Table Of ContentLARGE DEVIATION PRINCIPLE FOR A STOCHASTIC ALLEN–CAHN
EQUATION
MARTINHEIDAANDMATTHIAS RÖGER
Abstract. In this paper we consider the Allen–Cahn equation perturbedby a stochastic flux
term and prove a large deviation principle. Using an associated stochastic flow of diffeomor-
phismstheequationcanbetransformedtoaparabolicpartialdifferentialequationwithrandom
coefficients. We use this structure and first provide a large deviation principle for stochastic
flows in function spaces with Hölder-continuity in time. Second, we use a continuity argument
5 and deducea large deviation principle for thestochastic Allen–Cahn equation.
1
0
2
n
1. Introduction
a
J
The deterministic Allen–Cahn equation
6
1 1
ε∂tu = ε∆u− εW′(u) (1.1)
]
R
is one prominent example of a mesoscopic model for the dynamics of a two-phase system driven
P
by a reduction of surface energy and not conserving the total mass of the phases. Here W
.
h denotes a suitable double-well potential with equal minima in 1. The two phases correspond
t ±
a to regions where u is close to +1 or 1, respectively, and are separated by a thin transition layer
m of approximate width ε. The Allen−–Cahn equation (1.1) is characterized as the accelerated L2
[ gradient flow of the Van der Waals–Cahn–Hilliard energy
1 ε 1
v Eε(u) = ˆ 2|∇u|2+ εW(u)dx. (1.2)
7 U
1
By the famous Modica–Mortola theorem [23, 22] this energy approximates the perimeter func-
9
3 tional as ε 0. Besides its motivation from phase separation theory the Allen–Cahn equation is
→
0 also intensively studied because of its connection to geometric flows: in the sharp interface limit
1. ε 0 solutions uε of (1.1) converge to a family of phase indicator functions u(,t) that move
→ ·
0 according to mean curvature flow [6, 9, 17].
5
Stochastic perturbations of (1.1) have been introduced to include for example thermal effects
1
or any other unresolved degrees of freedom and to describe nucleation and growth phenomena.
:
v
A perturbation of the Allen–Cahn equation by additive noise leads to the formal stochastic PDE
i
X
1
r ε∂tu = ε∆u W′(u)+σζ, (1.3)
a − ε
where σ > 0 is a small noise intensity parameter and ζ denotes a space-time white noise (in
higher space dimensions spatially regularised). Such type of evolutions were studied in the
one-dimensional case in [12, 3] and in the higher-dimensional case in [13, 25, 21]. In higher
dimensions the Allen–Cahn equation with space-time white noise is in general not well-posed
and the introduction of spatial correlations by a kind of smoothing procedure are necessary.
In order to better understand the behavior of solutions to (1.3) extensions of the Freidlin–
Wentzell theory for randomly perturbed dynamical systems have been used to study the small
noise limit σ 0 of (1.3). In [10] for one space dimension, and in [11],[18] for higher dimensions
→
the Allen–Cahn action functional was identified and leads (for zero spatial correlation length) to
T 1 1
(u) := ε(∂ u)2+ ε∆u+ W′(u) 2dxdt, (1.4)
Sε ˆ0 ˆRn t ε − ε
(cid:0) (cid:1)
see also [16] for a recent analysis of coupled limits σ 0 and spatial correlation length to zero.
→
1
2 M.HEIDAANDM.RÖGER
Here we study an alternative stochastic perturbation of the Allen–Cahn equation in the form
of a Stratonovich stochastic partial differential equation (SPDE)
1
du = ∆u W′(u ) dt+ u X (x, dt), (1.5)
ε ε− ε2 ε ∇ ε· σ ◦
(cid:16) (cid:17)
where X is a vectorfield valued Brownian motion. Such an evolution was introduced in [24]
σ
where the existence of unique Hölder-continuous strong solutions, the tightness of the solutions
(u ) of (1.5), and the convergence to an evolution of (random) phase indicator functions
ε ε>0
u(t, ) BV(U) has been shown.
· ∈
In this contribution our goal is to first take the noise intensity to zero and to consider the
limit process σ 0. More precisely we study the large deviation problem associated to (1.5),
→
where the driving force is given by a vector-field Brownian motion
∞ t t
X (t,x) = √σ X(l)(s,x) dB (s)+ X(0)(s,x)ds, (1.6)
σ ˆ ◦ l ˆ
0 0
l=1
X
see below for the precise assumptions on the coefficients. The large deviation theory developed
for the stochastic Allen–Cahn equation with additive noise does not apply here. Instead, we
exploit the particular structure of (1.5). Following the approach by Kunita [19] we consider the
Stratonovich flow associated to X , that is the solution of the stochastic differential equation
σ
−
dϕ (x) = X ( dt,ϕ (x)) (1.7)
s,t σ s,t
− ◦
ϕ (x) =x,
s,s
and use the resulting family of diffeomorphism to transform (1.5) into a partial differential
equation with random coefficients R and S
ϕ ϕ
1 1
∂ w R :D2w S w w+ w3 = 0, (1.8)
t − ϕ − ϕ·∇ − ε2 ε2
(forthe details seeSection 5.1below). This approach hasbeenusedin [24]toprove theexistence
of solution. Here we also take advantage from the same transformation and deduce a large
deviation result for (1.5) from a suitable large deviation principle for (1.7) and a continuity
result for the mapping ϕ w.
