ARTICLE IN PRESS
Stochastic Processes and their Applications 116 (2006) 381–406
www.elsevier.com/locate/spa
Infinite dimensional stochastic differential equations
of Ornstein–Uhlenbeck type
Siva R. Athreyaa, Richard F. Bassb,!,1, Maria Gordinab,2,
Edwin A. Perkinsc,3
a
Statmath Unit, Indian Statistical Institute, 8th Mile Mysore Road, R.V. College P.O., Bangalore 560059, India
b
Department of Mathematics, University of Connecticut, Storrs, CT 06269, USA
c
Department of Mathematics, University of British Columbia, Vancouver, B.C., Canada V6T 1Z2
Received 28 February 2005; accepted 18 October 2005
Available online 22 November 2005
Abstract
We consider the operator
Lf ðxÞ ¼
1
1
X
1X
q2 f
qf
aij ðxÞ
ðxÞ $
li xi bi ðxÞ
ðxÞ.
2 i;j¼1
qxi qxj
qxi
i¼1
We prove existence and uniqueness of solutions to the martingale problem for this operator under
appropriate conditions on the aij ; bi , and li . The process corresponding to L solves an infinite
dimensional stochastic differential equation similar to that for the infinite dimensional Ornstein–
Uhlenbeck process.
r 2005 Elsevier B.V. All rights reserved.
MSC: primary 60H10; secondary 35R15, 60H30
Keywords: Semigroups; Hölder spaces; Perturbations; Resolvents; Elliptic operators; Ornstein–Uhlenbeck
processes; Infinite dimensional stochastic differential equations
!Corresponding author. Tel.: +1 8604863939; fax: +1 8604864238.
E-mail address: bass@math.uconn.edu (R.F. Bass).
Research supported in part by NSF Grant DMS0244737.
2
Research supported in part by NSF Grant DMS0306468.
3
Research supported in part by NSERC Research Grant.
1
0304-4149/$ - see front matter r 2005 Elsevier B.V. All rights reserved.
doi:10.1016/j.spa.2005.10.001
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1. Introduction
Let li be a sequence of positive reals tending to infinity, let sij and bi be functions defined
on a suitable Hilbert space which satisfy certain continuity and non-degeneracy conditions,
and let W it be a sequence of independent one-dimensional Brownian motions. In this paper
we consider the countable system of stochastic differential equations
dX it ¼
1
X
j¼1
sij ðX t Þ dW it $ li bi ðX t ÞX it dt;
i ¼ 1; 2; . . . ,
(1.1)
and investigate sufficient conditions for weak existence and weak uniqueness to hold. Note
that when the sij and bi are constant, we have the stochastic differential equations
characterizing the infinite-dimensional Ornstein–Uhlenbeck process.
We approach the weak existence and uniqueness of (1.1) by means of the martingale
problem for the corresponding operator
Lf ðxÞ ¼
1
1
X
1X
q2 f
qf
aij ðxÞ
ðxÞ $
li xi bi ðxÞ
ðxÞ
2 i;j¼1
qxi qxj
qx
i
i¼1
(1.2)
P
operating on a suitable class of functions, where aij ðxÞ ¼ 1
k¼1 sik ðxÞsjk ðxÞ. Our main
theorem says that if the aij are nondegenerate and bounded, the bi are bounded above and
below, and the aij and bi satisfy appropriate Hölder continuity conditions, then existence
and uniqueness hold for the martingale problem for L; see Theorem 5.7 for a precise
statement.
There has been considerable interest in infinite dimensional operators whose coefficients
are only Hölder continuous. For perturbations of the Laplacian, see Cannarsa and Da
Prato [6], where Schauder estimates are proved using interpolation theory and then applied
to Poisson’s equation in infinite dimensions with Hölder continuous coefficients (see also
[14]).
Similar techniques have been used to study operators of the form (1.2). In finite
dimensions see [17–19,12]. For the infinite dimensional case see [7–11,14,23]. Common to
all of these papers is the use of interpolation theory to obtain the necessary Schauder
estimates. In functional analytic terms, the system of equations (1.1) is a special case of the
equation
pffiffiffiffiffiffiffiffiffiffiffi
dX t ¼ ðbðX t ÞX t þ F ðX t ÞÞ dt þ aðX t Þ dW t ,
(1.3)
where a is a mapping from a Hilbert space H to the space of bounded nonnegative selfadjoint linear operators on H, b is a mapping from H to the nonnegative self-adjoint linear
operators on H (not necessarily bounded), F is a bounded operator on H, and bðxÞx
represents the composition of operators. Previous work on (1.3) has concentrated on the
following cases: where a is constant, b is Lipschitz continuous, and F & 0; where a and b
are constant and F is bounded; and where F is bounded, b is constant and a is a
perturbation of a constant operator by means of a Hölder continuous nonnegative selfadjoint operator. We also mention the paper [13] where weak solutions to (1.3) are
considered. In our paper we consider Eq. (1.3) with the a and b satisfying certain Hölder
conditions and F & 0. There would be no difficulty introducing bounded F ðX t Þ dt terms,
but we chose not to do so.
