AN APPLICATION OF THE MALLIAVIN CALCULUS
TO INFINITE DIMENSIONAL DIFFUSIONS
by
Laura E. Clemens
B.A., University of Colorado
(1977)
SUBMITTED TO THE DEPARTMENT OF MATHEMATICS
IN PARTIAL FULFULLMENT OF THE
REQUIREMENTS
FOR THE
DEGREE OF
IN
DOCTOR OF PHILOSOPHY
MATHEMATICS
at the
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
O
Laura E. Clemens
1984
Signature of Author..
of Mathematics
Department
`-gust 10, 1984
Certified by........
. L
.
Richard M. Dudley
Thesis Supervisor
Accepted by.... .
.........
.......... ......................
Nesmith C. Ankeny
Committee
Graduate
Departmental
Chairman,
ARCHIVES
FE
iNSTITU
MASSACHUSETTS
OF TECHNOLOGY
NOV 2 0 1984
LIBRA!ir3
2
AN APPLICATION OF THE MALLIAVIN CALCULUS
TO INFINITE DIMENSIONAL DIFFUSIONS
by
LAURA E. CLEMENS
Submitted to the Department of Mathematics on August 10, 1984
in partial fulfillment of the requirements for the Degree
of Doctor of Philosophy in Mathematics
ABSTRACT
Diffusions on the infinite product of a compact manifold are defined, and their
finite dimensional marginals studied. It is shown that under reasonable
hypotheses, the marginals possess smooth densities. An estimate on the
densities is obtained which is independent of the number of dimensions with
respect to which the marginals are taken. An application to statistical
mechanics is discussed.
3
I Introduction
The study of the regularity of infinite dimensional diffusions presents
certain difficulties.
One cannot expect the transition probability functions
to admit densities, even in the independent context.
It is clear, however, in
this context, that the finite dimensional marginals will, under reasonable
hypotheses, admit densities.
Equally clear is the fact that the uniform norms
of the densities become unbounded as the number of dimensions with respect to
which the marginal is taken goes to infinity.
For certain applications it is
necessary to have an estimate on the marginals which remains bounded as the
number of dimensions increases.
To see what sort of estimate may work, we will
look at the independent case in more detail.
Let M be a compact Riemannian manifold, and a its Riemannian measure,
which we may assume is a probabilitty measure.
Suppose that xk(t), k
Z
are independent diffusions on M whose transition probability functions
Pk(t,*) admit densities pk(t,*) with respect to
.
Then the
transition probability function for the diffusion (xklk Z is
P(t,*) =;1Pk(t,*),
and if p(N)(t,*) denotes the density for the
kcZ
marginal of P(t,*) with respect to a
P(N)(t, ) =
-N
< k
p (t,*).
(k: IkI
< N)
Although sup((N)(ta))---
N
, then
as N---,
the ratio
PgradP(N)(t,')/p(N)(tq) =gradIPk(t
remains bounded as N
.
k)/Pk(t
k
The dimension independent estimate which we obtain
in a more general setting involves this ratio.
The method of proof for all the results is the Malliavin calculus.
Techniques of partial differential equations do not lend themselves to the
infinite dimensional setting, because, as we have seen, we have to take
4
marginal distributions, and these marginals do not, in general, satisfy any
autonomous equation.
On the other hand, Malliavin's calculus is well suited to
the study of marginal distributions.
Furthermore, ratios like the one in the
preceding paragraph occur naturally when integrating by parts, on R, if
p(dx)
C0. (R).
p(x)dx then
f'(x)g(dx) = -f(x)p'(x)/p(x)g(dx)
for f
i
The Malliavin calculus allows us to integrate by parts on
Wiener space.
The main ideas for the proofs are derived from [A], where the
same results are proved on the infinite dimensional torus.
Section II is devoted to a discussion of diffusions on M, section III to
the Malliavin calculus on M, and section IV to the regularity of diffusions on
M.
(Sections II - IV result from private communication with D. Stroock.)