7→
Large deviation principles for stochastic flows have been obtained by Budhiraja, Dupuis and
Maroulas [5] in suitable classes of time-continuous diffeomorphism, see Section 4 below. In order
to achieve an appropriate continuity result for the mapping ϕ w we however need a large
7→
deviation in parabolic Hölder spaces. Therefore one key part in our approach is to suitably
extend the corresponding results from [5].
The paper is organized as follows. In the next section we fix some notation and state the
preciseassumptionsandmainresultsforlargedeviationsofvectorfieldvaluedBrownianmotions
(Theorem 2.5) and of solutions to the stochastic Allen-Cahn equation (Theorem 2.6). In Section
3, we introduce suitable function spaces and derive some estimates that are crucial for our
calculations in the subsequent sections. Section 4 provides the proof of Theorem 2.5, while in
Section 5 we present the proof of Theorem 2.6.
Acknowledgement. This work was partially funded by the DFG-Forschergruppe 718 Analysis
and Stochastics in Complex Physical Systems. We thank Hendrik Weber for helpful discussions.
2. Notation and main results
We first introduce some notation.
Let U Rn be open and bounded with C∞-boundary and for some fixed time interval [0,T]
⊂
letQ := [0,T] U. WedenotebyGm thesetofCm-diffeomorphismsonRn. SinceU isbounded,
×
the spaces
Cm(U) := u Cm(U¯;Rn) : u = Id ,
id ∈ |∂U
C0m(U) := (cid:8)u∈ Cm(U¯;Rn) : u|∂U = 0 (cid:9),
Gm(U) := u Cm(U) is a Cm-diffeomorphism ,
id (cid:8){ ∈ id (cid:9) }
3
equipped with the Cm(U¯;Rn) norm are Banach spaces.
For a Banach space we define
B
∞
l ( ) := (x ) : (x ) < , (x ) 2 := x 2
2 B k k∈N ⊂ B k k kkl2(B) ∞ k k kkl2(B) k kkB
n o Xk=1
and l := l (R). Further, for any compact subset K Rm and 0 < α < 1 we denote by
2 2
⊂
C0,α(K; ) the space of Hölder-continuous functions on K with values in .
B B
2.1. Stochastic flows. Here we follow Kunita [19] and introduce Brownian motions with a
spatial parameter. Throughout the paper we fix a filtered probability space (Ω, ,P, ).
t
F {F }
Assumption 2.1. We assume that we are given numbers k N with k 4, 0 < α < 1 and two
∈ ≥
mappings a, b such that
a : [0,T] Rn Rn Rn×n is Ck,α in the second and third component and continuous
• × × →
in time with a(,x,y) = 0 whenever (x,y) U U,
b : [0,T] Rn · Rn is Ck,α in space, andbo6∈und×ed andmeasurable intime with b(t,x) = 0
• × →
for almost all t and x U.
6∈
In what follows, we fix β < α and consider a continous stochastic process X(t) which
{ }t≥0
is a Ck,β-Brownian motion on Rn with X(t,x) = 0 on Rn U with local characteristics (a,b) as
\
above, in the following sense (see [19]):
X(0), X(t ) X(t ), i = 0,1,...,m 1 are independent Ck,β(Rn) valued random
i+1 i
• − −
variables whenever 0 t < t < < t T,
0 1 m
for each x Rn, the≤random var·ia·b·le M(≤t,x) := X(t,x) tb(r,x)dr is a continuous
• ∈ − 0
martingale, ´
for the corresponding quadratic variation we have for all (x,y) Rn×n that
• ∈
t
M(,x),M(,y) = a(r,x,y)dr.
hh · · iit 0
´
According to Kunita [19] for any local characteristics (a,b) given as above and for any 0 <
β < α such a Ck,β-Brownian motion exists and can be represented in the form
∞ t t
X(t,x) = X(i)(r,x) dB (r)+ X(0)(r,x)dr, (2.1)
ˆ ◦ i ˆ
0 0
i=1
X
where (B ) is a family of i.i.d. Brownian motions and
l l∈N
X(l) L2(0,T;C3,β(U)). (2.2)
l∈N ⊂
(cid:16) (cid:17)
By the above characterization, we find
∞
a(t,x,y) = X(i)(t,x)X(i)(t,y)T , b(t,x) = X(0)(t,x) (2.3)
i=1
X
and
T ∞ 2
sup X(i)(r,x) dr T sup a(t, ) < . (2.4)
x∈Uˆ0 Xi=1(cid:12) (cid:12) ≤ t∈[0,T]k · kCikd,α(U) ∞
(cid:12) (cid:12)
If a, b satisfy the above conditi(cid:12)ons, we fin(cid:12)d that X(t,x) = 0 for x U.