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The paper most closely related to this one is that of Zambotti [23]. Our results
complement those of [23] as each has its own advantages. We were able to remove the
restriction that the aij ’s be given by means of a perturbation by a bounded nonnegative
operator which in turn facilitates localization, but at the expense of working with respect to
a fixed basis and hence imposing summability conditions involving the off-diagonal aij .
See Remark 5.10 for a further discussion in light of a couple of examples and our explicit
hypotheses for Theorem 5.7.
There are also martingale problems for infinite dimensional operators with Hölder
continuous coefficients that arise from the fields of superprocesses and stochastic partial
differential equations (SPDE). See [20] for a detailed introduction to these. We mention
[15], where superprocesses in the Fleming–Viot setting are considered, and [4], where
uniqueness of a martingale problem for superprocesses on countable Markov chains with
interactive branching is shown to hold. These latter results motivated the present approach
as the weighted Hölder spaces used there for our perturbation bounds coincide with the
function spaces S a used here (see Section 2), at least in the finite-dimensional setting
(see [1]).
Consider the one dimensional SPDE
qu
1 q2 u
_,
ðt; xÞ ¼
ðx; tÞ þ AðuÞ dW
qt
2 qx2
_ is space-time white noise. If one sets
where W
Z 2p
X jt ¼
eijx uðx; tÞ dx; j ¼ 0; '1; '2; . . . ,
(1.5)
0
2
then the collection fX i g1
i¼$1 can be shown to solve system (1.1) with li ¼ i , the bi
constant, and the aij defined in an explicit way in terms of A. Our original interest in the
problem solved in this paper was to understand (1.5) when the coefficients A were bounded
above and below but were only Hölder continuous as a function of u. The results in this
paper do not apply to (1.5) and we hope to return to this in the future.
The main novelties of our paper are the following.
(1) C a estimates (i.e., Schauder estimates) for the infinite dimensional Ornstein–Uhlenbeck
process. These were already known (see [14]), but we point out that in contrast to using
interpolation theory, our derivation is quite elementary and relies on a simple real
variable lemma together with some semigroup manipulations.
(2) Localization. We use perturbation theory along the lines of Stroock–Varadhan to
establish uniqueness of the martingale problem when the coefficients are sufficiently
close to constant. We then perform a localization procedure to establish our main
result. In infinite dimensions localization is much more involved, and this argument
represents an important feature of this work.
(3) A larger class of perturbations. Unlike much of the previous work cited above, we do
not require that the perturbation of the second order term be bounded by an operator
that is nonnegative. The price we pay is that we require additional conditions on the
off-diagonal aij ’s.
After some definitions and preliminaries in Section 2, we establish the needed Schauder
estimates in Section 3. Section 4 contains the proof of existence and Section 5 the
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uniqueness. Section 5 also contains some specific examples where our main result applies.
This includes coefficients aij which depend on a finite number of local coordinates near ði; jÞ
in a Hölder manner.
We use the letter c with or without subscripts for finite positive constants whose value is
unimportant and which may vary from proposition to proposition. a will denote a real
number between 0 and 1.
2. Preliminaries
We use the following notation. If H is a separable Hilbert space and f : H ! R, Dw f ðxÞ
is the directional derivative of f at x 2 H in the direction w; we do not require w to be a unit
vector. The inner product in H is denoted by h(; (i, and j ( j denotes the norm generated by
this inner product. C b ¼ C b ðHÞ is the collection of R-valued bounded continuous
functions on H with the usual supremum norm. Let C 2b be the set of functions in C b for
which the first and second order partials are also in C b . For a 2 ð0; 1Þ, set
jf jC a ¼
sup
x2H;ha0
jf ðx þ hÞ $ f ðxÞj
jhja
a
and let C be the set of functions in C b for which kf kC a ¼ kf kC b þ jf jC a is finite.
Let V : DðV Þ ! H be a (densely defined) self-adjoint nonnegative definite operator such
that
V $1 is a trace class operator on H.
