 (cid:19) (cid:18)0≤t≤t0 0≤t≤T0 (cid:19)
≤P(KT0 >ρ/2)+P sup kYtxkV >ρ/2,G
(cid:18)0≤t≤t0 (cid:19)
=P(K >ρ/2). (3.20)
T0
The above inequality, together with (3.19) and (3.20), implies that
P sup kXtxkV ≥ρ ≤P(KT0 >ρ/2)≤C(α,T0)/ρ. (3.21)
(cid:18)0≤t≤t0 (cid:19)
Step 3. Define the stopping time
τx :=inf{t>0;kXtxkV ≥ρ}.
By (3.21), we obtain that for all t [0,t ],
0
∈
Px(τx ≤t)=P sup kXsxkV ≥ρ ≤C(α,t)/ρ. (3.22)
(cid:18)0≤s≤t (cid:19)
Since Eqs. (2.1) and (3.15) both have a unique mild solution, for all t [0,τ ), we have
x
∈
Xρ,x =Xx a.s.. (3.23)
t t
Lety Vbesuchthat x y V 1andchooseρ>0besufficientlylargesothatmax x V, y V
∈ k − k ≤ {k k k k }≤
√ρ. For any t (0,t0], it holds that
∈
P f(x) P f(y) = E[f(Xx)] E[f(Xy)] =I +I +I , (3.24)
| t − t | | t − t | 1 2 3
where
I := E[f(Xx)1 ] E[f(Xy)1 ],
1 | t [τx>t] − t [τy>t] |
I := E[f(Xx)1 ],
2 | t [τx≤t] |
7
I := E[f(Xy)1 ].
3 | t [τy≤t] |
It follows from (3.22) that
C f C f
∞ ∞
I k k , I k k . (3.25)
2 3
≤ ρ ≤ ρ
It remains to estimate I . It follows from (3.18), (3.22) and (3.23) that
1
I = E[f(Xρ,x)1 ] E[f(Xρ,y)1 ]
1 t [τx>t] − t [τy>t]
≤(cid:12)(cid:12)|E[f(Xtρ,x)]−E[f(Xtρ,y)]|+ E[f(Xtρ,x(cid:12)(cid:12))1[τx≤t]]|+|E[f(Xtρ,y)1[τy≤t]]
≤Ct−α1−θ−21kfk∞·kx−ykV+(cid:12)(cid:12)2Ckfk∞/ρ. (cid:12)(cid:12) (3.26)
For all ε>0,t (0,t ], choosing
0
∈
ρ max 12Ckfk∞,2 x 2V+2 , δ = ε tα1+θ−21,
≥ ε k k 2C
(cid:26) (cid:27)
by Eqs. (3.24), (3.25) and (3.26), we obtain that for all x y V δ,
k − k ≤
P f(x) P f(y) <ε.
t t
| − |
As t <t T , it follows from the Markov property and the strong Feller property above that
0 0
≤
P f(y) P f(x)=P [P f](y) P [P f](x) 0,
t − t t0 t−t0 − t0 t−t0 →
as y x V 0.
kT−he pkroo→f is complete. (cid:4)
4 Irreducibility
The main result of this part is the irreducibility of the stochastic dynamics.
Theorem 4.1 Assume that α (1,2). For any initial value x H, the Markov process X =
{Xtx}t≥0,x∈H to Eq. (2.1) is irred∈ucible in H. ∈
Remark 4.2 By the well-known Doob’s Theorem (see [2]), the strong Feller property and the irre-
ducibility imply that X admits at most one unique invariant probability measure.
4.1 Irreducibility of stochastic convolution
Let S be the space of all increasing and ca`dla`g functions from (0, ) to (0, ) with lim l = 0,
s→0+ s
∞ ∞
which is endowed with the Skorohod metric and the probability measure µS so that the coordinate
process S (l):=l is an α/2-stable subordinator.
t t
Consider the following product probability space
(Ω, ,P):=(W S, (W) (S),µW µS)
F × B ×B ×
and define
L (w,l):=w .
t lt
WeshallusethefollowingtwonaturalfiltrationassociatedwiththeL´evyprocessL andtheBrownian
t
motion W :
t
W
:=σ L (w,l);s t , :=σ W (w);s t ,
Ft { s ≤ } Ft { s ≤ }
and denote by ES and EW the partial integrations with respect to S and W, respectively.
8
For any l S, let Zl solve the following equation:
∈ t
dZl+AZldt=Q dW , Zl =0. (4.1)
t t β lt 0
It is well known that
t
Zl = e−A(t−s)Q dW = β z (t) e ,
t β ls k k · k
Z0 kX∈Z∗
where
t
zk(t)= e−λk(t−s)dWlks.
Z0
Notice that for any fixed l ∈S, {zk}k∈Z∗ are independent by the independence of {W·k}k∈Z∗.
We claim that for any γ (0,θ′ 1) with θ′ defined above Definition 2.1,
∈ −
EW sup kAγZtlkH ≤Cγ,θ′ lT. (4.2)
(cid:20)0≤t≤T (cid:21)
p
Indeed, upper to a standard finite dimension approximation argument, using integration by parts we
get
t
Zl =Q W Ae−A(t−s)Q W ds, (4.3)
t β lt − β ls
Z0
which clearly implies
t
sup kAγZtlkH ≤ sup kAγQβWltkH+ sup kA1+γe−A(t−s)QβWlskHds
0≤t≤T 0≤t≤T 0≤t≤TZ0
t
≤ sup kAγQβWltkH+ sup kA1+γ−γ′e−A(t−s)k·kAγ′QβWlskHds
0≤t≤T 0≤t≤TZ0
t
≤ sup kAγQβWltkH+C sup kAγ′QβWltkH· sup (t−s)1+γ−γ′ds
0≤t≤T 0≤t≤T 0≤t≤TZ0
sup AγQβWt H+CTγ′−γ sup Aγ′QβWt H,
≤ k k k k
0≤t≤lT 0≤t≤lT
where γ′ (γ,θ′ 1). Hence, by the martingale inequality we get (4.2).