6∈
WeassociatetoaCk,β-Brownianmotion X(t) asabovetheStratonovich flow(ϕ ,s t)
{ }t≥0 s,t ≤
and Ito¯ flow (φ ,s t), which satisfy the Stratonovich respectively Ito¯ initial value problem
s,t
≤
dϕ (x) = X( dt,ϕ (x))
s,t s,t
− ◦
∞
= X(i)(t,ϕ ) dB (t) X(0)(t,ϕ )dt, (2.5)
s,t i s,t
− ◦ −
i=1
X
ϕ (x) = x.
s,s
4 M.HEIDAANDM.RÖGER
and
dφ (x) = X(dt,φ (x))
s,t s,t
−
∞
= X(i)(t,φ )dB (t) X(0)(t,φ )dt, (2.6)
s,t i s,t
− −
i=1
X
φ (x) = x.
s,s
We remark that ϕ (x) = φ (x) = x for all x U and thus ϕ ,φ Ck,β(U) almost surely.
s,t s,t 6∈ s,t s,t ∈ id
Remark 2.2. By [19, Theorem 4.6.5] we can assume w.l.o.g. that the flows (ϕ ,s t) and
s,t
≤
(φ ,s t) are stochastic flows of Ck-diffeomorphisms in the sence of [19].
s,t
≤
2.2. C0,α-Large Deviation Principle for stochastic flows. We briefly recall the notions of
good rate functions and large deviation principle.
Definition 2.3. Let be a Polish space. A function I : [0,+ ] is called a good rate
E E → ∞
function on , if for each M < the sublevel set x : I(x) M is a compact subset of .
E ∞ { ∈ E ≤ } E
For every Borel-measurable A , we define I(A) := inf I(x).
x∈A
⊂ E
Definition 2.4. Let I be a good rate function on . A sequence uσ > 0 is said to satisfy the
E { }σ
large deviation principle (LDP) on with good rate function I if the following large deviation
E
upper and lower bounds hold:
For each closed subset F of ,
• E
limsupσlogP(uσ F) I(F).
∈ ≤ −
σ→0
For each open subset G of ,
• E
liminfσlogP(uσ G) I(G).
σ→0 ∈ ≥ −
We next describe a suitable large deviation principle for stochastic flows associated to a Ck,α-
Brownian motion. Let X(k) be given as in Assumption 2.1. For any σ > 0 we define X as in
σ
(1.6)
∞ t t
X (t,x) = √σ X(l)(s,x)dB (s)+ X(0)(s,x)ds (2.7)
σ ˆ k ˆ
0 0
l=1
X
and we associate to X (t,x) the stochastic flows φσ and ϕσ according to (2.5) and (2.6).
σ s,t s,t
Next we define for f L2(0,T;l ) controlled vector fields
2
∈
∞ t
b (t,x) := f (t)X(l)(t,x)+X (t,x), X0,f(t,x) := b (s,x)ds (2.8)
f l 0 ˆ f
0
l=1
X
0,f
and controlled associated flows φ that are given as the unique solution of
t0,t t∈(t0,T)
(cid:16) (cid:17)
t
φ0,f(x) = x+ b (s,φ0,f (x))ds t [t ,T], x U. (2.9)
t0,t ˆ f t0,s ∀ ∈ 0 ∀ ∈
t0
OurfirstresultisalargedeviationprincipleinspaceswithHölderregularityintime. Asdescribed
in the introduction this extends results from [5], where a large deviation principle in spaces of
time-continuous functions has been proved.
Theorem 2.5. For (ϕσ,X ) and (φσ,X ) defined above and for any 0 < γ < 1,
σ σ>0 σ σ>0 2
η (0,1) and η (0,β] the family (ϕσ,X ) and (φσ,X ) satisfy LDPs in the spaces
φ ϕ σ σ>0 σ σ>0
∈ ∈
C0,γ([0,T];Ck−1,ηϕ(U)2), C0,γ([0,T];Ck−1,ηφ(U)2) respectively with the good rate function I∗
W
defined by
1 T
I∗ (ϕ,X) = inf f(s) 2 ds : f L2(0,T;l ) s.t. (φ0,f,X0,f)= (ϕ,X) . (2.10)
W 2 ˆ k kl2 ∈ 2
(cid:26) 0 (cid:27)
We will give a proof of this theorem in Section 4.