(2.1)
Then there is a complete orthonormal system f!n : n 2 Ng of eigenvectors of V $1 with
corresponding eigenvalues l$1
n , ln 40, satisfying
1
X
l$1
n o1;
n¼1
ln " 1; V !n ¼ ln !n
(see, e.g. Section 120 in [21]). Let Qt ¼ e$tV be the semigroup of contraction operators on
H with generator $V . If w 2 H, let wn ¼ hw; !n i and we will write Di f and Dij f for D!i f and
D!i D!j f , respectively.
Assume a : H ! LðH; HÞ is a mapping from H to the space of bounded self-adjoint
operators on H and b : H ! LðDðV Þ; HÞ is a mapping from H to self-adjoint nonnegative
definite operators on DðV Þ such that f!n g are eigenvectors of bðxÞ for all x 2 H. If aij ðxÞ ¼
h!i ; aðxÞ!j i and bðxÞð!i Þ ¼ li bi ðxÞ!i , we assume that for some g40
X
aij ðxÞzi zj Xgjzj2 ; x; z 2 H,
g$1 jzj2 X
i;j
g$1 Xbi ðxÞXg;
x 2 H; i 2 N.
(2.2)
We consider the martingale problem for the operator L which, with respect to the
coordinates hx; !i i, is defined by
Lf ðxÞ ¼
1
1
X
1X
aij ðxÞDij f ðxÞ $
li xi bi ðxÞDi f ðxÞ.
2 i;j¼1
i¼1
(2.3)
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Let T be the class of functions in C 2b that depend on only finitely many coordinates and
T0 be the set of functions in T with compact support. More precisely, f 2 T if there exists
n and f n 2 C 2b ðRn Þ such that f ðx1 ; . . . ; xn ; . . .Þ ¼ f n ðx1 ; . . . ; xn Þ for each point ðx1 ; x2 ; . . .Þ
and f 2 T0 if, in addition, f n has compact support. Let X t denote the coordinate maps on
the space Cð½0; 1Þ; HÞ of continuous H-valued paths. We say that a probability measure P
on Cð½0; 1Þ; HÞ is a solutionR to the martingale problem for L started at x0 if PðX 0 ¼
t
x0 Þ ¼ 1 and f ðX t Þ $ f ðX 0 Þ $ 0 Lf ðX s Þ ds is a martingale for each f 2 T.
The connection between systems of stochastic differential equations and martingale
problems continues to hold in infinite dimensions; see, for example, [16, pp. 166–168].
We will use this fact without further mention.
There are different possible martingale problems depending on what class of functions
we choose as test functions. Since existence is the easier part for the martingale problem
(see Theorem 4.2) and uniqueness is the more difficult part, we will get a stronger and more
useful theorem if we have a smaller class of test functions. The collection T is a reasonably
small class. When aðxÞ & a0 and bðxÞ & V are constant functions, the process associated
with L is the well-known H-valued Ornstein–Uhlenbeck process. We briefly recall the
definition; see Section 5 of [1] for details. Let ðW t ; tX0Þ be the cylindrical Brownian
motion on H with covariance a. Let Ft be the right continuous filtration generated by
W. Consider the stochastic differential equation
dX t ¼ dW t $ VX t dt.
(2.4)
x
There is a pathwise unique solution to (2.4) whose laws fP ; x 2 Hg define a unique
homogeneous strong Markov process on the space of continuous H-valued paths (see, e.g.
Section 5.2 of [16]). fX t ; tX0g is an H-valued Gaussian process satisfying
for all h 2 H,
EðhX t ; hiÞ ¼ hX 0 ; Qt hi
(2.5)
and
CovðhX t ; gihX t ; hiÞ ¼
Z
t
0
hQt$s h; aQt$s gi ds.
(2.6)
The law of X started at x solves the martingale problem for
L0 f ðxÞ ¼
1
1
X
1X
a0ij Dij f ðxÞ $
li xi Di f ðxÞ.
2 i;j¼1
i¼1
(2.7)
R1
We let Pt f ðxÞ ¼ Ex f ðX t Þ be the semigroup corresponding to L0 , and Rl ¼ 0 e$ls Ps ds
be the corresponding resolvent. We define the semigroup norm k ( kSa for a 2 ð0; 1Þ by
jf jSa ¼ sup t$a=2 kPt f $ f kC b
(2.8)
t40
and
kf kSa ¼ kf kC b þ jf jSa .
Let Sa denote the space of measurable functions on H for which this norm is finite.
b=2
For x 2 H and b 2 ð0; 1Þ define jxjb ¼ supk jhx; !k ijlk and
H b ¼ fx 2 H : jxjb o1g.
(2.9)
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3. Estimates
We start with the following real variable lemma.