The regularity of finite dimensional marginals of diffusions on M
covered in section V, and the dimension -
is
independent estimate on these
marginals is covered in the following section.
Finally, an application to
statistical mechanics is discussed in the last section.
5
II. Diffusions on Manifolds
0
Let M be a compact manifold and V,
j < d be vector fields on M.
Set
d
(1)
(Vj)z + VO
L =
j=l
In this section, we will describe what is meant by 'the diffusion
on M
generated by L'.
Denote
of M, and for some D > m let i: M--
by m the dimension
imbedding. Then there are smooth functions
W: RD--R
RD be an
D,
j = O,...,d satisfying
For x 6 i(M),
i)
(2)
.()
(x)))
= i (V.(i
.
ii)
Wj
and each of its derivatives
0 < j
Set
=
< d and
1
< i
6 C([O,-),Rd):
@(t) = [Oj(t): 1
j
measure on (,G).
For x
for
01) and let
(0)
= ao((s): 0 < s < t) and
Set At
on R
< D.
be the position
d
is bounded
=
of
at time t > 0.
(0O(s):0 ( s).
Let W be Wiener
RD , let x(*,x) be the unique solution to the
equation
d
(3)
x(T,x) = x
T
Lemma(4)
Proof.
For i > 1
If x
WW(x(t,x))dt.
+.
j=l
T
0
0
i(M) then x(t,x)i
i(M) for all t
0.
The following set up will be useful in this proof and elsewhere.
let U i , U i '
and Ui 1 (
i (
r be precompact
open subsets
6
of
D
so that
i)
U.
·
(5)
ii)
iii)
(Uifi)
U'
co
U' C U.
i
0i
U.
RP
j
r(
U.
i(M)
i
For each i
i
so that
b()
is a coordinate chart in R
and
~m1
M(i
ni-D.
i
RD let n(x)
i > 1,
and
1
1
For x
r
1i
= min{n
-x
1: x
on =U. I
U ).
- -
. = inf{t >
=
Define T0 = 0 and for
~i~l))
n'r
'
i-1: x(tx);
U n((
))
By the strong Markov property, it will suffice to show that if x
X(tAAl)
Ei(M) for t
z(T,z) = z +
,1'
Set z = n(x)x.
iW.(z(t,z))edJ(t)+
where W. =
Wjn(z(tAlY
Then, by Ito's formula,
(z(t,z))dt,
J
j=lf0
T <
0.
(a /ax)W.. Since
z)) = 0 for zM,
0 < j < d and m+l
< n < D, the
proof is complete.
Now, set
= C([O,-),M) and for wfl
of w at time t > 0.
iI=
(o,M)
Set /' t =
,(i(s): 0 < s).
(n(s):
let
(t,w)
M be the position
0 < s < t) and
For nq*M we can define the measure P
as the distribution of i
Oy(*,i())
under
W.
on
M then
7
Theorem(6)
For all of M, P
described above is the unique
such that P ((0O)=i)=1
probability measure on (,W)
(f(n(t))-ftLf(i(s))ds,
C (M).
Finally,
,P )
the family
is a martingale
(P :
for every
EM) is Feller
and
f in
continuous
and
strong Markov.
Proof.
That
P
satisfies the desired conditions is clear.
The
rest of the proof can be taken from chapter 6 of [3].
Remark(7)
We can define diffusions on non - compact M in the same way if we
assume that there are W., 0
Remark(8)
MCRD
j
< d satisfying (2).
From now on we will assume, for notational convenience, that
and i is the identity.
V/
8
III. The Malliavin Calculus on Manifolds
In sections IV - VI we will prove certain regularity results about the
transition probability functions for diffusions on manifolds.
We first need
some results about the Malliavin calculus.
Let W : RD--R D,
be the solution
= ((
A(tx)
satisfy ii) of (2) and let x(t,x)
j0,...,d
to the system
(3).