5
2.3. Large deviation principle for the stochastic Allen-Cahn equation (1.5). Without
loss of generality, we set ε = 1 as the original problem can always be reduced to that case using
a parabolic rescaling. In the following we choose the standard quartic double-well potential
W(r)= 1(1 r2)2. WenowdescribeourmainresultconcerningthesolutionsoftheStratonovich
4 −
stochastic Allen-Cahn equation (1.5), that is
t t
u(t,x) = u (x)+ ∆u W′(u) + u(s,x) X ( ds,x), (2.11)
0 ˆ − ˆ ∇ · σ ◦
0 0
u ν = 0 on (0,(cid:0)T) ∂U , (cid:1) (2.12)
U
∇ · ×
where u are fixed, smooth deterministic inital data, and where X was defined in (2.7). Under
0 σ
the assumptions stated above existence of unique continuous C3,β(U)-valued semimartingale
solutions u to (2.11), (2.12) has been shown in [24, Theorem 4.1].
For a deterministic control f L2([0,T];l ), initial data u C3,β(U) and b as defined in
2 0 f
∈ ∈
(2.8) we consider the following differential equation for u C([0,T];C3,β(U)),
∈
t t
u(t,.) = u + u(s,.) b (s,.)ds+ ∆u(s,.) W′(u(s,.)) ds
0 ˆ ∇ · f ˆ − (2.13)
0 0
u ν = 0 on (0,T) ∂U , (cid:0) (cid:1)
U
∇ · ×
for all t [0,T].
∈
Theorem 2.6. The family (u ) satisfies a large deviation principle in C([0,T];C2,β(U))
σ σ>0
∩
C0,β([0,T];C1,β(U)) for σ 0 with good rate function
↓
1 T
Iˆ(u) = inf f(s) 2 ds : f L2([0,T];l ) satisfies (2.13) (2.14)
2 ˆ k kl2 ∈ 2
(cid:26) 0 (cid:27)
We give the proof of this theorem in Section 5.
3. Preliminaries
3.1. Function spaces. To obtain suitable continuity properties in Section 5.1 it is most con-
venient to work in parabolic Hölder spaces. Therefore, we introduce for any bounded subset
U Rd and any l > 0 the Hölder spaces Hl(U) of [l]-times continuously differentiable functions
⊂
with the [l]-th deriative being in C0,l−[l](U). We denote the corresponding norm by . Fur-
|·|U,l
thermore, we denote Hl/2,l(Q) the set of functions u satisfying DrDsu C(Q) for 2r +s l
and DrDsu being C0,l−[l] in space and C0,21(l−[l]) in time. The cotrresxpo∈nding norm is deno≤ted
t x
.
|·|Q,l
Working in Hölder spaces has the drawback that these spaces are not separable, which causes
some additional difficulties in the proof of the large deviation principle for stochastic flows. To
circumvent this problems we introduce the following subspaces of C0,α(K; ) for 0 < α < 1,
K Rd compact and a suitable Banach space: B
⊂ B
u(x) u(y)
λ0,α(K; ) := u C0,α(K) : lim sup k − kB =0 ,
B ( ∈ δ→0|x−y|<δ x y α )
| − |
λm,α(K; ) := u Cm(K; ) : Dmu λ0,α(K; ) .
B { ∈ B ∈ B }
Furthermore we define the spaces
W := C([0,T];Cm(U)), Wˆ := C([0,T];Gm(U)),
m m
(3.1)
Wγ := λ0,γ([0,T];λm,2γ(U)), Wˆ γ := λ0,γ([0,T];Gm(U)),
m m
and set λm,γ(U) := λm,γ(U;Rn) Cm(U), λm,γ(U):= λm,γ(U;Rn) Cm(U) and
id ∩ id 0 ∩ 0
Wγ (Q) := λγ([0,T];λm,2γ(U)), Wγ (Q) := λγ([0,T];λm,2γ(U)),
m,id id m,0 0
(3.2)
Wˆ (Q) := C([0,T];Gm(U)).
m id
6 M.HEIDAANDM.RÖGER
Remark 3.1. By [2, Theorem 1] the space λ0,α(K; ) is separable if is separable.
B B
Given two Banach spaces and ′, such that ֒ ′ is compact and ′ is separable, the
B B B → B B
embedding C0,α(K; ) ֒ λβ(K; ′) is compact for β < α. For any γ < 1 the embedding
B → B 2
C0,γ([0,T];C0,2γ(U)) ֒ Hγ,2γ(Q) is continuous.
→
3.2. Inequalities and embeddings. We now derive some useful inequalities and embeddings
and start with a generalization of the Garsia-Rodemich-Rumsey Lemma [14] to Banach-valued
functions.