Lemma 3.1. Let A40; B40. Assume K : C b ðHÞ ! C b ðHÞ is a bounded linear operator such
that
kKf kC b pAkf kC b ;
(3.1)
f 2 C b ðHÞ,
and there exists v 2 H such that
kKf kC b pBkDv f kC b ,
(3.2)
for all f such that Dv f 2 C b ðHÞ. Then for each a 2 ð0; 1Þ there is a constant c1 ¼ c1 ðaÞ such
that
kKf kC b pc1 jvja jf jC a Ba A1$a
for all f 2 C a .
Proof. Assume (3.1) and (3.2), the latter for some v 2 H. Let fpt : tX0g be the standard
Brownian density on R. If f 2 C a , set
Z
pe * f ðxÞ ¼
f ðx þ zvÞpe ðzÞ dz; x 2 H.
R
Since a change of variables shows that
Z
Z
pe * f ðx þ hvÞ $ pe * f ðxÞ ¼
f ðx þ zvÞpe ðz $ hÞ dz $
f ðx þ zvÞpe ðzÞ dz,
R
it follows that
Dv ðpe * f ÞðxÞ ¼ $
this is in C b ðHÞ and
Z
R
f ðx þ zvÞp0e ðzÞ dz;
" Z
"
"
"
0
"
jDv ðpe * f ÞðxÞj ¼ "$
f ðx þ zvÞpe ðzÞ dz""
"
"Z
"
"
0
"
¼ " ðf ðx þ zvÞ $ f ðxÞÞpe ðzÞ dz""
Z
jzj
a
a
jzja pe ðzÞ dz
pjf jC jvj
e
where c2 ¼
R
¼ c2 jf jC a jvja eða$1Þ=2 ,
jzjaþ1 p1 ðzÞ dz. We therefore obtain from (3.2) that
kKðpe * f ÞkC b pc2 Bjf jC a jvja eða$1Þ=2 .
Next note that
Z
jf ðx þ zvÞ $ f ðxÞjpe ðzÞ dz
Z
pjf jC a jvja jzja pe ðzÞ dz
jpe * f ðxÞ $ f ðxÞjp
¼ c3 jf jC a jvja ea=2 ,
(3.3)
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where c3 ¼
R
387
jzja p1 ðzÞ dz. By (3.1)
kKðpe * f $ f ÞkC b pc3 Ajf jC a jvja ea=2 .
(3.4)
Let c4 ¼ c2 _ c3 and e ¼ B2 =A2 . Combining (3.3) and (3.4) we have
kKf kC b pc4 jf jC a jvja ea=2 ½A þ Be$1=2 +
¼ 2c4 jf jC a jvja Ba A1$a :
&
Set
hðuÞ ¼
(
ð2uÞ=ðe2u $ 1Þ;
ua0;
1;
u ¼ 0;
and
jwjt ¼
X
!1=2
w2i hðli tÞ
i
pjwj.
Recall
Qt w ¼
1
X
e$li t wi ei .
i¼1
We have the following by Propositions 5.1 and 5.2 of [1]:
Proposition 3.2. (a) For all w 2 H, f 2 C b ðHÞ, and t40, Dw Pt f 2 C b ðHÞ and
kDw Pt f kC b p
jwjt kf kC b
pffiffiffiffi .
gt
(3.5)
(b) If tX0, w 2 H, and f : H ! R is in C b ðHÞ such that DQt w f 2 C b ðHÞ, then
Dw Pt f ðxÞ ¼ Pt ðDQt w f ÞðxÞ;
x 2 H.
In particular,
kDw Pt f kC b pkDQt w f kC b .
(3.6)
We now prove:
Corollary 3.3. Let f 2 C a , u; w 2 H. Then for all t40, Dw Pt f and Du Dw Pt f are in C b ðHÞ
and there exists a constant c1 ¼ c1 ða; gÞ independent of t such that
kDw Pt f kC b pc1 jwjt jf jC a tða$1Þ=2 pc1 jwj jf jC a tða$1Þ=2
(3.7)
and
kDu Dw Pt f kC b pc1 jQt=2 ujt=2 jwjt=2 jf jC a t2$1 pc1 jujt=2 jwjt=2 jf jC a t2$1
a
pc1 jujjwjjf jC a t2$1 .
a
a
ð3:8Þ
Proof. That Dw Pt f is in C b ðHÞ is immediate from Proposition 3.2(a). By (3.5) and (3.6) we
may apply Lemma 3.1 to K ¼ Dw Pt with v ¼ Qt w, A ¼ jwjt ðgtÞ$1=2 and B ¼ 1 to conclude
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for f 2 C a
kDw Pt f kC b pc2 jQt wja jf jC a jwj1$a
ðgtÞ$ð1$aÞ=2
t
pc2 gða$1Þ=2 jwjt jf jC a tða$1Þ=2 .