∈ −
The following lemma is concerned with the support of the distribution of Z ,Z .
t 0≤t≤T T
{ }
(cid:0) (cid:1)
Lemma 4.3 For any T > 0,0 < p < , the random variable ( Z ,Z ) has a full support in
t 0≤t≤T T
Lp([0,T];V) V. More precisely, for a∞ny φ Lp([0,T];V),a V{,ε}>0,
× ∈ ∈
T
P Zt φt pVdt+ ZT a V <ε >0. (4.4)
Z0 k − k k − k !
Proof: The proof is divided into several steps.
Step 1. (Finite dimensional projection)For any N N, let H be the Hilbert space spanned by
N
e ,andletπ :H H betheorthogonalpro∈jection. Noticethatπ isalsoanorthogonal
k 1≤|k|≤N N N N
p{ro}jection in V. Define →
πN :=I π , HN :=πNH.
N
−
Thenforanygivenl S,π Zl andπNZl areindependent. Thus,foranyφ Lp([0,T];V)anda V,
N
∈ ∈ ∈
we have
T
P Zt φt pVdt+ ZT a V <ε
Z0 k − k k − k !
9
T
=ES PW Z0 kZtl−φtkpVdt+kZTl −akV <ε!(cid:12)l=S!
(cid:12)
T (cid:12) ε
≥ES PW kπN(Ztl−φt)kpVdt+kπN(ZTl −a)(cid:12)kV < 2p+1,
(cid:18) (cid:18)Z0
T ε
kπN(Ztl−φt)kpVdt+kπN(ZTl −a)kV < 2p+1
Z0 (cid:19)(cid:12)l=S(cid:19)
=ES PW T kπN(Ztl−φt)kpVdt+kπN(ZTl −a)kV < 2pε+1 (cid:12)(cid:12)(cid:12)
(cid:18) (cid:18)Z0 (cid:19)(cid:12)l=S
(cid:12)
T (cid:12)ε
×PW Z0 kπN(Ztl−φt)kpVdt+kπN(ZTl −a)kV < 2p(cid:12)+1!(cid:12)l=S(cid:19).
(cid:12)
For any γ (1,θ′ 1), by the spectral gap inequality, Chebyshev inequality an(cid:12)(cid:12)d (4.2), we have for
∈ 2 −
any η >0
PW sup kπNZtlkV >η ≤PW sup kπNAγZtlkH >ηλγN−12
(cid:18)0≤t≤T (cid:19) (cid:18)0≤t≤T (cid:19)
≤PW sup kAγZtlkH >ηλNγ−21
(cid:18)0≤t≤T (cid:19)
≤Cθ′,γ lTη−1λN12−γ.
Hence, p
T
P Zt φt pVdt+ ZT a V <ε
Z0 k − k k − k !
≥ES PW T kπN(Ztl−φt)kpVdt+kπN(ZTl −a)kV < 2pε+1 × 1−Cθ′,γ lT2p+1ε−1λN12−γ
≥ES(cid:20)PW(cid:18)Z0T kπN(Ztl−φt)kpVdt+kπN(ZTl −a)kV < 2pε+1(cid:19)(cid:12)(cid:12)(cid:12)(cid:12)l=S ×(cid:16)1−Cθ′,γpT2p+1ε−1λN12−γ ,(cid:17)ST(cid:12)(cid:12)(cid:12)(cid:12)l=≤S(cid:21)T2
≥21E(cid:20)S PW(cid:18)Z0 T kπN(Ztl−φt)kpVdt+kπN(ZTl −a)kV < 2pε+(cid:19)1(cid:12)(cid:12)(cid:12)(cid:12)l=S ,S(cid:16)T ≤T2 , (cid:17) (cid:21)
(cid:20) (cid:18)Z0 (cid:19)(cid:12)l=S (cid:21)
(cid:12)
as N is sufficiently large. (cid:12)
(cid:12)
As long as we prove that for any ε>0
T
ES PW kπN(Ztl−φt)kpVdt+kπN(ZTl −a)kV <ε ,ST ≤T2 >0, (4.5)
(cid:20) (cid:18)Z0 (cid:19)(cid:12)l=S (cid:21)
(cid:12)
the proof is complete. (cid:12)
(cid:12)
Step 2. It remains to prove (4.5). Since π Zl = z (t)e with z (t) being independent
N t |i|≤N i i { i }i
stochastic processes, it suffices to prove (4.5) for one dimensional case, i.e., for any φ Lp([0,T];R),
a R and ε>0, P ∈
∈
T
ES PW z(t) φ(t)pdt+ z(T) a <ε ,S T2 >0, (4.6)
T
" Z0 | − | | − | !(cid:12)l=S ≤ #
(cid:12)
where z(t)= te−λ(t−s)dw with λ>0 and w being a one dim(cid:12)(cid:12)ensional Brownianmotion. To prove
0 ls t
(4.6), we only need to show that
R
T t p T
ES PW eλsdw eλtφ(t) dt+ eλsdw eλTa <ε ,S T2 >0, (4.7)
" Z0 (cid:12)(cid:12)(cid:12)(cid:12)Z0 ls − (cid:12)(cid:12)(cid:12)(cid:12) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z010 ls − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) !(cid:12)(cid:12)(cid:12)(cid:12)l=S T ≤ #