Lemma 3.2. Let be a Banach space, α 0 and p 1. There is a constant C 1 such that
α
B ≥ ≥ ≥
for any f C([0,T]; ) with the property that the right hand side in the following inequality is
∈ B
bounded, we have f C0,α([0,T]; ) and
∈ B
1
f(x) f(y) f(x) f(y) p p
sup k − kB C k − kB dxdy . (3.3)
x,y∈[0,T] x y α ≤ α ˆ[0,T]2 x y αp+2 !
| − | | − |
Remark 3.3. We can compare (3.3) to a classical Sobolev inequality: Considering the space
Ws,p(0,T) of R-valued functions with the norm
T T u(x) u(y) p
u := | − | dxdy,
k ks,p ˆ ˆ x y 1+sp
0 0
| − |
we find that Ws,p(0,T) ֒ C0,δ[0,T] in case s 1/p δ. Compared with (3.3), this corresponds
→ − ≥
to s = 1+αp and δ = α.
p
Proof. For simplicity, we expand f in a continuous way by constants outside [0,T]. We follow
the proof of Lemma 4 in [15]. Consider ψ(x) := x p and P(x) := x α+2/p as functions R R.
| | | | →
Furthermore, set R := f(x) f(y). We then find by convexity of ψ for any measurable sets
xy
−
A,B [0,T]
⊂
dxdy R dxdy
R P(d(A,B)/4)ψ−1 ψ k xykB
ˆ k xykB A B ≤ ˆ P(d(x,y)/4) A B
A×B | || | (cid:18) A×B (cid:18) (cid:19) | || |(cid:19)
U
P(d(A,B)/4)ψ−1 , (3.4)
≤ A B
(cid:18)| || |(cid:19)
where d(A,B) = sup x y and U = ψ kRxykB dxdy.
x∈A,y∈B| − | [0,T]2 P(d(x,y)/4)
´
Let R(t,r ,r ) := du dv R(cid:16) for r ,(cid:17)r > 0, with R(t,0,r ) :=
1 2 B(t,r1) |B(t,r1)| B(t,r2) |B(t,r2)| uv 1 2 2
dv R and R´(t,r ,0) simil´arly. Note that R is continuous on [0, )3 if one sets
B(t,r2) |B(t,r2)| tv 1 ∞
´
R(t,0,0) = 0 for all t 0. We choose s,t [0,T], s < t, define λ := t s and λ through
0 n+1
≥ ∈ −
P(λ ) =2P(λ ), inductively. Then, by monotonicity of P,
n n+1
P((λ +λ )/4) P(λ ) = 2P(λ )
n n+1 n n+1
≤
= 4P(λ ) 2P(λ ) = 4[P(λ ) P(λ )] .
n+1 n+1 n+1 n+2
− −
We find, using equation (3.4)
U
R(t,λ ,λ ) P((λ +λ )/4)ψ−1
n+1 n B ≤ n n+1 λ λ
(cid:18) n n+1(cid:19)
(cid:13) (cid:13) U
(cid:13) (cid:13) 4[P(λ ) P(λ )]ψ−1
n+1 n+2
≤ − λ λ
(cid:18) n n+1(cid:19)
λn+1 U
4 ψ−1 dP(r).
≤ ˆ r2
λn+2 (cid:18) (cid:19)
For any sequence of variables (x ) R, we find
i i∈N ⊂
n
R = R + R ,
xx0 xxn+1 xi+1xi
i=0
X
7
and averaging with respect to x over B(t,λ ) for i = 0,...,n+1 leads to
i i
n
R(t,0,λ ) = R(t,0,λ )+ R(t,λ ,λ ).
0 n+1 i+1 i
i=0
X
Since R is continuous and R(t,0,0) = 0, taking the limit n yields
→ ∞
∞ λi+1 U t−s U
R(t,0,λ ) 4 ψ−1 dP(r) 4 ψ−1 dP(r). (3.5)
0 B ≤ ˆ r2 ≤ ˆ r2
i=0 λi+2 (cid:18) (cid:19) 0 (cid:18) (cid:19)
(cid:13) (cid:13) X
Similarly, w(cid:13)e find (cid:13)
t−s U
R(s,0,λ ) 4 ψ−1 dP(r),
0 B ≤ ˆ r2
0 (cid:18) (cid:19)
(cid:13) (cid:13)
and a corresponding estima(cid:13)te for R(t,λ(cid:13)0,0). We use Rst = Rsx+Rxy +Ryt, and average over
the balls B(s,λ ) in x and B(t,λ ) in y to obtain
0 0
dxdy
R = R(s,0,λ )+ R +R(t,λ ,0)
st 0 ˆ xy 4λ2 0
B(s,λ0)×B(t,λ0) 0
Note that we can estimate the second term on the right hand side by P(3)P(λ )ψ−1(U/λ2).