ð3:9Þ
This gives (3.7).
By Proposition 3.2, Dw Du Pt f ¼ Dw Pt=2 DQt=2 u Pt=2 f , and the latter is seen to be in C b ðHÞ
by invoking Proposition 3.2(a) twice. Using (3.5) and then (3.9) we have
kDw Du Pt f kC b ¼ kDw Pt=2 DQt=2 u Pt=2 f kC b
pjwjt=2 ðgt=2Þ$1=2 kDQt=2 u Pt=2 f kC b
pjwjt=2 ðgt=2Þ$1=2 c2 gða$1Þ=2 jQt=2 ujt=2 jf jC a ðt=2Þða$1Þ=2 .
This gives (3.8).
&
Remark 3.4. We often will use the fact that there exists c1 such that
kf kC a pc1 kf kSa .
(3.10)
This is (5.20) of [1].
Corollary 3.5. There exists c1 ¼ c1 ða; gÞ such that for all l40, f 2 C a , ipj, we have
Di Rl f ; Dij Rl f 2 C b , and
kDi Rl f kC b pc1 ðl þ li Þ$ðaþ1Þ=2 jf jC a ,
(3.11)
kDij Rl f kC b pc1 ðl þ lj Þ$a=2 jf jC a ,
(3.12)
kDi Rl f kC a pc1 ðl þ li Þ$1=2 kf kC a ,
(3.13)
kDij Rl f kC a pc1 kf kC a .
(3.14)
a
Proof. Corollary 3.3 is exactly the same as Proposition 5.4 in [1], but with the S norms
replaced by C a norms. We may therefore follow the proofs of Theorem 5.6 and Corollary
5.7 in [1] and then use (3.10) to obtain our result. However, the proofs in [1] can be
streamlined, so for the sake of clarity and completeness we give a more straightforward
proof.
From (3.7) and (3.8) we may differentiate under the time integral and conclude that the
first and second order partial derivatives of Rl f are continuous. To derive (3.12), note first
that by (3.8),
kDij Pt f kC b ¼ kDji Pt f kC b pc2 jQt=2 !j jj!i jjf jC a t2$1
a
a
¼ c2 e$lj t=2 jf jC a t2$1 .
ð3:15Þ
Multiplying by e$lt and integrating over t from 0 to 1 yields (3.12).
Next we turn to (3.14). Recall the definition of the Sa norm from (2.8). In view of (3.10)
it suffices to show
kDij Rl f kSa pc3 kf kC a .
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Since
kPt Dij Rl f $ Dij Rl f kC b p2kDij Rl f kC b pc1 jf jC a ðl þ lj Þ$a=2
by (3.12), we need only consider tpðl þ lj Þ$1 .
Use Proposition 3.2(b) to write
Pt Dij Rl f $ Dij Rl f ¼ ½e$li t e$lj t Dij Pt Rl f $ Dij Pt Rl f +
þ ½Dij Pt Rl f $ Dij Rl f +.
Recalling that li plj , we see that the first term is bounded in absolute value by
Z 1
a=2
lj e$ls kDij Ptþs f kC b ds
c4 ðlj tÞa=2 kDij Pt Rl f kC b pc5 ta=2
pc5 t
ð3:16Þ
0
a=2
jf jC a ,
using (3.15).
The second term in (3.16) is equal, by the semigroup property, to
Z 1
Z 1
e$ls Dij Ptþs f ds $
e$ls Dij Ps f ds
0
0
Z 1
Z t
e$ls Dij Ps f ds $ elt
e$ls Dij Ps f ds.
¼ ðelt $ 1Þ
0
0
a=2
lt
and the bound for the second term in (3.16) now
Since ltp1, then e $ 1pc6 ðltÞ
follows by using (3.15) to bound the above integrals, and recalling again that ltp1.
The proofs of (3.11) and (3.13) are similar but simpler, and are left to the reader (or refer
to [1]). &
4. Existence
Before discussing existence, we first need the following tightness result.
Lemma 4.1. Suppose Y is a real-valued solution of
Z t
Y t ¼ y0 þ M t $ l
Y r dr,
(4.2)
0
where M t is a martingale such that for some c1 ,
hMit $ hMis pc1 ðt $ sÞ;
spt.
