LIDS-P-146 4
May 1985
On the Rate at which a Homogeneous Diffusion
Approaches a Limit, an Application of the
Large Deviation Theory of Certain Stochastic Integrals
Daniel W. Stroock*
Summary:
Let X(T) be the solution to a stochastic differential equation whose
coefficients are homogeneous of degree 1 (e.g., a linear S.D.E.).
Under mild
conditions, it is shown that limits like
limT
Log P(IX(T)I/IX(O)I > R)
T
exist and a formula is provided for their computation.
The techniques
developed apply to a broad class of situations besides the one treated here.
Running Head:
Large deviations of stochastic integrals
AMS Classification:
60J60, 60F10, 60H05
The research contained herein was sponsored by N.S.F. grant MCS8310642.
and by Army Research Office Grant No. DAAG29-84K-005.
1.Some Prel iminaries and
tatement of the Results:
N > 2
The notation introduced below will be used throughout.
-
are fixed integers; {VO,...DVd}
C (R
l\(O;
R
is
a collection of vector
fields each of which is homogeneous of degree 1 (i.e.,
and
(B(t)
=
))
(Bl(t),...,d(t
, 7t, P)
d > I
and
Vk(x)
=
IxlVk(
))
;
is a d-dimensional Brownian motion.
V
V, it will often be useful to identify
N
I Vi
which it determines.
with the directional derivative operator
When dealing with a.vector field
il
Vf _
Thus, for example,
C Vi
il
.-.
and
t
2
V f - V-Vf .
dt.
(1.1) Lemma:
For each
x E R \{0}
there is a P-almost surely unique,
t > O}-progressively measurable function
right-continuous, {[7:
that
will be used some-
*dS (t)
convenience when writing stochastic integrals,
times to denote
Also, for notational
i
P(X(t,x) E R \{0}
t > 0) - 1
for all
and
X(*,x)
X(-,x)
such
satisfies the
Stratonvich stochastic integral equation:
(1.2)
Moreover,
(1.3)
X(T,x)
if
p(T,x)
p(T,x)-
d
T
I I Vk(X(t,x))'dSk(t),
k-0 0
x +
log(lX(T,x)l/lxl)
-
d
I
T
d
T
and
T > O .
8(T,x)
-
X(T,x)/lX(T,x)l
f ak(S(t,x))-dB$(t)
k-0 0
I
k-1O
T
aok(
8
(t,x))dlSk(t) + f Q((t,x))dt ,
0
and
d
(1.4)
e(T,x) =X
+
d
T
k l f Wk((tx)).dS(t)
k=O
-
k
,
T > 0
T > 0 ,
,
then
2
where
Ok(e)
(,Vk(8))RN
I Wk(k)(d)
+
for
,
Wk(e) ' Vk(e) -
aO
0 (9)
k(e)e , and Q(9)
.
e ES
By the standard theory of stochastic integral equations, there is
Proof:
no problem about the existence and uniqueness of
time
X(-,x)
hits
that
P(.,x)
and
O .
X(-,x)
up until the first
Moreover, up until that time, it is easy to check
9(.,x)
satisfy (1.3) and (1.4), respectively.
Finally,
inf 1X(t,x)I/[xl > 0 (a.s.,P) for each
O<t<T
Hence, P-almost surely, X(.,x) never hits 0 in a finite time.
from (1.3), it is clear that
T > 0.
Q.E.D.
As a consequence of (1.4), it is clear that, for each
S i 'i
(.*,x) is the diffusion on
(1.5)
Let
k'Il W2Sk
k-i
L.'
P(T,8,-)
, (T,9)
E(0,)
by
x S
and generated by
, denote the transition probability
Henceforth it will be assumed that
Lie(Wl,...,Wd)(9)
(Lie (W1 ,...,d )
X
0
function for this diffusion.
(1.6)
starting at
x ER \{O) ,
T 8(S I),
8E S
1
.
denotes the Lie algebra of vector field on
{Wl,...,Wd} .)
SN-1
generated
In particular, by a renownedtheorem of L. HRrmander [21,
(1.6) guarantees that there is a smooth map (T,O,n) E (0,*) x SI-xSN-i1
such that
measure on
P(T,$,dn) - p(T,9,n)dn , where
S-
.
1
.
dn denotes the normalized Lebesgue
Moreover, by the strong Maximum principle (cf. Theorem
(6.1) in [3]), one can easily see that
x SN-t x St
p(T,,9n)
p(T,O8,)
> 0
for all
Hence, by Doeblin's Theorem, there is a unique
(the probability measures on
S
) such that
(T,O8,q) E (0,0)
m E M1 (SN)
3
!
T+
Since
m =
f
T
0.
)<
log( sup IP(T,9,.)-ml
var
sN-1
P(T,9,-)m(dO) , T > 0 , it is obvious from the preceding
discussion about
p(T,9,-)
positive everywhere on
sN
m(dn) -
that
1
.
In the future,
*E C(S
where
l(n)dn
f
fdm
)
is
will be denoted by
f
f ELl(m) .
for
The goal of this article is to prove several results about the behavior
P(p(Tx)/T E r)
of
,
x E RN\{O}
and
r6E
R ,
as
statement is a rather abstract existence assertion.
T +
'
.
The first
Subsequent statements
provide more concrete information.
(1.7)
i:
Theorem:
R 1 + [O,-) U {(}
There is a lower semi-continuous, convex function
r E
such, that, for each
R
:
(1.8)
inf I(p)
lim 1 log(inf P(p(T,x)/T E r)) > pEin*r
x
T
T+--
(1.9)
lim T log(sup P(p(T,x)/T E r)) < - inf I(p) ,
pEr
Tts
and
wbere it is to be understood that
x
varies over
I , it will be useful to have some
In order to describe the function
additional notation.
S
Define the function
a and the vector field
by
·-
and
respectively; set
R\{O} .