4 0 0
Thus, from the last equation together with (3.4) and (3.5) we get
t−s U
R 10 ψ−1 dP(r).
k stkB ≤ ˆ r2
0 (cid:18) (cid:19)
Using the definition of ψ and P, this finally proves the claim. (cid:3)
Lemma 3.4. Consider a Banach space , p,q 1, p > 2q and a random function f with values
B ≥
in C([0,T]; ) such that there is a fixed constant Λ with
B
Ekf(t)−f(s)kpB ≤Λ(t−s)2pq ∀s< t and EkfkpL∞(B) ≤ C.
Then, for every α < 1 1 we have
2q − p
E f p C (Λ+1),
k kC0,α([0,T];B) ≤ α
where C is a constant depending only on α,p,q.
α
Proof. By Lemma 3.2 we find
E f(t) f(s) p
E f p C k − kB dsdt +E f p
k kCα,0([0,T];B) ≤ α ˆ[0,T]2 t sαp+2 ! k kC([0,T];B)
| − |
p
t s 2q
C Λ | − | dsdt+E f p
≤ α ˆ[0,T]2 t s αp+2 k kL∞(B)!
| − |
and the last integral exists iff α < 1 1. (cid:3)
2q − p
We will need a generalized version of the Arzela-Ascoli theorem and of Kolmogorovs tightness
criterium:
Theorem 3.5. Given two Banach spaces and ′ with ֒ ′ compactly and ′ separable,
B B B → B B
let v C([0,T]; ) be a sequence of functions with sup sup v (t) < and for
{ n}n∈N ⊂ B n t∈[0,T]k n kB ∞
some α > 0 let
sup v (t ) v (t ) C(t t )α.
k n 1 − n 2 kB ≤ 1− 2
n
Then, v is compact in λ0,γ([0,T]; ′) and C0,γ([0,T]; ′) for any γ < α.
{ n}n∈N B B
Proof. By the Arzela-Ascoli theorem for Banach space valued continuous functions, v is
{ n}n∈N
compact in C([0,T]; ′). Furthermore, we find equiboundedness of v in C0,α([0,T]; ′),
B { n}n∈N B
which yields the desired result for any γ < α through embedding of Hölder spaces and Remark
3.1. (cid:3)
8 M.HEIDAANDM.RÖGER
Theorem 3.6. Given two Banach spaces and ′ with ֒ ′ compactly and ′ separable, let
B B B → B B
ψ be a sequence of random fields with values in C([0,T]; ). Assume for any p > 1 there
{ n}n∈N B
is a positive constant C such that
p
Ekψn(t)−ψn(s)kpB ≤ Cp|t−s|p2 ∀s,t∈ [0,T], (3.6)
E ψ (t) p C t [0,T] (3.7)
k n kB ≤ p ∀ ∈
for any n N. Then, ψ is tight in λ0,γ([0,T]; ′) for any γ < 1 1.
∈ { n}n∈N B 2 − p
Proof. We follow the proof of [19] Theorem 1.4.7 For arbitrary q N, q > 1, we represent any
real number t as t = ∞ a q−k, where a N , k = 0,1,2,... a∈re non-negative integers and
k=0 k k ∈ 0
a <q for all k > 0. Let t = N a q−k and say t is q-adic of length N if t = t for some N.
k P N k=0 k N
We introduce ∆ the set of all q-adic rationals of length N and for f C([0,T]; ) the values
N
P ∈ B
∆ (f)= max f(s) f(t) ,
N s,t∈∆N,|s−t|≤q−Nk − kB
∆γ (f)= ∆ (f)/ q−N γ .
N N
We infer from [19], Lemmas 1.4.2 and 1.4.3(cid:0)(note(cid:1)the different meaning of γ in this reference)
that for γ < 1 1, there holds
2 − p
∞
ψ (s) ψ (t) 4q ∆γ (ψ ) s t γ ,
k n − n kB ≤ N n | − |
!
N=1
X
∞ p
supE ∆γ (ψ ) < .
n (cid:12) N n (cid:12) ! ∞
(cid:12)NX=1 (cid:12)
(cid:12) (cid:12)
For any ε > 0, Chebyschev’s(cid:12)inequality yi(cid:12)elds existence of a >0 such that for all n N
(cid:12) (cid:12) ∈
∞
ε
P ∆γ (ψ ) >a < ,
N n 2
!
N=1
X
ε
P( ψ (0) > a) < .
k n kB 2
Let
∞
γ
K := f C([0,T]; ) : ∆ (f)< a, f(0) < a .