(4.3)
R t $lðt$sÞ
$lt
dM s . Then Z t ¼ Y t $ e y0 and for each q4e$1 ,
Let T40, e 2 ð0; 1Þ. Let Z t ¼ 0 e
there exists a constant c2 ¼ c2 ðe; q; TÞ such that for all d 2 ð0; 1+,
"
#
deq$1
2q
E
sup
jZt $ Z s j pc2 ðe; q; TÞ ð1$eÞq .
(4.4)
l
s;tpT;jt$sjpd
Proof. Some elementary stochastic calculus shows that
Z t
e$lðt$sÞ dM s ,
Y t ¼ e$lt y0 þ
0
which proves the first assertion about Z.
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Fix s0 ot0 pT. Let
K t ¼ ½e$lðt0 $s0 Þ $ 1+e$ls0
Z
t
elr dM r
0
and
Lt ¼ e$lt0
Z
t
elr dM r .
s0
Note
Z t0 $ Z s 0 ¼ K s 0 þ L t0 .
Then
hKis0 ¼ ½e$lðt0 $s0 Þ $ 1+2 e$2ls0
Z
s0
0
pc3 ½e$lðt0 $s0 Þ $ 1+2 e$2ls0
e2lr dhMir
e2ls0 $ 1
2l
pc3 ½e$lðt0 $s0 Þ $ 1+2 l$1
ð1 ^ lðt0 $ s0 ÞÞ
.
pc3
l
Considering the cases lðt0 $ s0 Þ41 and p1 separately, we see that for any e 2 ð0; 1Þ this is
less than
ðt0 $ s0 Þe
.
l1$e
Now applying the Burkholder–Davis–Gundy inequalities, we see that
c4 ðeÞ
EjK s0 j2q pc5 ðe; qÞ
ðt0 $ s0 Þeq
q41.
(4.5)
; q41.
lð1$eÞq
Combining (4.5) and (4.6) we get
(4.6)
lð1$eÞq
;
Similarly,
1 $ e$2lðt0 $s0 Þ
2l
pc6 ðl$1 ^ ðt0 $ s0 ÞÞ
ð1 ^ lðt0 $ s0 ÞÞ
.
¼ c6
l
This leads to
hLit0 pc6
EjLt0 j2q pc7 ðe; qÞ
ðt0 $ s0 Þeq
EjZ t0 $ Z s0 j2q pc8 ðe; qÞ
jt0 $ s0 jeq
.
lð1$eÞq
It is standard to obtain (4.4) from this; cf. the proof of Theorem I.3.11 in [2].
Recall the definition of H b from (2.9).
&
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391
Theorem 4.2. Assume aij : H ! R is continuous for all i; j, bi is continuous for all i, (2.2)
holds, and for some p41 and positive constant c1
lk Xc1 kp ;
(4.7)
kX1.
Then for every x0 2 H, there is a solution P to the martingale problem for L starting at x0 .
Moreover if b 2 ð0; 1Þ, then any such solution has supeptpe$1 jX t jb o1 for all e P-a.s. If in
addition x0 2 H b for some b 2 ð0; 1Þ, then any solution P to the martingale problem for L
starting at x0 will satisfy
sup jX t jb o1
(4.8)
for all T40; P $ a.s.
tpT
Proof. This argument is standard and follows by making some minor modifications to the
existence result in Section 5.2 of [16]. We give a sketch and leave the details to the reader.
Fix x0 in H. Using the finite dimensional existence result, we may construct a solution
X nt ¼ ðX n;k
t : k 2 NÞ of
" Z
¼ x0 ðkÞ þ 1ðkpnÞ $
X n;k
t
t
n
lk X n;k
s bk ðX s Þ ds þ
0
n Z
X
j¼1
0
t
#
snk;j ðX ns Þ dW js .
Here fW j g is a sequence of independent one-dimensional standard Brownian motions and
sn ðxÞ is a symmetric positive definite squareProot of ðaij ðxÞÞi;jpn which is continuous in
x 2 H (see Lemma 5.2.1 of [22]). Then X nt ¼ nk¼1 X n;k
t !k has paths in Cð½0; 1Þ; HÞ and we
next verify this sequence of processes is relatively compact in this space. Once one has
relative compactness, it is routine to use the continuity of the aij and bi on H to show that
any weak limit point of fX n g will be a solution to the martingale problem for L starting
at x0 .
By our assumptions on bk , each bkRis bounded above by g$1 and below by g. We perform
t
n;k
n;k
n
a time change on X n;k
be the inverse of An;k
t , and let
t : let At ¼ 0 bk ðX s Þ ds, let tt
n;k
n;k
n;k
Y t ¼ X tn;k . Then Y t solves the stochastic differential equation
t
Y n;k
t
# Z
¼ x0 ðkÞ þ 1ðkpnÞ $
0
t
lk Y n;k
s
ds þ
M n;k
t
$
,
n;k
n;k
where M n;k
t is a martingale satisfying jhM it $ hM is jpc2 jt $ sj, and c2 is a constant not
depending on n or k.