))
<(xk(t,x),(t,x
Let
< k
Malliavin covariance matrix. (For
4YS
P
-
, where
p+SP
D
< n < D be the
Cp,YiE
(see [1l]),<',.>
(PM)
=
Then, for
is the Ornstein - Uhlenbeck operator.)
n EC-c (R ,R )
b
))1
n(s(t,))>
< n(s(t,()),
(xD,1
k
< n
<
< D
an/ax(x(t.x))A(t,x)a*/ax(x(t,x)).
Hence, A(t,x)TT
x(tx) (RD) .
(If N is a manifold and p
N,
T (N) means the tangent space to N at p, and T* (N) means the
P
P
and W. is an extension to ED
cotangent space.) We will show, when xM
of a vector field V. on M, 0
j
d, that A(t,x)
We can make a selection of the map xx(*,x)
(t,x)
(t,)
[O,T] X RD-
x(t,x)6
= (( (axi/a8)(t,x)
d
(9)
C0' ([O, =]
))1
< ij
RD).
< D'
T
(
M)
2
.
so that for T ) 0,
Then, setting
we have
T
X(T,x) = I +
(aw./ax)(x(t,x))X(t,x)o°de(t)
j=l
T
+
J
(aWo/ax)(x(t,x))x(t,x)dt
0
Thus, X is invertible.
Furthermore,
(see [1])
d T
(10)
A(T,x)
J
=
j=l
[X(t,T,x)W.(x(t,))]
J
dt
where X(t,T,x) = X(T,x)X-
and W (x(tx))E
Tx
1
(tx).
) (M), A(T.x)
Since
X(Tx):
T (M)-4T
(Tx(T x)(M)):.
x (t(Tt,x)
(M)
10
IV. Regularity
Suppose that M is a compact Riemannian manifold and that its Riemannian
gM is the same as the metric
metric,
Let
from RD.
it inherits
be
the positive measure on M associated with the Riemannian structure.
We may as
well assume that a is a probability measure.
Vkj
Let
kj
(11)
0
< j
0
< d be a collection of vector fields on M so that
d
j
in M,
for all
< j < d spans
V
the tangent
space
to M at .
)
= P o((t)
Let P(t,io,)
for OM and PsM.
The goal of this section is to show that, for t>O, P(t.,1o,dq) admits a
Fix 1 < i < r and set U = U., U
Ui' U i '
and
.
with respect to a which is smooth in
density p(t,9 oj)
i are defined in (5).
n
= U.' and
.,
=
where
We will show that
admits a density on MU.
P(t,1,#*)
0
Choose
U
a smooth
C[p = 1]C¢Csupp(p)CC
z , ... ,z
Z(m) =
M
.
z(m)_
],
on R
by
where z = nx and
Clearly we will be
admits a smooth density.
finished once we show that
suffices
Define the measure
U'.
)
=- EW[p(x(t0
r
g(tno~
p on R D so that 0 < p < 1 and
function
to show that for all f
In order to do so, it
C0 (R ) and all
multi-indicesa - Nm ,
0 8 d4)I
IfDaf(Q)g(t,
< C (t)llfltl.
Da= alal/( a 1
Fix 1
x(t,ql 0 )
(aF/az
aam)
k < m, and set F(z) = f(Z(m)).
U', <F(z(t,qo 0 )),z
q) ( z( t
where
O)
) Aq' p ( t
q
(t,1q0 )> =
=
O) where A(t, o)
0
Then,
for 1
p
D and
11
D
L
aar/ax(x(tq))A(t,1iO~an/8x*(r~t'nO)).
(So, by
q=l
section III,
APq(t
x(t,10
jP'q(t,1no)
= <z(t ,
= 0 if m+l
1 0 )
< p
0
),Zq(tn
0
)> and
Thus,
D and x(t0)U'.)
q
OF/az q (Z(t,.n ))=
)U',
m
(A(m))p1q<F(z(t,0))
)
,z(t,
0
p=l
where (( (A())p
q
))1 < p,q
m
A
= (A(M))
)-1
, and A(M )
is the upper left hand m by m submatrix of A.