I
k-I
o2
W
on
4
a
(1.10)
inf{ I
f (ak-wk,)2d: *E C (S l)} ;
k-1
and define the bilinear operation
<,.>
by
d
<412
>'
k-i(Wk1 )'(Wk02)
Assume that
(1.11)
The'orem:
(1.12)
I(p) = sup inf[(p -
where 4 varies over C (S
when e
(ak
{(Q
or p
-
a > 0 .
Then
(Q - L2)d4) /2
), U varies over M1 (S
(ak
Wk)
du]
); and it is understood that,
=
Wkt)2du = 0, the ratio is 0 or - according to whether
- L)du.
(1.13)
1''2 E Cm(SN1-)
(Q
In particular, there is an A E (0,o) such that:
A(p -_Q)2 < I(p) < (p -Q)
2/2a, pe R'
and so, I E C(R'), I(Q)' = 0, and I is strictly increasing (decreasing) on
(Q,Q)
((-_,Q)).
(1.14)
f E C (S
Theorem:
)
Assume that
such that
f0
0 .
a
O and
Then there is a unique
Wkf I-
1< k < d .
Moreover, if
Q - Q- Lf , then
(1.15)
I(P)
inf{JO( u ): U E M(S
1)
and
p - f Qdu},
where
J.o()
0l~rb) - -
(1.16)
inf{f (22 <*,.>
and it is to be understood that
satisfying
then
I
I
du - p .
is continuous on
+
L)d:I + E C (SN1)}
I(P) - -
In particular, if
if there is no
q. -
u E M (S- l)
sup{±Q(M):
(q,q+) and is infinite off of
E SN-1
[q.Sq
,
.e Finally
L)djl
I(
Lf
is
I
and so
Remark:
2=
k)
Wk(a
(1.19)
Thus, Q
If either
Corollary:
R: (0,-) * (0,-)
|T
lim sup
T~4 x
<0
inf
xz
a - 0
and
observe that
0 E (q,q+) , then
and
a - 0
lim T logR(T) - 0:
satisfying
M
noreover,
.
if
> R(T))
P(IX(T,x) I/Ix >R(T))
(1.22)
Acknowedgement:
me by Mark Pinsky.
> R(T))) - Il(C)
|
O
0
a
and
Q
> 0,
then
0 .
q, < 0 , then
lin sup 1 log
To x
-
The origin of this paper was a question posed to
What he wanted to know is whether, at least in the case
Vl(8),...,Vd(8)}
span I
lim T log(P(X(T,x)l > R))
R > 0 .
or
a > 0
P(X(Tx)I/Ix
Tlog
(1.21)
when
R1 ,
Q>O
if
(o), if
Finally, if
p
WffO .
W
a
log(P(IX(Tax)j/jxj
]I()
liu
T+
for all
T·#·
0o
(1.20)
2
and strictly decreasing on o(-,Q)
Referring to Theorem (1.14),
+ Wf .
for any function
I(p) > A(p-:q
such that
strictly increasing on (,9°)
(1.17)
(1.18)
where
A> 0
and there is an
' 0
for each
8 E SN1 ,
exists and is independent of
x E RN\{O
and
I profitted greatly from Pinsky's own work [31 on this problem; and
it is a pleasure to acknowledge here his contribution to the present article.
2
Proofs:
The proof of Theorem (1.7) follows the same pattern as that used in
Chapter 6 of (4].
, T > 0 . and
x E RW\{0)
Given
from (1.3)
Note that,
and (1.4),
r'E fR
set
P(p(T,x)/T E r).
F(T,x,r)
F(T,x,r) - F(T,-w, r)
and that for all
T 1 ,T 2 > 0
(2.1) p((p(T 1 +Tx)
a
for all
> 0,
r(
where
)
Proof:
BE [1,-)
for all
r E R,
T > 2 , and
F(T,x,r) < A(F(T,y,r
'(2.3)
E rIFT)
There exist constants
Lemma:
(2.2)
p(Tl,x))
-
( 6))
F/T 2 ) (a.s.,P).
- F(T,e(Tix),
2
and
A E (O,-)
E
> 0
(x,y) e (R\{0L) 22
+ exp(-_c2 T
))
,
R1: dist(p,r) < 6)
{p
First note that, by standard estimates and (1.3),
c > 0
and an
8 > 0
such that
and
such that
T > 0.
and observe that for all
there is .a
sup P(IP(1,x)I/T > 8/2)< B exp(-c62 T 2 )
Second, define
su - i
x,y E RN\{0}) and
p(l ' o'
s,9',r
sup{vpl,,):
M-
r) '
f E B(S N - ):
ef(0(1,x))] < ME[f(e(1,y))]
Using this in conjunction with (2.1), one now sees that:
F(T,x,r) < P((p(T,x) - p(1,x))/T E r(6
/2 ))
· B exp(-c(8T) 2 )
E[F(T-I,e(l,x), T-1 r/2))] *+ B exp(-C(T) 2 )
< ME[F(T-1,
e (1,y),
-T- r ( (8/2))]
- MP((p(T,y) - p(l,y))/T E
r(
/ 2 ))
< MF(Ty,r ( 6 ) + B(M1) exp(-c(8T)
+ B exp(-c(8T) 2 )
. B exp(-c(8T) 2 )
)
S
}
for all
A
T > 2,
B(M+)
0 < 6 < 1,
and
x,y E RN\{0) .
Thus (2.3)-holds with
.
Q.E.D.
For
T >0
and
r E PR ' set
,(T,r)
- inf F(T,x,r)
X
R1
for
where
and
6 > 0,
define
B(p,6) - (p-6,p+f).; define
G - {p
and for
p E R
R1
(S6
> O)t(p,6) '
a}
;
, define.
1(p) - sup{L(p,3): 6 > } .
(2.4)
Lena:
If
(2.5)
lim -
logs(T,B(p,6))
In particular,
Finally, if
(2.6)
for all
I: R
p 4 G , then for all
+ [0,-) U {,}
IQI - max
|Q(9)1
and
6 >0
- l(p,6)
is lower semi-continuous and convex.
lal - maxja(8)i
, then
sup P(Jp(T,x)l/T > R) < 2 exp(-T(R-IQI) 2/21al)
T >
Proof:
and
R > I
.
and T1 T 2 > :
First note that by (2.1),
for any
p
R1 , r > 0 , x E R\{}
8
F(T 1 +T 2 ,x,B(P ,r))
p(TI+T x)-p (T1 ,X)
E2 B(P,r))
> P(p(Tl,x)/T E B(p,r) ,
, p(Tx)/T1 E B(p,r)]
- E[F(T 2 , e(T,x) , B(p,r))
(T2,B(p,r))F(Tl,
s>
x
, B ( p,
.
))
Hence,
for all
for
B ( p, r ) )
4P(Ti+T 2 ,B(p,r)) >S (T
(2.7)
p E R
T1 ,T2 > 0
r > 0 , and
Now let
p B G and
T > 0 .
By (2.7)
Thus, the equality
is subadditive.
S
6,
8
r
log(T,B(p,6))
S(T) --
be given, and set
8 > 0
with
B (p, r ) )
9( T 2,
S(T) - inf - S(T) will follow once. it is shown that there exist
T>O
· To this end, note
0 < T < T2 < - such that sup{S(T): T E [T 1,T 2 3} <
li_
-
that since
there is a
p
(To,B(p,6/2)) - B .
for all
Hence by (2.7) with
r - 6/2 ,
so that
T 1 - noTo
S '1
be a fixed element of
T e [T1 ,T]2
.
>B
B(nTO,B(p,8/2))
>
Then, since
.
there is a
T2 > T 1
80
Hence, by (2.3) with
y/2 < F(T,e ,B(P,6/2))
r -
is lower
T * F(T,8 0 ,B(p,6/2))
B(p,6/2)
and
8/2
for
> y/2
F(T,O0,B(p,6/2))
such that
n
0
T -
and
2
(cf. (2.3) for the definition of A and c), and let
> 4Aexp(-e(6Tl/2)2)
semi-continuous,
no > 1
Choose
n > 1 .
such that
B E (0,1]
and a
To > 0
in place of
8 ;
< A9(T,B(p,6)) + A exp(-c(ST/2) 2 )
< AVT,B(P,8)) + y/4
for all
T E [T1 ,T2]
.
Clearly this proves that
The lower semicontinuity of
it
suffices to consider
p s FC1 + (1-C)P2
P 1 ,P
and choose
2
4
I
G .
6' > 0
is obvious.
Given
sup[S(T): T E [TT1 T2 ]} <
To prove that
C E (0,1)
So that
CB(pl,8')
and
I
.
is convex,
8 > 0 , set
+ (1-)R(p92,
')
9
C B(p,6) .
Then, just as in the derivation of (2.7), one can show that
9(T,B(p,;)) >
T > 0 .
for all
P 1,P 2 4 G
In particular, since
T log-(9T,B(pl,,'))
lim
)
.(;T,B(pl,9'))9((1-)T,B(P2,-6') )
and therefore
'-9(p1, ')
< -
and
lim
it
follows that
p
log9((1-9)T,B(Pl,,'))
-
4G
< "
(1-0)(p 2 ,6')
and that
L(P,6) < 9l(p 1 ,6')
+ (1-)l(P
,'
2
Clearly, this completes the proof that
) <
(1-C)I(P
.
2)
is convex.
I
Finally, from (1.3), the derivation of the estimate in (2.6) is
standard..
*Q.E.D.
To prove (1.8),
prove (1.8) and (1.9).
suppose that
> -1()
0 <6
.
In view of Lemma (2.4), we need only
Proof of Theorem (1.7):
(2.8)
If
p E r .
If
I(p) <
< 0-0 be given.
I(p) -
,
let
r
be an open subset of
then it is clear that
, choose
6 0 >0
Then, since
p
Gt
l(p,6) + I(p)
'- inf{I(p): p E r} .
that
r
is a compact subset of
Given
t(p,28(p)) > y - B if
E r F G. ,choose
=-
.
(p,)
8 + 0 , this completes the proof of (1.8).
as
Next suppose that
and let
and therefore (2.5) holds:
lim I log9(T,r) > lim T log(T,B(p,S))
-T
T-
Since
and
logP(T,r)
ttm
B(O, 0) C r
so that
R
/(p) > 0
B> 0
y < so that
and
and
R
p E r n G
, and set
t(p,26(p)) > 1/B
L(p,28(P))
'.
6(p) > 0
, choose
if y ' - .
Since
r
so
If
is compact,
10
n
there exists an
v
6(P
v
)
.
n > 1
and
PI,...,P n fr
Thus, by (2.3) with
so that
8'61
...
r
c
U B(P ,6 V )
where
A 6
n
z(T,x,r) <
I F(T,x,B(p .v))
< 2nA max{IGT,>(pv,26)v
for all
x E RN\{O} .
T_> 2 and
Vexp(-(
T)2):
<v
< n}
Note that
T log(A(T,B(pv,26 v )) V exp(-c(6T)2))
lti
T.~o
-
-
iff
v E G-
-(P_,26 v)
if
P,
G.
Hence,
lim T log(sup F(T,x,r)) <
-,Y
+ B if
Thus (1.9) is now proved in the case when
To complete the proof of (1.9), let
and for
R > IQI
'
x
T-w
define
r
Yl< -
is bounded.
r
be a given closed subset of
r R = r n B(OiR) .
Then, by the preceding plus
R
(2.6):
Mm-i log2sup
TU
x
F(T,x,r)).< (-infI(p)).V
i-rR
< (-inf
(p))
-('R-IQI)2/21al)
V (-(R-IQ) 2 /21al)
OEr
for all
R > IQI .
Clearly (1.9) follows after one lets
R +
Q.E.D.
I(')
Lemna:
(2.9)
1 )
0 E C(R
satisfies
-
0
lim
and
Moreover,
if
0 , then
(1(p)0/p
lim
I(p)/p 2 > 0 .
IP-I
(2.10)
lim sup I| log E[exp(T*(P(Tx)/T))] T-
where
A() I -0
x
A(#) _ sup(e(p) - I(P)) E K1 .
Proof:
T(n) - 0 , note that, by the ergodic theorem and
To prove that
standards estimates apntlied to (1.4):
lim I F(T,e,B(Q,6))m(de) - 1
T+40
for each
6 > O .
0--
Hence,
li .s
by (1.9):
(l1og(f F(T,e8,(Q,8))m(de)))
~pEB(Q, )
T-0-
lip
I(o)/p2 > 0 ,
let
R > IQI
be given.
Then, by (1.8) and (2.6):
(R- IlQII)2/2tat < - lim ! 1og inf F(T~x,BCO,R) c)
<
Thus,
for
I > 21QI ,
Equation (2.10) is
[6].
(P)
inf
> (P-21QI)
I(p) < I(2R) .
/811all.
a variation on a lemrna first proved by Varadhan in
First nt$e that, from the preceding,
continuous function which tends to
-
as
p * *(p) - I(p)
is an upper semi-
IPI * o and is finite at
~
.
12
Thus there exists a
j
p0
()
G such that
1
A)
-( )
.
> 0 , note that
E[exp(T§(P(T,x)/T))] > E[exp(T#(P(T,x)/T)) , p(T·x)/T
exp(T
>
B(pO,6)]
V
#(p) )(T,B(pO,6.))
inf
B(PO',)
Thus, by (1.8),
8 > 0:
for every
lim T log(inf E[exp(T¢(p(T,x)/T))])
T-
X
B(po,6)
-B(Po.a)
>
Since
*
I(p) -
inf
I(P)
inf
*(p) -
inf
>
(P
o)
is continuous, this proves- that
lis
Tlog(inf E[exp(T#(p(T,x)/T))])
T
>
(P o ) -
(P o ) - A( )
To complete the proof, first choose
< (1/41al)p
2
for
IPI
>
.
E[exp(Ti(p(T,x)/T))]
0
so that
)> IQI
R > R:
Then, for
, jp(T,x) /T < Rj
- E[exp(T#(p(T,x)/T))
+
I
!(P)
[exp(T9(P(T,x)/T))
, iP(T,x)l/T > R]
By (2.6):
Z[exp(T(PC(T,x)/T))
, !P(T,x)l/T > R] < 2
o<
forsome
KX
(0,)
and
)> 0
R
exp(- iT
21QI ((p-! Q
esp(-TR
Thus, for all
R >
2
-
2
p2
Given
13
;i
<i
C
T
I
log(sup E[exp(T9(P(T,x)/T)) ])
Ip(T,x)I/T < R])
1 log(sup E[exp(T§(P(T,x)/T)),
x
-
V (-XR 2 )
it
Thus,
R > R
o < 8 < R
Choose
for all
R > R:
limiT log sup E[exp(T4(p(T,x)/T)) , IP(T,x)I/T < R] < A(M)
x
T~"
S > 0 , choose
10(p)( . Given
be fixed and set M - max
*(2.11)
Let
to prove that
suffices
80so that
E B(,R)
P 1 ,...,p
B(O,R)S; UB(pV,6)
E[exp(TO(p(T,x)/T)),
.Hence,
} ( B .
Then
.
t T(Pv )
n
JP(T,x) I/T < R] <I
Ia-pi <
and
jal V fji < R
sup{[(o) - t(p):
so that
BT
.F(TxB(pV'6))
by (1.9),
rm
.ii
t
T
log(sup E[exp(T(:.p.(T,x)/T))
<S +
max [()
l<v<n
< 2s + sup [(P)
-
IP(Tx) I/T < R])
',
inf
U(p ,)
- U(P)]
1(p)]
= 20 + A()
.
Q.E.Do
14
(2.12)
Lemma:
For each
XE R
is a continuous convex function o
(2.13)
R
set
Then
A
,
tim sup I T log(E[exp(XP(Tx))])
T+
sup[1a - I(p)] .
A(M)
-
A()
0
x
and
(2.14)
1(p) - sup[Xp - A()] .
Proof:
lrom its definition it is clear that
continuous convex function.
I e R'
Moreover,
and satisfies (2.13).
Finally,
A
is
A(l)
E R
for all
A must be continuous.
I ; and so, since
the Legendre transform of
is
is a lower semi-
by Lema (2.9),
In particular,
semi-continuous and convex, I
A
the Legendre transform of
I
is
lower
That is,
A .
(2.14) holds.
Q.E.D
(2.15)
Lemma:
There is a
2 (_ < I
(-2.16)
K E ( ,)
*
>dm ,
<+
such that
)
E C"(S
.
L t )
Define
Wroof:
adjoint of the operator
,
for alt
is
Wk
N-
in
)
L2 (m)
)
L2(S
)
*1*2E C (S
syannetric in
L (
.
Then
"
·
.
Thus,
if
L
-
Ldm-
f <> +d>m,
Thus (2.16) is equivalent to the existence of a
1+m2
Wk
J-
2
kldm
- 1
on
C (SN-)
,
then
L
and
.J
(2.16')
f +1lWku2 d m
< -2K f L+*dm,
E C
SN-)
K E (O,")
+ 6 c'(s-l)
with
such that
*
0 .
L (a)
Noting
I~otikthat
Wk - Wk
kht6+ ck
where
ck E C (S')
, recalling that (1.6) holds
15
and applying HRrmander's Theorem and the strong maximum principle, one
2
con: laelas that L is essentially self-adjoint in L (m) and tlhat its selfsatisfies:
L
adjoint extention
is a positiv.
p
where
(O,nr) f SN- 1 x SN
1
,
an
L (SN
n >0 ·
follow once it
Then
L(m)
at
value
maximmn
l!
C(S N- )
tl(,)O
)
00
Thus there exist
.
{%n}
0
-0
80 , then,
> 0 .
!
,
such that
(2.16')
* 1 (e 0 )
* which clearly contradicts
f
with
But if
,
0 9
1 (n)P(l,8
- I
If l
Li(a)
-Xnn
2K - 1/X1
In
and
'
will
suppose not.
°1
)m(dn)
and
...
Ln
X1 > 0
To show that
and exp(L)#l - °1 *
from
<
X1
(
0 -
C C-(SN-i)
may be chosen to be. 1,
is shown that
,-
I
t > O ,
and, for each
is a symmetric doubly stochastic kernel.
(t,p,;n)
)-orthonormal basis
Because
C ((O0,)xS N - )
element of
is a co;npsct self-adjoint operator, all of whose eigen-
particular, exp(L)
functions are in
E C'($S- 1 ),
(n)p(t,-,n)m(dn) ,
exp(tL)(*) - I
achieves its
,
one has
' 01
Q.E.D.
16
Before proceeding,
L
on
C(SN- )
f or
U E M(S
for
JX(u)
fN-iI
)
Writing
.
du: u E C (SN-)
u - e
sup{- f (I < * *> + LX)du:
(2.18)
(2.18)
Lemma:
Lemna:
Jg~m)'O
JW(m) - 0
(2.19)
JO(U) > AE(fd
if
(2.20)
p E Ml(S N 1)
C (S
and for each EC
_ T)2
is given by
E C (SN-l)}
m N-l
(S N- ) there is -an A E(O,c)
( N Il
p(de) - g(O)m(dd)
'here
g
is a
) , then
J0(u) < x~L t*(g)l I22( )/2minfg(e):
is the constant in (2.16) and L
Proof:
-
}
S
is the adjoint of
L
in L2(sN-l )
First note that, by (2.17):
0
given WC (S
N-1
Lh = 4 - i and h = 0.
),
Jo ()m)
let
X f R1 -
h
sup (--f
<6,>dm) < 0.
N-1
be the unique element of C (S
)
Then, by (2.17):
J0(u) >
for all
u > 0}
one sees that an equivalent expression
J (u)
K
and
is
positive element of
Next,
L + XW
(2.17)
where
X E R1 , define
and
JA(y) -- inf {f u
Mboreover,
For
some more notation is required.
2 .f <h,h>du +(
In particular,
9d
*d
-
)
satisfying
such that
17
JO(U) > (f
dU -
> (f
du -I)
'2/2f <h,h>du
2/2 1<h,h>l
C(s
)
Thus (2.19) is proved.
To prove (2.20),
If
* E C"(SN- 1 )
let
L|dulF - If L (g)'(O-O)d-nI
be given.
< 1 -
Then, by (2.16):
2 ( 1L)
IL
L2 (a) 'J
L(0)
2
L2(m)
< K1/24i.L*(g)l
2
(f <4,">dm)"/2
'
L (m)
Rence, if
(
<
}
min{g(e): 6 E SN-
e
>
L*)d
, then:
f <,>d
*(g)
--
2
L (m)
<%.%>d m)/2J
*L (4gl 2
/2c ;
L ()
< K-
and so (2.20) follows from (2.17).
Q.E .D.
(2.21)
Remark:
Although it will not be used in the present article,
may want to note that if
g E C(SN-1) , then
ui E M1 (S
J 0(0)
< -
N-i
)
is given by
p(do) , g(iO)(de)
as soon as there exists an
A
one
with
E (0,-)
for
which
(2.22)
holds.
I|f Wo
_< A'(f <*,.>dl)l/2 ,
0 du
+E C(SN-
Before proving this, observe that if
IJ
Wodul
-
If
(WOg)-(f-')dmj
<
> 0 , then by (2.16):
g >
IWO
)
l2
-
L2 ((I
< K 1/21WOgl
(I <C
,
>dm) 1/2
-g
0 l2
-- (n)
_
!
< (K/c) 1 /21W0 I 2 n(f
.>du)
<g
cf<,>i),
18
where
is defined as in the proof of Lemma (2.15).
W0
with
A
{bk}'
every
U EM (S
if g > e .
Also, note that if
40 -
bkWk
1
A
bC
g(e)m(d)
with
) , then (2.22) 1holds with
C(S
where
(2.22) holds
d
1/2
W0 gl 2
L (m)
- (K/c)1
Thus,
2
for
1)
To prove that
J 0 (u)
<
(dB)
when w
g E C(SSN-1 +
and (2.22) holds, observe that:
IJ Ldul< I I
Jf
I kg.gWkdm+
is defined as in the proof of Lea
where
Wk
where
c k e C"(SN 1):
W·Oduj .
(2.15).
Since
I I Wk-Wk*dl < B1 f g <*,'> 1/2dm * t <g,g>1/2<,,
where
B1-
ck )
I(
1
k
N-I
Because
g >
,
(Wg)
2
Wk +k c
1>/2d,
< 2W 2kg
)
C(S
~
Wk
)
'c(s
and so
<gg>12
where
82 '
(2 1
WkgI
2'1
NIl
c(s '
)/2
1/2 .
Combining these with (2.22),
one easily
)
arrives at
(If (<4,+>
+ L+)duA< -2
/
1
f < a+>du ' + (BB2+)(f <*,*>du)
<_
( 1 + B+2
AU) /2
2
19
For each
Lema:
(2.23)
tog(E[exp(X
1im supil
(2.24)
T""°
x
--
X!. +
:
T
+ I H(e(t,x))dt)])
by
T
7
+I 0 T
(WO+)(8X(t,x))d
dT
kl0f Wk(OX(t,x))edek(t)
1
Then, by the Canmron-Martin
)
a)du - JX(<)]| = 0.
8X(*,x) , xE RN\{O},
Define
d
Ox(T,x)
H E C(S
and
d T
I f ak(9.(t,x))dSk(t)
1O0
sup[/ (H + 2
-
Proof:
R1
X
formula:
, T_> 0
t
T
dT
E[axp(X I t ak(e(t,z))dBk(t) + f H(C(t,x))dt)]
0
10-
,
- E exp(f R (6(tx))dt)]
0
where
at
X
r
R + 2
a .
At the same time,
and generated by
eX(',x)
is the diffusion starting
LI ; and, because of (1.6), H1rmander's Theorem,
P;(T,S,dn)
and the maximum principle, the transition probability function
for this diffusion is given by
CC(0,)xSN'
lxS
1
) .
and yields (2.24).
(7.21),
Hence,
px(T,O,rl)dl
with
pX
a positive eletm,-nt of
the theory of Donsker and Varadhan
(See Chapters 6 and 7 of [4],
I] applies
in particular Corollary
for details.)
(2.25)
Proof of Theorem (1.11):
Applying (2.24) with
Assume that
R -
Q , one sees fromr
A()
- sup[J (IQ +
(cf.
a > 0
(2.13) and (2.17)
a)du - J
that:
|(U)]
M Sup inf[f (IQ + X- a)du + f
2
(1.10)).
2
<.
+ LX*)du]
20
Hence, by (2.14):
(XQ
I(p) = sup inf sup[Ap
a)dV - J(1/2<,4> + LhX)du]
+2
If X # 0 (after replacing ~ by XA):
inf sup[Ap
=
-
I(XQ
+
a)dw -
inf sup[X(P
(Q-
(1/2<4,4> + Lx )du]
)d)
d
2
At the same time:
0 < inf sup[-
F(1/2<0,4>
+ LO)dU] < sup[-1/2
<4,>dml]
0
Thus
I(P) = sup inf sup[X(p -
(Q-L4)di)
2
(°kWk)Zd1] ;
and so, after two applications of the mini-max theorem:
I(P) = sup inf sup[X(p -
fL)d)
(k - Wk)
(Q
d].
The expression for I(p) given in (1.12) follows immediately from the preceding
one.
Starting from (1.12), one has:
I(p)
<
sup((p -
J(Q
-
L)dV) /2
J(
Wk)
d
< (P - Q)2/2·
On the other, choosing h for Q as in the proof of (2.19), we see that
(p) > inf[(p - Q) /2
VI
ak + Wkh) d] > A(p
Q)
21
d
where 1/A
(a
(k
+ Wkh) 211
E (0,oo).
N-1
1
C(S
Thus, (1.13) has now been proved.
)
The rest of Theorem (1.11)
now follows imuediately from (1.13)
and
standard facts about lower semicontinuous convex function.
Q.E .Do
(2.26)
fE C(SN
Lemnma:
)
n +"
Wk
f
for each
tlet.
f - O
C (S -1)
so
n~ 1
~
(1.10)),
and
Wkf -
I < k <d .
fn
hat
~
and
n1
(
I
f E L2( )
To prove existence,
Wkfn
k
k
such that
liem
(W#).fdm
fn + f
ak
and
E C"(S
1 < k <d
% E C*EC(S
(S
)) ,,where
where
.
N-
.i
) . In particular,
To prove that
ggin~~~-(SN-1-" Wk(O k) f C"(S
f E C (S
f (LO)-fdm-
- l)
.
Hence
-
if
f E
By Lemma (2.26),
*a dm
N C(S
) , define
Jf
-gdm
Lf - g
L,
.·
Proof of Theorem (1.14):
f
-N-
of distributions, and so by Hormanlder's Theorem applied to
unique.
L 2 (m)
in
n
(Wk).-fndm - lir - | *Wkf dm
proof of Lemma (2.15) and observe that
(2.27)
as
as in the proof of Lemna (2.15) and note that
1 <k < d
Wkf
then there is a unique
immediate from (2.16).
By (2.16), there exists an
Define
then
(cf.
The uniqueness is
{f }l
.
a - 0
satisfying
Proof:
choose
If
f
)
L
,
as
for all
in the sense
f E Cf(SN
l) o
.(Q.E.D.
exists and1 is
Rence, by Ito's formula:
T.
o(T,x) = f(O(T,x)) -
f(.)
+
fI
(e(t,x))dt
0
Applying (2.24) (with
X - O) to (2.13) and using the above exprssion for
p(T,x) , one sees that:
A(1)
sup[XI .Qdu - Jo(d)]
UI'
--·
in
22
lence, by (2.14) and the mini-mx theorem:
J Qd) + Jo(u)]
I(p) - sup inf[X(p u
X
- inf sup[X(p - f Qdz) + JO(u)]
Ak
u
, let
Lh
f Qdm - Q
=
_-2 2 .>.
o
, u
-Q
oJ(a)>&A(fQdi
this proves that
Next, suppose that
du
that
=
P ; and so
then there Ls a positive
by (2.20)
Nence,
(1.15),
I(p) <
To cogaplte
(2.9)) that
(2.2R)
, Jo(Y)
M1(S
N- )
, for some
*
"
g E C (S
< -
)
i vw
In view of
.
)
U E Mi(S
such
p E (q_,q+)',
f Qgdm = p.
and
1
f gdm
such that
y(d9) - g(O)m(de)
the proof of Theorem (1.14),
=
In particular,
by
it
suffices to recall (cf. Lemma
.
Q.E.D.
Proof of Coroliarvy (1.18):
log R(T)| <6
.
On the other hand, if
.
N-1
when
Then there is no
.
[^_,q,]
I(P)
A E (0,)
2
I(P) > A(P
P
h , one sees that
in place of
h
.
I(Q)
tim T log R(T) '
for some
By repeating the argument
- Q .
used to prove (2.19), only this tii4e using
(1.15),
I(P) > A(pQ) 2
be chosen as in the proof of Lemma (2.18) and set
h
Then, since
h - h+f .
Qdp - p)
To prove that
Thus, (1.15) has been proved.
A E (0,')
f
inf{Jo(u):
-
0
be given.
when
T >T
For
.
R: (O,-) * (O,-)
6 ) 0 , choose
T; > 0
x
R \{(O
Then, for any
P(p(T,x)/T > 6) < P(IX(T,x)l/Ix
and so, by (1.8) and (1.9):
Let
satisfying
so that
and
T > T6
> R(T)) < P(p(T,x)/T > -6);
23
< Mi 1 log(sup P(IX(T,x)l/Ixj > R(T)))
(2.29)
T+29-T
x
6> 0
Since this is true for every
O (_ q ,q+),
and
0
.
I()
< -inf
a
> R(T)))
log(inf P(IX(Tx)l/l x
- iaf I(p) < lim
and because, when either
I('Q) --
where
inf l(p) .
But, if
Q <
0
> R(T))) - i(
, then
is increasing on
I
On the other hand, if
.'
()
and so, in this case, ](') - -
> 0 .
O when
0(')
< I(-) - 0 ; and so
0o< inf I()
0 , one concludes that
is continuous at
in( supX log(P(lX(T,x)l/lxl
or
a > 0
Q'
[0,')
0 , then
Thus (1.19) is
p>O
proved .
0
- O.
Finally, suppose that
If
Q)
0 , then (2.29),with
0 <
< Q ,
implies that
O = - inf I(p) < lim
p>6
T-
On the other hand, if
and so (1.20) follows.
0 < a < -q+ ,
log(inf P(iX(T,x) !lx|
x
> R(T)))
0 ) q+ , then (2.29), with
implies that
inf I(p) - -= ;
p>-6
lim-log(sup P(IX(T,x)l/ixI > R(T))) <x
T~
from.which (1.21)
is inunediate.
Q.E.D.
(2.30)
Remark:
It
is
seldom true that
implies both that there is no
RN
and that there is some
0 ES
8 E Sq 1
-1
a-
0 .
for which
at which
a
a-
For example,
[Vl(8),...,Vd(6)
vanishes.
}
0
spans
To see these,
24
first suppose that
80 E SN
1
a - 0
Then by Lemna (2.26), there is an
Wkf - a
for
Wkf(%o)
ak(90) '
(00,Vk(80))
1, < k< d , where
n -
0
Then,
(2.31)
Vk(x)
B
)
N
x ERN\(0}
I-0
Let
- 0 ,
vwhich is obviously impossible.
0 EfS
a
1 < k <d ; and so
x
f E C (S
be a point at which
f
)
is
a(e 0 ) - O .
In [3], Pinsky dealt with vector fields
0 < k <d
RN , this
,Vd (80)spans
{V l (8O)...
0 , again use Lemma (2.26) to find
a
Wkf(
Remark:
,
(rn,0)
1 < k < d.
Wkf
maximal.
But, since
and that
Second, assuming that
k
and note that
f(x) - f()
(S-1) .
for some
satisfying
Define
r - gradf(8 0 ) E T
means that
f E C(S N 1)
RN
1I < k < d .
(n,V k( o))-
with
span({Vl(8 0 ),...,Vd(80 )})
and that
R\{0} , where the
Bk
Vk
NxN
given by
matrices.
The additional structure in this case gives rise to several interesting
features.
In the first place, the condition (1.6) becomes the condition that
= )
(8.,8)8: B E Lie(Brla.gd)}
spaen({8
where
Lie(Bl,**,Bd)
1 < kt < d
X(*,x)
A(.)
-
sN
Bk
(i.e., the Lie product here is the comrnmtator corresponding to
Secondly, and more important,
is the observation that
of (1.2) is now given by
X(T,x) - A(T)x ,
where
t(Sh1), 8 E
is the Lie algebra generated by the matrices
matrix multiplication).
the
T
(T,x) E [0,.") x (Rt{O}l),
is the matrix valued stochastic process determined by:
d
(2.32)
A(T) - I +
T
f BkA(t)d(t)
k=0 0k
I
(
it is therefore natural to transfer questions about
about the norm of
A(T) .
Because,
T > 0
Ix(T,,)l/IxI
for.the present purposes,
to ones
the choice of
25
denote the
IA(T)I
norm is inconsequential, let
uilbert-Schmidt norm of
A(T)
and set
K(T) - loglA(T)I .
(2.33)
Fix an o.n. basis
{ 9 l''...
R
in
N}
and observe that
p(T, 1 ) < K(T) <1 log N + max P(T, O)
~l<i<N
2
.
Hence, by (1.3) and the ergodic theorem:
lia K(T)/T - '(a.s.,P)
(2.34)
and, by Theorem (1.7):
- inf I(p) < lim T log P(K(T)/T > d)
T~+
0>6
(2.35)
<
-
S
ms 1 log P(K(T)/T >
R: (O,,)
(2.36)
i
liz
, 6 E R1
.pT
In particular, by Corollary (1.18), if
then for any
) < - inf I(o)
* (O,")
*a>0 or
satisfying
a - 0
and
lim - log R(T) -0
0 E (q_,q+)
:
log P(K(T)/T > R(T)) - RI(Q)
T-iO
where
1I(i)
is the same as it was in that corollary.
For purposes of comparison, it is interesting to look at
log(det(A(T)))
.
Indeed, by Ito's formula for Stronovich integrals:
T
d
det(A(T)) - 1 +
where
b-
A(T)
Trace Bk
Hence
I
f bkdet(A(t))-d$k(t) ,
k-0 0
T > 0
26
d
det(A(T)) - exp( I bkSk(T))
T > ,
,
k-l
and so
d
- bkSk(T) + bT ,
X
k-l
A(T)
T> 0
In particular:
rim A(T)/T - bo(a.s.,P) ,
(2.37)
and, after an elementary computation:
(2.38)
lm,1
T-
log P(A(T)/T >) )
b0)2/2H
-(8-
,
8 > bo
if
I bk > 0 .
k-l
Noting that
A(T)/N < K(T) ,
T )> 0
one concludes from (2.34) and (2.37) that:
(2.39)
and,
> 0b /N
H > 0 , from (2.35) and (2.38):
so long as
(2.40)
(In
;
I(p)
< (No - b ) 2 /2H
the derivation of (2.40),
particular,
(1.,14):
if
R >)0 ,
then
,
p >
recall that
I(p) <
I
is
for all
increasing on
p >
EQi).)
In
and so, by Theorem
27
(2.41)
a >0
if R > 0
.
Note that (2.41) leads to the following statement about matrices:
if {B
- (e,Be)e: B E Lie(B1 ....
and if Trace Bk $ 0
such that
f(x)
f(x) If f
(83B8
(
X
.
),
for some
W)
,Bd)
N-
1
for each
1 <.k < d , then there is no
(grad f(8),Bk ) N
a x f R \{0}) .
T (S
spans
}
for all
9 E SN
8
SN-
f E C (SN-)
(where
Surely there is a more direct route to this
fact than the one given above.
(2.42)
^
0 E(qq+)
Remark:
Assume that Q < 0 and that either a > 0 or a = 0 and
^
Let R : (0,0)
1
-t (0,0-) with lim 1 log R(T) = 0 be given.
Then:
T-,.
(2.43)
lim sup |1 log(P(suplX(t,x) /lxl
T-*o
x
> R(T)) + I(O)
=
.0
t>T
In view of (1.19), checking (2.43) comes down to showing that
lim sup4T log(P(supjX(t,x)I/[xj > R(T)) <T-*O x
t>T
(0) .
To this end, note that
co
P(sup|X(t,x)|/lxl
t>T
> R(T)) < I Jn(T,x)
o
where
J (T,x) = P( sup
IX(t+n,x) /lx|
> R(T))
T<t<T+l
Clearly,
3 /4 )
J (T,x) < P(p(T+n,x) > log R(T) - (T+n)
+ P(
sup
p(t,x) - p(T+n,x) > (T+n)3 /4 )
T+n<_tC+En+l
;
and, by standard estimates, there exist C E (0,oo) and A E (0,oo) such that
28
P( sup
p(s+t,x) - p(s,x).> M) < C exp(-M 2/2A)
0<t<l
for all (s,x) E [0,c)
choose GT
>
'
x (RN\{0}) and M > 0.
> X. Next, choose TX > (2XA)
O so that I(-6X
)
Ilog R(T) ) v (l/T1/
(
Now let X E (0,I(O)) be given and
4)
so that
< i/2
and (cf. (1.19))
sup P(P(T,x)/T > -6)
-XT
< e
x
for all T > T I.
Then, so long_as T > TX:
Jn(T,x) < e-
(T+n)
+ C exp(-(T+n)3/4/2A)
< (C+l) eX(T+n)
for all n > 0.
Hence:
sup P(sup IX(t,x)l/lxl > R(T)) < [(C+l)/(l-e- )] e
x
, T > TX
t>T
Since X was any element of (0,I(O)), (2.42) has now been proved.
(2.44)
Remark:
It must be clear that the analysis given in this article
applies equally well in a much broader setting.
For example, let M be a
connected, compact, Riemannian manifold and let W , ..., W d be smooth vector
fields on M satisfying Lie(W 1, ..., Wd) = T(M).
Next, let (8o("), ..., 8d("))
be as before and, for 8 E M, let 8(-,8) be the solution to de(t,8) =
d
° ) the transition
I Wk(e(t,6))odBk(t) with e(0,e) = e and denote by P(t,9,
probability function determined by {8(',8) : e E M}.
o, ..., ia E C (M) and set
Finally, let
29
ak(e(t,e))OdSk(t)
p(T,e) =
k=Oi
=
0
ak(e(t,e))dak(t) +
X
Q(e(t,e))dt, T
>
O ,
o
o
d
=
Wkak. ' Then, with no essential changes, the analysis given
1Wkk
and conclusions drawn in this article can be transferred to the study of
where Q
o
+ 1/2
~0
log P(p(T,x)/T E F) as T
-
a.
Actually, with more work, it is possible to get away from the compact case
if one is willing to impose a sufficiently strong ergodicity assumption (e.g.,
something on the order of hypercontractivity).
Such extensions allow one to
study the analogue of Pinskey's problem even when the vector fields are not
homogeneous.:
References
Ill
Donsker, M. and Varadhan, S.R.S., "Asymptotic evaluation of certain
Markov process expectations for large time, I," Comnm. Pure Appl. Math.
vol. 27 (1975), pp. 1-47.
(2]
RHrmander, L., "Hypoelliptic second order differential equations," Acta
Math., 119 (1967), pp. 147-171.
[31
Pinsky, M., "Large deviations for diffuil'or
processes," Stochastic
Analysis, ed. by A. Friedman & M. Pinsky, publ. by Academic Press, N.Y.
(1978), 271-284.
[4]
Stroock, D., An Introduction to the Theory of Large Deviations,
Universitext Series of Springer-Verlag (1984).
[5]
Stroock, D. and Varadhan, S.R.S., "On the support of diffusion processes,
with applications to the strong maximum principle," Proc. Sixth Berkeley
Symp. in Prob. & Stat. (1971), vol. III, pp. 333-359.
[6]
Varadhan, S.R.S., "Asymptotic probabilities and differential equations,"
Coam. Pure Appl. Math., vol. 19 (1968), pp. 261-286.