∈ B N k kB
( )
N=1
X
If γ < 1 1, we can repeat the arguments of Lemma 1.4.2, Lemma 1.4.3 and Theorem 1.4.7 of
2 − p
[19], to derive
f(t) f(0) + f(0) f(t) a+4aqtγ,
k kB ≤ k kB k − kB ≤
f(s) f(t) 4aq s t γ
k − kB ≤ | − |
for all f K. Since γ < 1 1 was arbitrary, K is compact in λ0,γ([0,T]; ′) by Theorem 3.5.
∈ 2 − p B
Finally, note that P(ψ K)< ε and thus ψ is tight in λ0,γ([0,T]; ′) for all γ < 1 1. (cid:3)
n 6∈ n B 2 − p
3.3. Large deviation principles and continuous mappings. For the proof of our main
Theorems 2.5 and 2.6 we will finally need the following contraction principle.
Theorem 3.7 (Contraction Principle, [7] Theorem 4.2.1). Let and ˜ be Hausdorff topological
E E
spaces and F : ˜ be continuous. If I is a good rate function on , the function
E → E E
I˜(v) = inf I(u) : v = F(u)
{ }
is a good rate function on ˜. If u is a sequence of -valued random variables satisfying
E { σ}σ>0 E
a large deviation principle on with good rate function I, the sequence F(u ) satisfies a
E { σ }σ>0
large deviation principle on ˜ with good rate function I˜.
E
9
4. Large deviations for stochastic flows
The aim of this section is to prove Theorem 2.5. We will obtain this theorem as a consequence
of Theorems 4.5–4.7 below. We first introduce some notations.
Given the filtered probability space (Ω, ,P, ) from Section 2.1, we define
t
F {F }
[l ]:= φ φ φ : [0,T] R is - predictable for all i and
A 2 ≡ { i}i∈N | i → {Ft}
T
(cid:8) φ(s) 2 ds < a.s. ,
ˆ k kl2 ∞
0 (cid:27)
T
S [l ]:= φ= φ L2(0,T;l ) : φ(s) 2 ds N ,
N 2 { i}i∈N ∈ 2 ˆ k kl2 ≤
(cid:26) 0 (cid:27)
[l ]:= u [l ] : u S [l ] almost surely .
N 2 2 N 2
A { ∈ A ∈ }
We equip S [l ] with the weak topology in L2(0,T;l ) such that S [l ] is a Polish space.
N 2 2 N 2
Now, for σ > 0 consider X , φσ given by (2.6) and (2.7) and let
σ s,t
φσ := φσ () : 0 s t T
s,t · ≤ ≤ ≤
be the forward stochastic Ito¯ flow of C(cid:8)k-diffeomorphisms assoc(cid:9)iated to X .
σ
The following theorem was proved in slightly more generality (i.e. replacing U by Rn) in [4].
Theorem 4.1. [4] The family (φσ,X ) satisfies a LDP in the spaces Wˆ W and
σ σ>0 k−1 k−1
×
W W with rate function
k−1 k−1
×
1 T
I (φ,X) = inf f(s) 2 ds : f L2(0,T;l ) s.t. (φ0,f,X0,f) = (φ,X) .
W 2ˆ k kl2 ∈ 2
(cid:26) 0 (cid:27)
Below, in Theorem 4.7, we generalize this theorem to Wγ (Q) Wγ (Q) (see the defi-
k−1,id × k−1,0
nition in (3.2)). However, we first need to show that (φσ,X ) have enough regularity:
σ σ>0
Lemma 4.2. For all σ > 0 and all 0 < γ < 1, the pair (φσ,X ) is in Wˆ (Q) Wγ (Q)
2 σ k−1 × k−1,0
and Wγ (Q) Wγ (Q) almost surely.
k−1,id × k−1,0
Proof. Theprooffollows[4,Prop. 4.10]. Introducingthenotation forthenormonWj,p(U),
k·kj,p
we note that according to [4] Lemmas 4.7-4.9, for each p >1 there exists C such that
p
supE φσ φσ p C t s p/2
k t − skk,p ≤ p| − |
σ
supE X (t, ) X (s, ) p C t s p/2
k σ · − σ · kk,p ≤ p| − |
σ
and due to the initial values X (0, ) = 0, φσ(x) = x we also have
σ · 0
supE φσ p +supE X (t, ) p C .
k tkk,p k σ · kk,p ≤ p
σ σ
Since the Sobolev embedding Wk,p(U) ֒ Ck−1,γ(U) is continuous if γ > 0 and 3 < 1 γ [1],
→ p −
also Wk,p(U) ֒ λk−1,γ(U) is continuous for all γ > 0 with 3 < 1 γ. Lemma 3.4 yields the
→ p −
desired Hölder-regularity in time.The other properties, i.e. φ being a diffeomorphism and the
boundary values follow from [19], Theorem 4.6.5 and Assumption 2.1. (cid:3)
4.1. Proof of Theorem 2.5. Let f with f = fl be a sequence in [l ] for some
{ n}n∈N n n l∈N AN 2
fixed N < and let f [l ]. Let σ be a sequence of positive numbers such that
∞ ∈ AN 2 { n}n∈N (cid:8) (cid:9)
σ 0 for n . For simplicity of notation, we set
n
→ → ∞
∞ t
M(t,x) := X(k)(s,x)dB (s), (x,t) Rn [0,T].