We may use stochastic calculus to write
n;k
n;k
Y n;k
t ¼ x ðtÞ þ Z t ,
where
xn;k ðtÞ ¼ ½1ðkpnÞ e$lk t þ 1ðk4nÞ +x0 ðkÞ
and
Z n;k
t
¼ 1ðkpnÞ
Z
0
t
e$lk ðt$sÞ dM n;k
s .
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Let T40 and sptpT. Choose e 2 ð0; 1 $ 1pÞ and q42=e. By Lemma 4.1 we have for
kpn and any d 2 ð0; g+,
E
"
sup
u;vpg$1 T;ju$vjpdg$1
#
n;k 2q
jZ n;k
pc2 ðe; q; g$1 TÞg$eqþ1
v $ Zu j
deq$1
lkð1$eÞq
.
Hence, undoing the time change tells us that
E
"
sup
s;tpT;js$tjpd
where
#
deq$1
n;k
n;k 2q
e
e
jX t $ X s j p1ðkpnÞ c3 ðe; q; g; TÞ ð1$eÞq ,
lk
$lk
n;k
Xe n;k
t ¼ 1ðkpnÞ ðX t $ e
Rt
0
bk ðX nr Þ dr
x0 ðkÞÞ þ 1ðk4nÞ x0 ðkÞ,
n;k
so that X~ tn;k ¼ Z n;k
t . Now for 0ps; tpT and jt $ sjpg,
t
%
%
%
%X
%
%
1=q
n
n
2q
n
n
2
n;k
n;k
2
ðEjXe t $ Xe s j Þ ¼ kjXe t $ Xe s j kq ¼ %
jXe t $ Xe s j %
%
% k
q
X
X
n;k
n;k 2
n;k
2q 1=q
e
e
e
p
kjX t $ X s j kq ¼
ðEjX t $ Xe n;k
s j Þ
k
k
X jt $ sje$1=q
pc3 ðe; q; g; TÞ1=q
,
l1$e
k
k
where k ( kq is the usual Lq ðPÞ norm.
By our choice of e this is bounded by c4 ðe; q; g; TÞjt $ sje=2 , and hence
sup EjXe nt $ Xe ns j2q pcq4 jt $ sjeq=2 ;
s; tpT; js $ tjpg.
n
It is well known ([5]) that this implies the relative compactness of Xe n in CðRþ ; HÞ.
We may write
X nt ¼ Xe nt $ U n ðtÞ,
where
U n ðtÞ ¼
n
X
k¼1
e
$lk
Rt
0
bk ðX nr Þ dr
x0 ðkÞek .
If sot, then
jU n ðtÞ $ U n ðsÞj2 ¼
p
n #
X
k¼1
n
X
k¼1
e
$lk
Rt
0
bk ðX nr Þ dr
$e
$lk
Rs
0
bk ðX nr Þ dr
ððl2k g$2 jt $ sj2 Þ ^ 1Þx0 ðkÞ2
$2
x0 ðkÞ2
(4.9)
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p
1
X
1ðlk pgjt$sj$1 Þ l2k x0 ðkÞ2 g$2 jt $ sj2
k¼1
þ
1
X
k¼1
1ðlk 4gjt$sj$1 Þ x0 ðkÞ2 .
Fix e40. First choose N so that
1
X
1ðlk 4gd$1 Þ x0 ðkÞ2 oe
N
X
l2k x0 ðkÞ2 g$2 d2 oe.
k¼1
393
P1
k¼N
ð4:10Þ
x0 ðkÞ2 oe, and then d40 so that
and
k¼1
If 0ot $ sod, then use the above bounds in (4.10) to conclude that
jU n ðtÞ $ U n ðsÞj2 p
N
X
k¼1
þ
o3e.
l2k x0 ðkÞ2 g$2 d2 þ
1
X
k¼1
1
X
k¼N
x0 ðkÞ2
1ðlk 4gd$1 Þ x0 ðkÞ2
This and the fact that U n ð0Þ ! x0 in H prove that fU n g is relatively compact in CðRþ ; HÞ.
The relative compactness of fX n g now follows from (4.9).