Assume,
that
U,((t,i 0))/det( A ( m)(t,
0
(12)
Then,
(see Lemma
)
0)
)I
L(W)
) )
p=l
3.4 in [2]), p'(x(t,B
m and any p'
1 < q,p
p' (x(t
for the moment,
(U').
C
0
))(A())q
If }E
p
setting
, we may define Hk(i*)
m
_
I[
<zq(tno)
/
(A(m))q,k
(t'0)
q=1
+ 2 (A(m)) q, ,0o)
k(tllo)z
Then, choosing
p'
C
supp(p)CC
{p' = 11),
EW[aF/az
)(z(t,
a(F(p'
EW[{aF(p'On
0o
))p(t
(t
(U') with 0 < p'
no
l
z(t-1
z(t
))Iaz
(z(t,ij
)]
1 and
=
0 ))P(t,
0 )]
]
=
for
=
if
12
W
EWF('z(t,n
o)
)p' (x(tn
O)
)k (p (tO))
]
We thus have the desired bound on I/Daf(4),(t.,,4)j
For general a,
for la
< 1.
the bound can be obtained by induction.
It remains to show (12).
By (11), there is a positive e with
(~1(t,T,no)V
>
d
j)l
((t,).q)
e(tT,%)
S
-1
Hence, if x(T,x)e U',
)
(x(t,%O))Xa(t,9TF
(A(m))
(To)
)*
<
T [(an -1/ax)(x(t,))*Y (t,T,nO)
(l/(eT2 )) E[
8
x
W(t,T,I)(an-/ax)(=(t,%))]
Setting Y(t,T,nO) = Xl(t,T,no)*Xl
But, Tr([(an /ax)(x(t,n
< CTr(Y(t,T,nO)) if
(t,TiO),
))*Y(t,T,nO)(an
(t,no)(U',
l//a(( t,no)
(
and Tr(Y(t,T,nO)) can
easily be estimated in LP(w).
We have now completed the proof of the following theorem.
Theorem (13)
m
Let X>1 and tO.
P(t,jo,dn) admits a density
13
p(t,no
0
)
which is smooth in
.
The uniform norms on p and
its derivatives can be bounded independent of 1/
( t <
and
0
M.
14
V. Infinite Dimensional Diffusions
We want
to extend
the results
of section
IV to diffusions
on MZ
However, to avoid certain technicalities, we will prove results about
M
'
where K is a large
K],
suffice
integer,
if we show that the results
C([O,_),(Rd
e = {
)[-KD).
:
Let RZ
, 1 < j
+
]
I
)
Iki ( K).
This will
on K.
=
and
11t,1
K} ,
:Ik
< d},
and W as before.
ki < K and 0
that for
. Suppose
Vk j:M [-K ]-
=
Define
[ K
:
do not depend
so obtained
= C([O,=),M -
let
In this context,
and [-K,K] = {k
j
< d,
TM satisfies
i)
Vk j ()
ii)
Vk j
MM
TM(^k) for
is smooth and Vk
j
and each of its derivatives is
(14)
bounded independent of k and j
iii) Vk,
j
depends only on {n}k-R
Then there are functions W
k,j.
i)
j . is
k,j
W
(15)
ii)
: (RD)[-KK]--
smooth and Wkj
kj
independent
< n < k+R
RD satisfying
and each derivative is bounded
of k and j
Wk, j depends only on {yn} k-R ( n < k+R for yC
D [-K,K]
iii)
For
M[K
K]
Wk,j(I( )) = i*Vk j(T) where
I = i[-K'K]
For y
K'
(RD) [ - I
, let y(t,y)
= {yk(ty)
:
Ikl < K} denote
unique solution to the system of equations
d
(16)
k(T) = yk+
T
Wj
(y(t))odek(t)+W
j=1
Theorem (6) yields the following.
(y(t))dt
the
I
15
Corollarv(17)
C (M[ -KK])
f
Let Vkj,
,
K
IkI<
0 _< j < d satisfy
(14).
For
, define
d
Lf(TI)=
{1/2
kl
< K
(Vk j)2f(+V
j=l
(Vk jf is formed by fixing n , nk
]
For riM[KK
function
of Ik.)
solution
to (16) with
Wk
j
P
on (i,1)
P
is the unique probability
kOf(0n)}.
and acting Vk,j on f as a
and y = I(n) let y(*,y) be
Ik[ < K,1
< j
< d satisfying
the
(15).
be the distribution of I-1 y(*,I(y)) under W.
P ((0)=)=l
on (,11)
measure
and (f(j(t))-fLf(i(s))dsDt,P)
[
martingale for every f in C (M
Finally, the family
P :
MZ]
- K'
,
)
Let
Then
such that
is a
.
is Feller continuous and strong
Markov.
Assume
that we are given
that (tfs>0) (
Vk,j for which
< K) (V i
lkl
M [-K'])
(14) holds
and so
TM*i(k)
(
d
<YVk
(18)
j(?k)>2 > elYI2.
j=1
Let P(N)(t'qo'0
)
( (N)(t)i r)
P
Theorem (19)
P(N)(t,qO')
Let
for nTiM- K ' K ] and -nC[ M NN.
>1 and t>O.
which is smooth in
P(N) and its derivatives
[-K,K]
Oi
M
P(N)(t,o,n)
.
admits
a density
The uniform norms of
can be bounded
independent
of 1/
(< t
<
and
16
D ) [ - K' K]
Proof. For y(R
= (yk)n,
sx
R[-KD+,(K+)
R
1 < n
be defined
by
yN ) and
...
...
km+n
(x(m))(N+l)m)
))-Km+l
BD+p'
for -K
qD+q'
(( (B
Let Ui' U.',
{k}i N < k< Nbe
k N
k < N
U = t
Ui
IkI
< N
rr(Y)k
< p,q
-
that UC(p
EW[(Y(N)
(t
Ik
< m.
Define
B(mN)
=
ri, 1 (
i < r be as in (5) and let
of
'.
integers between
1 and r.
Set
Define
< N
otherwise
' 1 with Ykz Uik.
k0
= 1} and 0 < p < 1.
0) )
Choose pg CO (U') so
]
,0 Or
=
for
Since, by section 6 of [21, xi(to0 ) is
(in the sense of Malliavin's
< i < (K+1)D,
(t
Define
]
r (x) (m,N)
(t,0))
R [- Nm +l(N+l) m
bounded
the proof
admits a smooth density if
(20)
)
=
for yz.Rt[
-KD+I
B(m
< K and 1 < p',q'
U.
=
Yk
- C
K and 1 < n < m.
Similarly, for a matrix
a sequence
k
k _<
< i,j < (N+l)m.
and
and U'
kD+n
< i,j < K(m+l) by (Bm)
i ))-mN+l
)))
x(m)f
x
< i,j < (K+1)D define
(( (B()))'J
by
(m,N) =
'.
((B
-KD+
Let
< D.
be defined by
Set Y(N) = (Y-N
((x(m))-Nm+l
< K,
Ik
x*[- K D+1,(K+1)DIbe defined
let
calculus)
in section
independent
IV will work here
we can show that
,
,(Y(N)(tlO))/det(A(m N)(t,
)
o0)
E i\
p=l
where A(t, 1 0) = (a/ax)(x(t,0))A(t,71)(a/ax)(x(t,))
LP(W),
of
and
to show that
17
<(P(t,n0),x
and A(t,i0 ) = ((
E
Define Wkj
Wkj
RZ
for
(tn0>))))-ED+
kl
<
UI
n=k and 1 < n'
0
Set Rk, j=Wk
j/ax
and lt
d
< (+1)D.
. O < j < d by
(Wkj )
nD+n'
< p,q
< D
otherwise
X be the unique solution to
T
Rk,j (y(sy))X(tsy)°dekI(s) +
X(t,T,y) = I +
Ikl < K j=l
t
Rk
<
Ik
(y(sy))X(ts.y)ds
K
d
Then A(T,y) =
Ik
Suppose
0
<
that
f(X(t,T,y)Wk j(y(t.y)))
dt.
j=1
ik}lkI
< K is an extension
{ik}lkI < N with 1 < ik < r for
Iki < K.
of the given
Then,
sequence
as in Theorem
(13),
using Lemma (2.18) from [A], if
x .flUi
U,
(t0)))
then (1/det(A(
N+)
1/(sI(2N+i)mT')f[Ir[((an/ax)(x(tn0))Y(t
-1
where Y(t,Tq 0 ) = X (t,Tq
-1
0
)*X
.T., 0 )(a/ax*)((t.n0)))(
(t,Ti
0 ).
.N)]dt
The
integrand is bounded by a constant times the trace of (Y(t,T,O0))(N),
where the constant depends only on {ik
lk
}
_<
N and
18
Tr(Y(t,T,.o)(N))
is estimated as in section-6 of
2].
19
VI A Dimension - Independent Estimate
The estimates on the marginals obtained in the preceding section are
dependent on the number of dimensions for which we are taking the marginal.
In
this section we obtain, under an additional hypothesis, an estimate which does
Specifically, for IkI < K, define
not have this dependence.
Gt ( , k) = lgradkp(t,1,4)/p(t,,~4)J1p(t,I,C)C[- K' ]
(18).
Ik
j
n'
of
< K
1/A
l,...d and n f k, Vk.
(14) and
j
is
< C < X
Gt (i,k)
sup
< t <
< d satisfy
> 1
Then
sup
sup
=
j
0
( K
Assume in addition that for j
independent
Iki
that V
Suppose
(21)
Theorem
( dt )
M[lK.ll
.
where C does not depend on K.
First we need two lemmas.
Proof.
Vk
Jki < K.
is independent
j
(Wk, jlkl
< K0
(RD)-[
K
K]
k.
and 1
< j < d satisfying (15).
p,q
See the proof
Proof.
0
k, and 0 < j < d
k, t
Then
0, n
D.
of Lemma
(3.8)
in [1].
Lemma (23)
Let F be a finite subset of the integers and set Wk,
X
for Ikl < K, where
c(k)WkO
satisfies
(15).
Let z be
k,
j < d and n
Let y be the solution to (16)
= 0 for n
<ykP(t.),yaq(t,)>
that for every
Suppose
of In and that for all n
does not depend on
V
for
Fix
(22)
Lemma
(Wkj}lkl
the solution
K,0
j
to the system
d
(16) with WkO
=
20
d
k 0. '
replacing
Set
ak =
Wk j
.
F,
Then for k
j=1
there is
kEC Cb
(R)
-I]
so that akc
R
k
d
= b = W
+ (1/2)
0
Wkj(Wkj)
j=l
(Wkj (W,):
(D)[-KIKl
RD is definedby
D
jP(a/akp)(wk
/k
[Wk j(Wk j)] =p
= exp[
If S(t)
)
<cW
jn).)
.
,j
t
+
<c,b>(z(s,))ds],
then (S(t),it,W)
martingale and so there is a probability measure P on
EW[R(t),A].
Finally,
EP[f(z(t,?))]
if y(*,I)
= Ef(y(t,n))]
is the solution
is a
with P(A)
to the system
=
(16) then
for all bounded measurable f on
D([-K,].
(R)
x
Proof.
Let U., 1 < i < r be as described
partition of unity subordinate to Ui.
in (5) and let
Choose ck.
so that
i be a
for
I
Yke Ui' a(I)ck.(1)
2
bk.
= bk(1).
Set ck()
=
i(k )ck().
Then c k is smooth and akok =
The rest of the lemma follows from Cameron - Martin - Girsanov theory.
(See [2].)
Fix
(5).
{~
kI < K and 1 < i < r, and define U., U.',
Set U =
M
- [
~ M [- K'I:
{E Ui},
and U' =
: k6 U i '). Define n on U' by
and n. as in
21
(n(4))n
n.(T)
n = k
4{n
otherwise
=
We will show
JX (4)lgradkP(t, ,e)/p(t
,4)2p(t,n,4)[-K'](d)(dg)
is bounded in the required manner.
j
Choose
o
j
{Wrj
d so that
or
Ir-k
(15) holds
and set
> R
Wk
otherwise
O
Denote by y(t,y) the solution th the system (16) and by w(t,y) the solution
with W j
{r:
Ir-kI
Let S and P be as in Lemma (23) witth F
replacing Wr..
< R}.
=
Define a(t,)
))1 < p,q
(( <wkP(t,),wk(t,)>
d
Then
< D.
t
2
f(ek(s,t,q)V
k (w(t)))
k(thl) =
dt
j=1
where ek(t,T,q) = I +
d
J (aWk,/awk)(w(sT))ek(s,T,T)odOek
j=1
For w(t,x)=U',
=
set ak(tq)
awk ) (k( t
(a/aw k ) (Wk(t, ))ak( t ,)(a*/
and
(( ((a k ) (m))p
q
) 1
p,q _ m
((ak)(m))
If FEC O (U') and p
P(wk(t,))((ak)(
))
EW[(aiazkn)(F°
)(z(t,)p(
-
r
£
C O (U') then
for
1 < p,q < m, and for
wk(t,1))
]
Cf
6,
=
22
(z(t,))Hkn(p(w(t
=-EW[Fon
)) )]
there k(l)
=
<zq(t)((a
>+
k )(m ))qon (t , q)
q=l
2((a) (m)q,n(t , )
z(t,
and z = nw.
CO (Ui') with 0 < p < 1 and U CC[p
Choose p
Define the measure L
U (dw) =
(w)(n
on [M-
' ]
=
}.
by
[-
)*(a/zn)p(t,,w)
N N]
,
(dw).
Then by Lemma (3.6) in [1] the theorem will be proved once we find a function
with the L2(P) norm of T bounded independent of the desired
L2()
quantities and with E [f] = EP[f(w(ti))']
fe C (M
-K'
for every
)
Integrating by parts, E [f] =
-EP[
8 8
/
= (Ig(z
zk((fp)°
k
)l
)
h)(z(t,q))(l/h(z(t,j)))], where h(z)
1/ 2
.
(Here g(*) is the Riemannian metric expressed in the
coordinates ni..) Thus, E [f] =
=
-EW[a/azk((fp)°n-lh)(z(t,))(S(t)/h(z(t,j)))
EW[(fp°n-lh)(z(t,q))Hkn((S(t)/h(z(t,q)))pl(wk(t,)))]
where
pl E C
(U i') is chosen
so that 0 < P1
1 and
[p1 = ])CC supp(p).
So. E [f
= EP[f(w(t,))
] with
(p(w(t,q))h(z(t,q))/S(t))Hkn((S(t)/h(z(t,q)))pl(wit,~)))
23
and
~
can be estimated
Remark (24)
For Ikl < N < K, set Gt(q,k,N) =
f grad
kp(N)(t
Then Gt(n,k,N)
Remark (25)
M[K ' K
for
in L (P).
(-tN,/p,(t,N](dt
I2P(N)(t'n4)a
'')/P(N)(tn
Gt(q,k).
The obvious analogues to the preceding results hold on
i
any positive integer, and with the same proofs.
24
VII Application
In section 4 of [A],
results are proved about the ergodic properties of a
certain class of diffusions on the infinite dimensional torus.
There, the
specific energy function is introduced and shown to be a Liapunov function.
on TZ
That is, denoting by h(p) the specific energy of a measure
+
PZ
, and by Pt the Markov semigroup associated with the diffusion,
h(Pt*p ) is nondecreasing for t > O, where Pt* is the adjoint of Pt.
The analogous results hold for a class of diffusions on MZ , and with
the same proofs, once we show that the specific energy function is finite.
A collection of smooth functions
IFI = cardinality(F)
<
) is called
=
{JZi MF:
a potential
R:
if JF(1)
C
depends
only
on ai, ke F, and 3F is invariant under permutations of the indices of
F.
is called finite range if there is an R EZ
a
=
and Ik-nl > R then
ZA, JF+k()
=
F(S-k)
Assume that 9
0.
=
Hkl()=
Fk
and kf
n-k
For kE
Hk(n) by
JF ( n)
a probability measure on MZ , define the specific free energy
For
as follows.
Set
Ak =
n:
Inl
<
A
on M
Z
F
is a shift invariant and finite range potential.
ZO, define the energy at site k,
h(P)
n
so that if kn
invariant if for any F
is shift
where (S-k)
+
k).
If the marginal
A
has a density with respect to a
(n) ) and set h (p) =
nn
, denote it by
density
of
25
I
JF( nA )
'A
M n FcA
S·
)o
(A
n
(dlA ) +
n
n
n
(g
)log(
l
(n)(lA
A
)
))o n(dl
n
n
n
M n
If not,
set h (r)
=
.
-)
Let c:MZ -(0,
Let h(I) = lim(2nR+l)
h (m).
n
nn
be a smooth function depending only on
coordinates Ik for IkI < R and define ck(
H ( )
Suppose that e
Ck(n)
= c(S-k ).
1)
depends only on 0.
Set
Hk (k)
Lf =
e·
dik(ck(()gradkf(
for f
C (Z
) which depend only on finitely many coordinates.
))
Denote by P(t,n,*) the transition probability function for the diffusion
associated with L.
a probability measure on MZ , set gt
For
(*)
=
fp(t,q,*)(di).
Theorem(26)
h(
) < c(t) < .
t
I
An
C (Mn), ff2(j)log(f(j))o
Proof. For f
(d)
A
( Cf/grad(f(n)) 2
n(di) +
A
ff2(j)a n(dr)log(f2()a
A
n(di))
where
on n.
C does not depend
A
M n.)
Thus, setting f(nA
(n)
(Rt
1/2
(nA )
n
)
=
(This
is the logarithmic
Sobolev
inequality
on
26
n)
M
log((n)
))A d
n
n
n
grad
Ot
n
M
n
A
(n)
(q ) 12/t
n
(n) (nA
)
n
n
F'A
n
A
since ff2(tA :)
(qA
n
) = 1.
Furthermore,
n
by Lemma (3.3) in [1],
(n)
Igradkt
and Theorem (21),
A
(A ) 12/(n) (nA
n
can be bounded independent of n.
(n)
A(
(qn )log(t
n
n
2
AI
F5A
JF ('n
lnn
n
)A(n
(-
)a
n
IF(.0(2nR~l)
MZ
))a
(d-
n
< Isup
T
(n) (
F 0
.
)
n
Since xlogx is convex,
n(dn
I
Also,
< C(2nR+I)1.
(dii
A
n
FAn
M n
)
)
n 1
)
n
0
<
I
27
Combining the preceding two statements, we obtain the desired bound on
h(t).
The following theorems are proved as in section (4) of [1].
Theorem (27) For O<t1 ( t2, h(
)
1
Theorem(28)
If
is shift
invariant
h(
).
1
(in terms
of the shift on Z)
stationary for the diffusion generated by L then
diffusion.
and
is reversible for this
28
REFERENCES
1
2
Holley, R., and Stroock, D., "Diffusions on an Infinite
Dimensional Torus," J. Fnal. Anal. 42, no. 1, (1981)
pp. 29-63.
Stroock, Daniel W., "The MaLliavin Calculus and its
Application to Second Order Parabolic Differential
Equations, parts I & II," Math. Systems Theory 14, (1981)pp.
25-65 & 141-171.
3
Stroock, D., and Varadhan, S.R.S., Multidimensional
Diffusion Processes, Grundlehren 233, Springer-Verlag
(Berlin, Heidelberg, New York), 1979.