ˆ k ∀ ∈ ×
0
k=1
X
10 M.HEIDAANDM.RÖGER
Then, we define the following quantities:
t t
X (t,x) = b (s,x)ds+√σ M(ds,x), (4.1)
n ˆ fn nˆ
0 0
t
X (t,x) = b (s,x)ds,
0 ˆ f
0
t t
φn(x) = x+ b (s,φn(x))ds+√σ M(ds,φn(x)), (4.2)
t ˆ fn s nˆ s
0 0
t
φ0(x) = x+ b (s,φ0(x))ds
t ˆ f s
0
where b , b are defined by (2.8).
fn f
Note that due to Lemma 4.2, (φn,X ), φ0,X Wγ (Q) Wγ (Q). We next specify
t n t 0 ∈ k−1,id × k−1,0
suitable notions of weak convergence in Wˆ W and Wγ (Q) Wγ (Q):
(cid:0)k−1× (cid:1)k−1 k−1,id × k−1,0
Definition 4.3. [4] Let Pˆn , Pˆ0 be the measures induced by (φn,X ), φ0,X respectively
k−1 k−1 t n t 0
on Wˆ W , i.e.
k−1× k−1 (cid:0) (cid:1)
Pˆn (A) = P((φn,X ) A) , Pˆ0 (A) = P (φ0,X ) A A (Wˆ W ).
k−1 n ∈ k−1 0 ∈ ∀ ∈ B k−1× k−1
The sequence (φn,X ) is said to converge(cid:0)weakly as Gk−(cid:1)1-flows to φ0,X as n if
{ t n }n∈N t 0 → ∞
Pˆn converges weakly as measures to Pˆ0 as n .
k−1 k−1 → ∞ (cid:0) (cid:1)
Definition 4.4. Let Pn , P0 be the measures induced by (φn,X ), φ0,X respectively on
k−1 k−1 t n t 0
Wγ (Q) Wγ (Q). The sequence (φn,X ) is said to converge weakly as Ck−1-flows
k−1,id × k−1,0 { t n }n∈N (cid:0) (cid:1) γ
to φ0,X as n if Pn converges weakly as measures to P0 as n .
t 0 → ∞ k−1 k−1 → ∞
N(cid:0)ote th(cid:1)at the last definition makes sense in view of Lemma 4.2 which guaranties (φn,X )
Wγ (Q) Wγ (Q). We find the following weak continuity property of the mappting fn ∈
k−1,id × k−1,0 7→
(φ ,X), which was proved in [4] for the Gk−1-case:
t
Theorem4.5. Letf converge tof indistributionasS [l ]-valued sequence ofrandomvariables.
n N 2
Then the sequence (φn,X ) converges weakly as Ck−1-flows and Gk−1-flows to the pair
{ n }n∈N γ
φ0,X as n for any γ < 1.
t 0 → ∞ 2
(cid:0) We p(cid:1)ostpone the proof of this theorem to Section 4.2.
Let R∞ := R be the usual product space. Then, = C([0,T];R∞) is a Polish space
n∈N S
and β = B is a random -valued variable (see [4]). As shown in [4], proofs of theorems in
{ i}i∈QN S
the spirit of Theorem 2.5 or Theorem 4.7 below basically reduce to applications of the following
result:
Theorem 4.6. [4,Theorem3.6] Let be aPolishspace, let σ be acollection ofmeasurable
E {G }σ≥0
maps from ( , ( )) to ( , ( )) and let Xσ = σ(√σβ). Suppose that there exists a measurable
map 0 : S B Ssuch thEatBfoEr every N < tGhe set Γ := 0( ·u(s)ds) : u S [l ] is a
G S → E ∞ N G 0 ∈ N 2
compact subset of . For f let ´
E ∈ E (cid:8) (cid:9)
·
:= u L2(0,T ; l ) : f = 0 u(s)ds .
Cf ∈ 2 G ˆ
(cid:26) (cid:18) 0 (cid:19)(cid:27)
Then, Iˆdefined by
1 T
Iˆ(f)= inf u(s) 2 ds , f ,
u∈Cf (cid:26)2 ˆ0 k kl2 (cid:27) ∈ E
is a good rate function on . Furthermore, suppose that for all N < and families uσ
E ∞ { } ⊂
[l ] such that uσ converges in distribution to some u [l ], we have that
N 2 N 2
A ∈ A
· ·
σ(√σβ + uσ(s)ds) 0( u(s)ds)
G ˆ → G ˆ
0 0