Assume now P is any solution to the martingale problem for L starting at x0 2 H and
let X it denote hX t ; !i i. Fix b 2 ð0; 1Þ and T41. Choose e 2 ð0; 1 $ bÞ. Using a time change
argument as above but now with no parameter n and d ¼ 1, we may deduce for any q41=e
and k 2 N
&
'
Rt
$l
b ðX Þ ds
$b=2
P sup jX kt $ e k 0 k s x0 ðkÞj4lk
tpT
pc5 ðe; q; T=gÞlbq$qð1$eÞ
.
k
The right-hand side is summable over k by our choice of e and (4.7). The Borel–Cantelli
lemma therefore implies that
Rt
$b=2
$l
b ðX Þ ds
sup jX kt $ e k 0 k s x0 ðkÞjplk
for k large enough; a.s.
(4.11)
tpT
If x0 2 H b , this implies that with probability 1, for large enough k,
sup jX kt jlk p1 þ x0 ðkÞlk p1 þ jx0 jb ,
b=2
b=2
tpT
and hence
sup jX t jb o1
tpT
a:s.
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For general x0 2 H, (4.11) implies
sup
T
$1
ptpT
jX kt jlk p1 þ e$lk gT lk jx0 jpc6 ðg; T; b; x0 Þ for large enough k; a.s.
b=2
$1
b=2
This implies supT $1 ptpT jX t jb o1 a.s. and so completes the proof.
&
5. Uniqueness
We continue to assume that ðaij Þ and ðbi Þ are as in Section 2 and in particular will satisfy
(2.2). Let y0 2 H and let P be any solution to the martingale problem for L started at y0 .
For any bounded function f define
Z 1
Sl f ¼ E
e$ls f ðX s Þ ds.
0
Fix z0 2 H and define
L0 f ðxÞ ¼
1
1
X
1X
aij ðz0 ÞDij f ðxÞ $
li xi bi ðz0 ÞDi f ðxÞ.
2 i; j¼1
i
(5.1)
Set B ¼ L $ L0 and let Rl be the resolvent for L0 as in Section 2.
To make this agree with the definition of L0 in Section 2 we must replace li by b
li ¼
bi ðz0 Þli and set a0ij ¼ aij ðz0 Þ. As gpbi ðz0 Þpg$1 , and the constants in Corollary 3.5 may
depend on g, we see that the bounds in Corollary 3.5 involving the original li remain valid
for Rl . We also will use the other results in Section 3 with b
li in place of li without further
comment. In addition, if we simultaneously replace bi by bbi ¼ bi =bi ðz0 Þ, then
Lf ðxÞ ¼
1
1
X
1X
b
li xi bbi ðxÞDi f ðxÞ,
aij ðxÞDij f ðxÞ $
2 i; j¼1
i¼1
L0 f ðxÞ ¼
and
bbi ðz0 Þ ¼ 1
1
1
X
1X
b
li xi Di f ðxÞ,
aij ðz0 ÞDij f ðxÞ $
2 i; j¼1
i¼1
for all i.
In Propositions 5.1 and 5.2 we will simply assume bi ðz0 Þ ¼ 1 for all i without loss of
generality, it being understood that the above substitutions are being made. In each case it
is easy to check that the hypotheses on ðbi ; li Þ carry over to ðbbi ; b
li Þ and as the conclusions
only involve L, L0 , Rl , and our solution X, which remain unaltered by these substitutions, this reduction is valid.
Let
1
X
Z ¼ sup
jaij ðxÞ $ aij ðz0 Þj.
(5.2)
x
i; j¼1
Set
Bi ðxÞ ¼ xi ðbi ðxÞ $ 1Þ.
As before, a will denote a parameter in ð0; 1Þ.
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Proposition 5.1. Assume
X
$a=2
jaij jC a lj o1,
395
(5.3)
ipj
X
li kBi kC b o1,
X
li
i
1=2
(5.4)
and
ð1$aÞ=2
i
jBi jC a o1.
(5.5)
There exists c1 ðlÞ ! 0 as l ! 1 and c2 ¼ c2 ða; gÞ such that for all f 2 C a , we have BRl f 2
C a and
kBRl f kC a pðc1 ðlÞ þ c2 ZÞkf kC a .
Proof. We have
jBRl f ðxÞjp
X
i;j
þ
jaij ðxÞ $ aij ðz0 ÞjjDij Rl f ðxÞj
X
i
li jxi jjbi ðxÞ $ 1jjDi Rl f ðxÞj
pZc3 jf jC a þ c4 ðlÞjf jC a ,
ð5:6Þ
where c4 ðlÞ ! 0 as l ! 1 by (5.4) and (3.11). In particular, the series defining BRl f is
absolutely uniformly convergent.
Let abij ðxÞ ¼ aij ðxÞ $ aij ðz0 Þ. If h 2 H, then
"
"X
"
½b
a ðx þ hÞDij Rl f ðx þ hