Directory UMM :Data Elmu:jurnal:S:Stochastic Processes And Their Applications:Vol91.Issue2.2001:

Stochastic Processes and their Applications 91 (2001) 239–254
www.elsevier.com/locate/spa

On the small time large deviations of diusion processes
on conguration spaces
T.S. Zhang ∗
Department of Mathematics, Faculty of Science, HIA, 4604 Kristiansand, Norway
Department of Mathematics, University of Manchester, Oxford Road, Manchester M13 9PL, UK
Received 26 October 1999; received in revised form 31 May 2000; accepted 1 June 2000

Abstract
In this paper, we establish a small time large deviation principle for diusion processes on
c 2001 Elsevier Science B.V. All rights reserved.
conguration spaces.
Keywords: Dirichlet forms; Intrinsic metric; Large deviations; Conguration spaces

1. Introduction
Let R be the space of all integer-valued Radon measures on R. The Poisson system
of independent Brownian particles can be formally written as
Xt =

∞
X

Bti ;

i=1

{Bti ;

i¿1} are independent Brownian motions. One way to construct the difwhere
fusion is to use the Dirichlet form theory (see, for example, Albeverio et al., 1997).
The geometric analysis on conguration space R was carried out in Albeverio et
al. (1996a, 1996b, 1997). The intrinsic metric of the associated Dirichlet form was
identied as the L2 -Wasserstein-type distance in Rockner and Schied (1999). In this
paper, we will establish a small time large deviation principle for the diusion process
Xt ; t¿0 starting from an explicitly described invariant set. The rate function turns out
to be the square of the intrinsic metric of the Dirichlet form.
The paper is organized as follows. Section 2 is the framework. In Section 3, we give
an explicit expression of an invariant set of the diusion. The diusion starting from
every conguration in this set never leaves the set. Section 4 is devoted to the upper

bound estimates and identication of the rate function. The lower bound estimates is
Correspondence address: Department of Mathematics, University of Manchester, Oxford Road,
Manchester, M13 9PL, UK.
E-mail address: tzhang@math.uci.edu (T.S. Zhang).
∗

c 2001 Elsevier Science B.V. All rights reserved.
0304-4149/01/$ - see front matter
PII: S 0 3 0 4 - 4 1 4 9 ( 0 0 ) 0 0 0 6 2 - 4

240

T.S. Zhang / Stochastic Processes and their Applications 91 (2001) 239–254

put in Section 5. Section 6 is concerned with a small time large deviation of an average
form.
Remark 1.1. The paper is written for the case where the base space is R and the
motion is Brownian motion, but the method clearly works for Rd and a large class of
diusions. Now, the natural problem is the large deviation principle on the path space
over conguration spaces. This will be studied in a forthcoming paper.

2. Framework
Let R be the space of all integer-valued Radon measures on the real line R. Then
equipped with the vague topology R is a Polish space. The set of all
∈ R such that
({x}) ∈ {0; 1} is called conguration space over R. For simplicity, we also call R
conguration space. The geometric analysis on conguration space has been carried
out in Albeverio et al. (1997). We recall here some results there. For f ∈ C0 (R) (the
set of all continuous functions on R having compact support), set
Z
X
f(x):
hf;
i = f(x)
(d x) =
R

x∈

Dene the space of smooth cylinder functions on

on R of the form
u(
) = F(hf1 ;
i; : : : ; hfn ;
i);

∈

R,

FCb∞ , as the set of all functions
(2.1)

R;

for some n ∈ N , F ∈ Cb∞ (Rn ), and f1 ; : : : ; fn ∈ C0∞ (R). For u as in (2.1) dene its
gradient ∇u as a mapping from R × R to R:
∇u(
; x):=

n
X
i=1

@i F(hf1 ;
i; : : : ; hfn ;
i)∇fi (x);

∈

R;

x ∈ R:

Here @i denotes the partial dierential with respect to the ith coordinate, and ∇ is the
usual gradient on R.
Let  denote the Poisson measure on R with intensity d x, i.e., the unique measure
on R whose Laplace transform is given by
Z


Z
ehf;
i (d
) = exp
(ef(x) − 1)d x
R

R

for all f ∈ C0 (R). Now we introduce a pre-Dirichlet form:
Z
h∇u; ∇vi
(d
); u; v ∈ FCb∞ ;
E0 (u; v) =

(2.2)

R

R
where h∇u; ∇vi
= R ∇u(
; x)
R · ∇v(
; x)
(d x). It has been shown in Rockner and
Schied (1999) that E0 (u; v) = R h∇u; ∇vi
(d
) is closable on L2 ( R ; ) and its closure, denoted by (E; D(E)), is a quasi-regular Dirichlet form. Thus, according to the
general theory of Dirichlet forms (see Ma and Rockner (1992)), there is a diusion process {
; Xt ; P
;
∈ R } associated with the Dirichlet form (E; D(E)). It was proved

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T.S. Zhang / Stochastic Processes and their Applications 91 (2001) 239–254

in Albeverio et al. (1997) that the diusion Xt can be identied as innitely many

independent Brownian particles, i.e.,
X
Bti ; t¿0;
Xt =
i

Bti ;

i = 1; 2; 3; : : : are independent Brownian motions, and x denotes the Dirac
where
measure at the point x. The diusion X itself is also called Brownian motion on the
conguration space.
For u ∈ D(E), set (u; u)(
) = h∇u; ∇ui
. Recall that the intrinsic metric of the
Dirichlet form (E; D(E)) is dened by
(
; ) = sup{u(
) − u(); u ∈ D(E) ∩ C(

R)

and

(u; u)61 -a:e: on

R}

for
;  ∈ R . It was shown in Rockner and Schied (1999) that the intrinsic metric is
given by the Wasserstein-type distance on R .
)
(sZ
d(x; y)2 !(d x; dy); ! ∈
× ;
(2.3)
(
; ) = inf
R×R

where
× denotes the set of ! ∈
1
2
2 |x − y| .

R×R

having marginals
and , and d2 (x; y) =

3. Identication of the congurations
The Dirichlet form theory tells us that the diusion {
; Xt ; P
;
∈ R } exists for
quasi-every starting points, but does not give us the exceptional set explicitly. It is
easy to see that the diusion process Xt ; t ¿ 0 can not start from every conguration
P∞
P

∈ R , for example, Xt = i Bti with initial value
= i=1 ln(i) will not be a Radon
measure on R. In this section, we will identify the exceptional set and nd an explicit
invariant set for the diusion. Let us rst recall the denition of capacity. For an open
subset G, the capacity is dened as
Cap(G) = inf {E1 (u; u); u ∈ D(E); u¿1 -a:e: on G};
where E1 (u; u) = E(u; u) + (u; u)L2 (
For an arbitrary subset A,

R;

) .

Cap(A) = inf {Cap(G); A ⊂ G; G open}:
Let g(x) = 1=(1 + x2 ). Dene
Lemma 3.1. Cap(

c
g)

g

= {
; hg;
i ¡ ∞}. We have

= 0.

Proof. Set u(
) = hg;
i. The lemma follows if we show that u is quasi-continuous and
belongs to D(E), since a quasi-continuous version of an element in the domain of the
Dirichlet form is nite quasi-everywhere. First we note that
Z
2 Z
Z
2
u(
) (d
) =
g(x) d x + g2 (x) d x ¡ ∞:
R

R

R

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T.S. Zhang / Stochastic Processes and their Applications 91 (2001) 239–254

Furthermore,
Z
Z
Z
′
2
h∇u; ∇ui
(d
) =
hg (x) ;
i(d
) =
R

R

R

4x2
d x ¡ ∞:
(1 + x2 )4

Combining these two yields u ∈ D(E). Now, take a sequence of functions n ∈ C0∞ (R);
n¿1, such that 06n 61, |′n (x)|61, n (x) = 1 on [ − n; n] and 0 on (−∞; −n − 1] ∪
[n + 1; ∞). Set gn (x) = g(x)n (x). Dene un (
) = hgn ;
i, for n¿1. Then it is easy to
see that un (
) converges to u(
) in the Dirichlet space and also pointwisely. Since un ,
n¿1, is continous, it follows from Theorem 2:1:4. in Fukushima et al. (1994) that u
is quasi-continuous.
Lemma 3.2.
P
(Xt ∈

g;

Proof. Fix
∈
∞
X

P

is an invariant set for the diusion {
; Xt ; P
;
∈

g

i=1

R

∀t¿0) = 1;

∀
∈

with hg;
i =

P∞

R },

i.e.,

g:

i=1

1
¡ ∞; ∀t¿0
1 + (Bti )2

1=(1 + (
i )2 ) ¡ ∞. We need to show that
!
= 1:

By the choice of
, there exists an integer m0 such that
i 6= 0 for i¿m0 . Since under
P
, Bti , i¿1 are independent Brownian motions with initial values
i , i¿1, we have
for any integer n,




∞
∞
X
X
1
|
i |
6
P
sup |Bsi −
i |2
¡ ∞;
P
sup |Bsi −
i | ¿
i )2
3
((1=3)
s6n
s6n
i=m
i=m
0

0

−
i |2 ] is independent of i. By the Borel–
where we used the fact that
Cantelli lemma it follows that
!
\[
|
i |
i
i
= 0:
sup |Bs −
| ¿
P
3
s6n
P
[sups6n |Bsi

k i¿k

Dene

′ =

\[ \ 
n

k i¿k

sup |Bsi −
i |6
s6n

|
i |
3



:

Then, P
(
′ ) = 1. The lemma follows since
!
∞
X
1
′
¡ ∞; ∀t¿0 :

⊂
1 + (Bti )2
i=1

4. Large deviation estimates: the upper bound
Let  denote the set of all signed Radon measures on R. Equip  with the vague
topology generated by
{Uf; x = { ∈ ; |hf; i − x| ¡ }; f ∈ C0 (R); x ∈ R;  ¿ 0}:

T.S. Zhang / Stochastic Processes and their Applications 91 (2001) 239–254

243

Then,  is a locally convex topological space with ∗ being identied as C0 (R). It is
known that R is a closed subspace of . Let
0 ∈ g be xed. Dene, for  ∈ ,




Z
1
2
(4.1)
I
0 () = sup
hf; i − sup f(y) − |x − y|
0 (d x) :
2
f∈C0 (R)
R y
Lemma 4.1. Let f ∈ C0 (R). Then
(f) =: lim  log P
0 [e(1=)hf; X i ] =
→0



1
sup f(y) − |x − y|2
0 (d x):
2
R y

Z

Proof. By the independence,
Y
P
i0 [e(1=)f(B ) ]
(f) = lim  log
→0

= lim

→0

i0 ∈
0

X

 log P
i0 [e(1=)f(B ) ]:

i

By the large deviation principle of Brownian motion (see, for example, Dembo and
Zeitouni, 1992), it follows that
lim  log P
i0 [e(1=)f(B ) ] = sup(f(y) − 12 |
i0 − y|2 ):

→0

y

Taking the limit inside the series, we get
 Z



X
1
1 i
2
2
sup f(y) − |
0 − y| = sup f(y) − |x − y|
0 (d x):
(f) =
2
2
y
R y
i

It now remains to justify taking the limit inside the series. We suppose that the support
of f is contained in [ − a; a]. Without loss of generality, we can also assume
i0 ¿ 2a.
With these assumptions, we have
E
i0 [e(1=)f(B ) ]6E
i0 [e(1=)||f||∞ [−a; a] (B ) ]
=e

(1=)||f||∞

Z

a

−(y−
i0 )2 =2

e

−a

1
√
dy +
2

Z

i 2

|y|¿a

e−(y−
0 ) =2 √

1
dy
2

61 + e(1=)||f||∞ P(B ¡ a −
i0 )
i 2

61 + e(1=)||f||∞ e−(a−
0 ) = C;
2

where C = E[e(B1 ) ] for some  ¿ 0.
′

i 2

61 + e− (
0 ) = C
for big enough i and an appropriate choice of ′ . By the Schwartz’s inequality,
(E
i0 [e(1=)f(B ) ])−1 6 E
i0 [e−(1=)f(B ) ]
′

i 2

6 E
i0 [e(1=)||f||∞ [−a; a] (B ) ]61 + e− (
0 ) = C:
Hence, it follows that for 61,
′

i 2

′

i 2

′

i 2

|log E
i0 [e(1=)f(B ) ]|6log(1 + e− (
0 ) = C)6e− (
0 ) = C6e− (
0 ) C:

244

T.S. Zhang / Stochastic Processes and their Applications 91 (2001) 239–254

P∞
P
(1=)f(B )
i 2
]
Since
i  log P
i0 [e
i=1 1=(
0 ) ¡ ∞, the above estimates show that
converges uniformly with respect to , which justies the limit inside the series.
Proposition 4.2. Let  be the law of X on  under P
0 . Then { ;  ¿ 0} is exponentially tight; namely; for any L ¿ 0; there exists a compact subset KL such that
lim sup  log P
0 (X ∈ KLc )6 − L:

(4.2)

→0

Proof. Observe that the set of the following type:
\
K = { ∈ ; ||([ − n; n])6Ln }

(4.3)

n

is relatively compact. Given L ¿ 0, we are going to choose appropriate Ln so that
2
(4.2) holds.
Let us x a positive number 0 such that E[e0 |B1 | ] ¡ ∞. Let mn = ]{i;
p
|
i0 − n|6 1=0 }. Recall from the proof of Lemma 4:1 that
i 2

E
i0 [e(1=)[−n; n] (B ) ]61 + Ce−(0 (n−
0 ) −1)= :

Hence,
P
0 (X ([ − n; n]) ¿ Ln )
P∞
6e−Ln = E
0 [e(1=) i=1 [−n; n] (B ) ]
=e

−Ln =

∞
Y

E
i0 [e(1=)[−n; n] (B ) ]

i=1

6e−Ln =

∞
Y
i 2
(1 + Ce−(0 (n−
0 ) −1)= )
i=1

i 2

Y

6e−Ln =

(1 + Ce−(0 (n−
0 )

−1)

)

0 (
i0 −n)2 −1¿0

6e

−Ln =

−Ln =

6e

(

exp C
(

X

|
i0 −n|6
−(0 (n−
i0 )2 −1)

e

i

0 n2 +1

exp Ce

Y

X

)

√

(1 + Ce1= )
1=0

(1 + Ce1= )mn

−((1=2)0 (
i0 )2 )

e

i



)

(2C)mn emn =


Ln − mn
0 n2
= exp −
+ C0 e
+ mn log(2C) ;

P
i 2
2
where C0 = Ce i e−(1=2)0 (
0 ) . Now take Ln = L + mn + C0 e0 n + mn log(2C) + n and
dene KL as in (4.3). We have
P
0 (X 6∈ KL ) 6

∞
X
n=1

P
0 (X ([ − n; n]) ¿ Ln )

245

T.S. Zhang / Stochastic Processes and their Applications 91 (2001) 239–254

6

∞
X



2
Ln − mn
+ C0 e0 n + mn log(2C)
c exp −


∞
X



L+n
6Ce−L= :
exp −


n=1

6

n=1

This implies (4.2), which ends the proof.
Combination of Lemma 4.1, Proposition 4.2 and Theorem 4:5:20 in Dembo and
Zeitouni (1992) yields the following upper bound estimates:
Theorem 4.3. Let  be the law of X under P
0 . Then; for any closed subset F ⊂
lim sup  log  (F)6 − inf I
0 ():

(4.4)

∈F

→0

R;

Next; we are going to identify the rate function I
0 () as√(
0 ; )2 . Let M (R) denote
the space of positive Radon measures on R. Set d(x; y)=(1= 2)|x−y|. Let !;
∈ M (R).
The L2 -Wasserstein-type distance (!;
) is similarly dened as
)
(sZ
d(x; y)2 (d x; dy);  ∈ !;
;
(!;
) = inf
R×R

where !;
stands for the set of  ∈ M (R × R) having marginals ! and
.
Now x
0 ∈ M (R). Dene h(
) = (
0 ;
)2 .
Proposition 4.4. h(
) is convex and lower semi-continuous on M (R) with the topology
of vague convergence.
Proof. Let
1 ;
2 ∈ M (R). For any 0 ¡  ¡ 1, 1 ∈
(1 − )2 ∈ M (R × R). Then,

0 ;
1 ,

2 ∈

0 ;
2 ,

dene  = 1 +

(d x; R) = 1 (d x; R) + (1 − )2 (d x; R) = 
0 (d x) + (1 − )
0 (d x) =
0 (d x);

(R; dy) = 
1 (dy) + (1 − )
2 (dy):

Hence  ∈

0 ; 
1 +(1−)
2 ,

and
Z
h(
1 + (1 − )
2 ) 6

d2 (x; y)(d x; dy)

R×R

=

Z

R×R

1

2

1

d (x; y) (d x; dy) + (1 − )

Z

d2 (x; y)2 (d x; dy):

R×R

(4.5)

2

Since  and  are arbitrary, it follows from (4.5) that
h(
1 + (1 − )
2 )6h(
1 ) + (1 − )h(
2 )
which completes the proof of the convexity. Next, we prove that h is lower semicontinuous. Let {
n ; n¿1} be a sequence of positive Radon measures on R converging
vaguely to
 ∈ M (R). We need to show
h(
)6

lim inf h(
n ):
n→∞

(4.6)

246

T.S. Zhang / Stochastic Processes and their Applications 91 (2001) 239–254

To this end, without loss of generality, we assume that the limit limn→∞ h(
n ) exists and is nite. By the denition of , there exists a sequence of Radon measures
n (d x; dy) ∈ M (R × R) such that n ∈
0 ;
n and
Z
d2 (x; y)n (d x; dy):
lim h(
n ) = lim
n→∞

n→∞

R×R

Let 1 be the projection operator from R × R to R dened by 1 (x; y) = x. For any
compact set K ⊂ R × R, we have
sup n (K)6 sup 1∗ n (1 (K)) =
0 (1 (K)) ¡ ∞:
n

n

This shows that the family {n ; n¿1} is relatively compact with respect to the topology
of vague convergence. Let {nk ; k¿1} be a subsequence converging vaguely to some
Radon measure 0 on R × R. Next, we prove that
0 (d x; R) =
0 (d x);

0 (R; dy) =
(dy):


Note that this is not the consequence of vague convergence. Because of the similarity,

we prove one of them, say, 0 (R; dy) =
(dy).
Choose a sequence {m (x); m¿1} of C ∞ functions satisfying 06m (x)61, m (x)=
1 on [ − m; m], m (x) = 0 on [ − m − 1; m + 1]c . Let any f ∈ C0 (R). Suppose
Supp[f] ⊂ [ − m0 ; m0 ] for some m0 . Then,
Z
Z
f(y)m (x)0 (d x; dy)
f(y)0 (d x; dy) = lim
m→∞

R×R

R×R

= lim lim

m→∞ k→∞

Z

f(y)m (x)nk (d x; dy):

(4.7)

R×R

But,

Z

 lim
k→∞

R×R

f(y)m (x)nk (d x; dy) − lim

k→∞


Z

=  lim
k→∞

R×R

Z

6 c lim

k→∞

6 c sup
k



f(y)nk (d x; dy)

[−m0 ; m0 ] (y)[−m; m]c (x)nk (d x; dy)

R×R

Z

{d2 (x;y)¿(m−m0 )2 }

Z

nk (d x; dy)

d2 (x; y)nk (d x; dy)6

R×R

Thus, it follows from (4.7) and (4.8) that
Z
Z
f(y)0 (d x; dy) = lim lim
m→∞ k→∞

R×R

k→∞

R×R



f(y)(m (x) − 1)nk (d x; dy)

1
6c
sup
(m − m0 )2 k

= lim

Z

Z

R×R

M
:
(m − m0 )2

(4.8)

f(y)m (x)nk (d x; dy)

R×R

f(y)nk (d x; dy) = lim

k→∞

Z

R

f(y)
nk (dy) =

Z

R

f(y)
(dy):


T.S. Zhang / Stochastic Processes and their Applications 91 (2001) 239–254

247

Since f is arbitrary, we conclude that 0 (R; dy) =
(dy).

Hence,
Z
d2 (x; y)0 (d x; dy)
h(
)
 6
R×R

6 lim

n→∞

Z

d2 (x; y)n (d x; dy) = lim h(
n ):
n→∞

R×R

The proof is complete.
Lemma 4.5. If f ∈ C0 (R); then g(x) = supy (f(y) − 21 |x − y|2 ) ∈ C0 (R).
Proof. Suppose Supp[f] ⊂ [ − M; M ] and C = supy |f(y)|. Then, for |x| ¿ M ,
06g(x) =

sup

(f(y) − 12 |x − y|2 ) ∨ sup (− 12 |x − y|2 )

y∈[−M; M ]

|y|¿M

6 sup (C − 21 |x − y|2 ) ∨ 0:
|y|6M

The lemma follows as sup|y|6M (C − 21 |x − y|2 ) ¡ 0 for big enough |x|.
Lemma 4.6. Let f ∈ C0 (R). Then

1
2
sup (¡ f;
¿ − (
0 ;
) ) = sup f(y) − |x − y|
0 (d x):
2
∈ R
R y∈R
Z

2



P∞
Proof. Suppose that
0 = i=1 
i0 . Due to the compact support, there exists integer
N0 such that f(
i0 ) = 0 and supy∈R (f(y) − 21 |
i0 − y|2 ) = 0 for i ¿ N0 . Let
∈ R with
(
0 ;
) ¡ ∞. According to Rockner and Schied (1999), the distance (
0 ;
) is reached
P∞
by some  ∈ M (R × R) having marginals
0 and
. Let us say  = i=1 
i0 ;
i . Thus,
P∞
P∞
= i=1 
i and (
0 ;
)2 = 21 i=1 |
i0 −
i |2 . Hence,
hf;
i − (
0 ;
)2 =

=

∞
X
i=1

∞ 
X
i=1

=

f(
i ) −

Z

∞

1X i
|
0 −
i |2
2
i=1


 X

∞
1 i
1 i
i 2
2
sup f(y) − |
0 − y|
f(
) − |
0 −
| 6
2
2
y
i

i=1





1
sup f(y) − |x − y|2
0 (d x):
2
y
R

On the other hand, for any  ¿ 0, there exist yi ; i = 1; 2; : : : ; N0 such that

N 
 Z


0

X
1 i
1


2
2
f(yi ) − |
0 − yi | − sup f(y) − |x − y|
0 (d x)



2
2
y
R
i=1

N 
 X


N0
0
X
1 i
1


f(yi ) − |
0 − yi |2 −
sup f(y) − |
i0 − y|2  6:
=


2
2
y
i=1

i=1

248

T.S. Zhang / Stochastic Processes and their Applications 91 (2001) 239–254

Dene
i = yi if i6N0 and
i =
i0 if i ¿ N0 . Then,
hf;
i − (
0 ;
)2
¿

N0
X
i=1

f(yi ) −

N0
X
1
i=1

2

|yi −

i0 |2 ¿




1
2
sup f(y) − |x − y|
0 (d x) − :
2
R y

Z

Hence,



1
2
sup (hf;
i − (
0 ;
) )¿ sup f(y) − |x − y|
0 (d x) − :
2
∈ R
R y
Z

2

Since  is arbitrary, the proof is complete.
Proposition 4.7. I
0 (
) = (
0 ;
)2 for all
∈

R.

Proof. By Lemma 4.6,
I
0 (
) = sup
f∈C0 (R)

2

!

ˆ − (
0 ;
)
ˆ ) :
hf;
i − sup (hf;
i
∈
ˆ

R

ˆ − (
0 ;
)
ˆ 2 )6(
0 ;
)2 , we have
Since for any f ∈ C0 (R), hf;
i − sup
∈
ˆ R (hf;
i
I
0 (
)6(
0 ;
)2 . On the other hand, by the general theorem on the inverse legendre transform of convex functions (see, e.g., Gross, 1980) and Proposition 4.4, we
obtain
!
(
0 ;
)2 = sup

f∈C0 (R)

ˆ − (
0 ;
)
ˆ 2) :
hf;
i − sup (hf;
i

(4.9)

∈M
ˆ
(R)

Since I
0 (
) is greater than the right-hand side of (4.9), the proposition follows.

5. Large deviation estimates: the lower bound
Let U be an open neighborhood described by
)
(
n
n
X
X
2
xi with
d(xi ; yi ) ¡  ;
U =
∈ R ;
(@Wr ) = 0
|Wr =
i=1

i=1

where Wr = {|x| ¡ r}.
Proposition 5.1. Let
0 ∈

g.

Then; for any 1 ¿ 0 and distinct integers i1 ; : : : ; in ;
n

1X
i
(yj −
0j )2 (1 + 1 ) −
lim inf  ln P
0 (X ∈ U ) ¿ −
→0
2
j=1

−

X

k6∈{i1 ; i2 ;:::; in }

d(
k0 ; Wrc )2 :




1
+1 
1

T.S. Zhang / Stochastic Processes and their Applications 91 (2001) 239–254

Proof. For any distinct integers i1 ; i2 ; i3 ; : : : ; in , dene

n

X
i
d(Bj ; yj )2 ¡ ;
Ai1 ; i2 ;:::; in = Bi1 ∈ Wr ; : : : ; Bin ∈ Wr ;

j=1

c
Bk ∈ W r ;

k 6∈ {i1 ; : : : ; in }

Then,
{X ∈ U } =

[




:



Ai1 ; i2 ;:::;in

i1 ; i2 ;:::; in

P
0 (X ∈ U ) =

∞
X

P
0 (Ai1 ; i2 ;:::;in ):

i1 ; i2 ;:::; in

Now,


P
0 (Ai1 ; i2 :::;in ) = P
0 Bi1 ∈ Wr ; : : : ; Bin ∈ Wr ;
×
=

Z

k6∈{i1 ; i2 ;:::; in }

Z
· · · Pn

j=1

×exp
×

Y

j=1

c

Y

−

n
X

P
0 (Bk ∈ W r )



i

d(Bj ; yj )2 ¡ 

(2)−n=2
d2 (xj ;yj )¡; xj ∈Wr

Pn

j=1 (xj

k6∈{i1 ; i2 ;:::; in }

2

i

−
0j )2

Z

√

|x|¿r

!

d x1 d x2 · · · d x n



(x −
k0 )2
1
dx
exp −
2
2

= ai1 ; i2 ; :::; in × bi1 ; i2 ; :::; in :
For any 1 ¿ 0, we have
 n

n
n
X
X
1 X
i
i
(xj −
0j )2 6 1 +
(xj − yj )2 + (1 + 1 )
(yj −
0j )2 :
1
j=1

j=1

j=1

Thus,
ai1 ; i2 ; :::; in ¿

Z

Z
· · · Pn

j=1

d2 (xj ;yj )¡;

×exp −(1 + 1 )

Pn



(1 + 1=1 )
;
(2)−n=2 exp −
2
xj ∈Wr

j=1 (yj

2

i

−
0j )2

!

d x1 d x2 · · · d x n :

249

250

T.S. Zhang / Stochastic Processes and their Applications 91 (2001) 239–254

Hence,
lim inf
→0

To treat

 log ai1 ; i2 ; :::; in ¿

bi1 ; i2 ; :::; in ,

1
−
2




n
1
1X
i
+ 1  − (1 + 1 )
(yj −
0j )2 :
1
2
j=1

we need the following

Lemma 5.2. There exists a constant K ¿ 0 such that if |x| ¿ 2r ∨ K; then
2

Px (|B | ¿ r)¿e−C=x ;

(5.1)

where C is a positive constant; B is a Brownian motion starting from x under Px .
Proof. We rst assume x ¿ 0. We have
Px (|B | ¿ r) = P0 (|B + x| ¿ r)



x
x
= P0 B1 ¿ − √
¿P0 (B ¿ r − x)¿P0 B ¿ −
2
2 
= P0
Thus,



x
B1 ¡ √
2 

C=x2

Px (|B | ¿ r)e



=

C=x2

¿e

Z

√
x=2 

e−y

2

=2

−∞

Z

√
x=2 

−∞

e−y

2

=2

1
√ dy:
2
1
√ dy:
2

It is therefore, sucient to show
Z x=2√
2
1
C=x2
e−y =2 √ dy¿1
FC;  (x) = e
2
−∞
for big enough x and suciently small .
Since FC;  (x) is decreasing in , we can let  = 41 . Put
Z x=2
2
1
4C=x2
e−y =2 √ dy:
F(x) = F 1 (x) = e
C; 4
2
−∞

(5.2)

Since limx→∞ F(x) = 1, to show (5.2), it is sucient to verify F ′ (x)60 for big
enough x. Dierentiating F, we get


Z
2
2
2
8C x
1
1
√ e−y =2 dy ;
F ′ (x) = e4C=x √ e−x =2 − 3
x −∞ 2
2
which is clearly negative when x is big enough.
Now, we assume x ¡ 0. In this case,

x
Px (|B | ¿ r) ¿ P0 (B ¡ − r − x)¿P0 B ¡ −
2


x
= P 0 B1 ¡ − √ :
2 
By the same reason as above, we can choose  = 14 . Then, for 6 14 ,
Z −x
2
1
√ e−y =2 dy:
Px (|B | ¿ r)¿
2
−∞

T.S. Zhang / Stochastic Processes and their Applications 91 (2001) 239–254

251

Dene
ˆ
F(x)
= e4C=x

2

Z

−x

−∞

2
1
√ e−y =2 dy:
2

ˆ
ˆ
We see that F(x)
= F(−x), which implies that F(x)¿1
for big enough |x|. The proof
is complete.
Lemma 5.3. We have
lim  log bi1 ; i2 ; :::; in = −

→0

Proof. From the denition,
X
 log bi1 ; i2 ; :::; in =

X

d(
k0 ; Wrc )2 :

(5.3)

k6∈{i1 ;:::; in }

 log P
k (|B | ¿ r):
0

k6∈{i1 ;:::; in }

By the large deviation principle of Brownian motion, we know that
lim  log P
k (|B | ¿ r) = −d(
k0 ; Wrc )2

→0

0

On the other hand, it follows from Lemma 5.2 that
| log P
k (|B | ¿ r)|6
0

M
:
(
k0 )2

Eq. (5.3) follows from the dominated convergence theorem.
Combination of Lemmas 5.3 and 5.2 gives Proposition 5.1.
Theorem 5.4. Let
0 ∈

g.

Then for any open set O ⊂

R;

we have

lim inf  ln P
0 (X ∈ O)¿ − inf (
0 ;
)2 :
→0

∈O

Proof. It is sucient to prove
ˆ2
lim inf  ln P
0 (X ∈ U )¿ − (
0 ;
)
→0

for any
ˆ ∈ O. Fix such
ˆ ∈ O. Without loss of generality, we can assume that
ˆ 2=
(
0 ;
)

∞

1X i
|
ˆ −
i0 |2 ¡ ∞:
2
i=1

Choose an increasing sequence Wrn of open balls such that
|
ˆ Wrn =

mn
X


ˆ i ;

(@W
ˆ
rn ) = 0:

S

Wrn = R and

i=1

Let m be a sequence of positive numbers converging to zero. Set
(
)
mn
mn
X
X
i 2
xi with
d(xi ;
ˆ ) ¡ m :
Un; m =
∈ R ;
(@Wrn ) = 0
|Wrn =
i=1

i=1

Since O is open, there exist n0 ; m0 such that Un; m ⊂ O for n¿n0 , m¿m0 .

252

T.S. Zhang / Stochastic Processes and their Applications 91 (2001) 239–254

By Proposition 5.1 it holds that for m¿m0 ; n¿n0 and any 1 ¿ 0,
lim inf  ln P
0 (X ∈ O)¿ lim inf  ln P
0 (X ∈ Un; m )
→0

1
¿−
2

→0

mn
X
i=1

i

(
ˆ −

i0 )2 (1

1
+ 1 ) −
2




∞
X
1
c
+ 1 m −
d(
k0 ; W rn )2 :
1
k=mn +1

First letting m → ∞ and then 1 → 0, we obtain that
mn
∞
X
1X
c
(
ˆ i −
i0 )2 −
d(
k0 ; W rn )2 :
lim inf  ln P
0 (X ∈ O)¿ −
→0
2
i=1

(5.4)

k=mn +1

c
By the choice of Wrn , d(
k0 ; W rn )6d(
k0 ;
ˆ k ) for k¿mn + 1. Hence, it follows from
(5.4) that
∞
1X i
(
ˆ −
i0 )2 = −(
0 ;
)
ˆ 2:
lim inf  ln P
0 (X ∈ O)¿ −
→0
2
i=1

The proof is complete.

6. Large deviations of an average form
In this section, we will prove a large deviation principle of an average form for
the Brownian motion on conguration space. See the related results for other type of
diusions in Aida and Zhang (1999) and Zhang (2000).
Set
Z
P
(·) d:
P (·) =
R

Let B and C be any Borel subsets of R . Dene


(B; C) = max ess inf (
; C); ess inf (
; B) :
∈B

∈C

Theorem 6.1. Let B; C be two Borel subsets with (B) ¿ 0; (C) ¿ 0. Then
lim sup  log P (X0 ∈ B; X ∈ C)6 − (B; C)2 :
→0

Proof. We proceed as in Fang (1994) and Fang and Zhang (1999). Without loss
of generality, we can assume (B; C) ¿ 0. Let  be any positive number such that
 ¡ (B; C). Then, ess inf
∈B (
; C) ¿  or ess inf
∈C (
; B) ¿ . We assume, for
example, the later holds. This implies that there exists a Borel set K ⊂ C with (K) =
(C) such that
(
; B) ¿ 

for all
∈ K:

(6.1)

Now x an integer n ¿ . Dene u(
)=(
; B)∧n. Applying Proposition 3:1 in Rockner
and Schied (1999), we have u ∈ D(E). Then, by the Lyons–Zheng’s decomposition (see
Lyons and Zhang (1996); Lyons and Zheng (1988)), under P the following holds.
1
1
u
(
t (!))) for 06s6t;
(6.2)
u(Xs ) − u(X0 ) = Msu − (Mtu (
t (!)) − Mt−s
2
2

T.S. Zhang / Stochastic Processes and their Applications 91 (2001) 239–254

where M u is a Ft = (Xs ; s6t)-square integrable martingale with
Z t
(u; u)(Xs ) ds
hM u it =

253

(6.3)

0

and
t is the reverse operator such that Xs (
t (!)) = !(t − s); 06s6t:
Remarking that  is an invariant measure of the diusion process, it follows that
P (X0 ∈ B; X ∈ C) = P (X0 ∈ B; X ∈ K)
6 P (u(X ) ¿ ; u(X0 ) = 0)6P (u(X ) − u(X0 ) ¿ )
= P ( 21 (Mu − Mu (
 (!))) ¿ )
6 P (Mu ¿ ) + P (−Mu ¿ )
64

Z

∞
√

= 

2
1
√ e−s =2 ds ;
2

(6.4)

where we have used the reversibility of the diusion and hM u it 6t which follows from
(6:3) and (u; u)(
)61. Thus we get from (6.4) that
lim sup  log P (X0 ∈ B; X ∈ C)6 − 2 :

(6.5)

→0

Letting  → (B; C), the assertion follows.
Theorem 6.2. Let B; C be two Borel subsets with (B) ¿ 0; (C) ¿ 0. If B or C is
open; then
lim inf  log P (X0 ∈ B; X ∈ C)¿ − (B; C)2 :
→0

(6.6)

Proof. Assume, for example, C is open. Let  = (B; C) and  ¿ 0. Since
ess inf
∈B (
; C) ¡ +, there exists a compact subset K ⊂ B∩ g with (K) ¿ 0 such
that (
; C) ¡  +  on K. For any
∈ K, appplying the large deviation estimates in
Theorem 5.4 it follows that
lim inf  log P
(X ∈ C)¿ − (
; C)2 ¿ − ( + )2 :
→0

(6.7)

Fix an arbitrary sequence n of positive numbers converging to zero. It suces to show
(6.6) for such sequences. First, we see that (6.7) implies
[\
{
; n log P
(Xn ∈ C) ¿ − ( + )2 }:
(6.8)
K⊂
m n¿m

T
Thus, there exists m0 such that K0 = n¿m0 {
; n log P
(Xn ∈ C) ¿ − ( + )2 } ∩ K has
positive measure. Hence, by Jensen’s inequality,
lim inf n log P (X0 ∈ B; Xn ∈ C)¿ lim inf n log P (X0 ∈ K0 ; Xn ∈ C)
n→∞

n→∞

= lim inf n log
n→∞

Z

K0


P
(Xn ∈ C)(d
)

254

T.S. Zhang / Stochastic Processes and their Applications 91 (2001) 239–254



1
¿ lim inf n log((K0 )) + lim inf n log
n→∞
n→∞
(K0 )
Z
1
n log P
(Xn ∈ C)(d
):
¿ lim inf
n→∞ (K0 ) K
0

Z

K0


P
(Xn ∈ C)(d
)

By (6.8),
1
¿
(K0 )

Z

K0

(−( + )2 )(d
) = −( + )2 :

Since  is arbitrary, the theorem is proved.

Acknowledgements
I am grateful to M. Rockner, A. Schied and J. UbHe for stimulating discussions
during the preparation of this work. My special thanks go to L. Gross for the stimulating
discussions and his hospitality during my stay at Cornell University. Actually, the proof
of Proposition 4.4 is based on a discussion with him. I also thank the anonymous referee
for the very careful reading and useful comments.
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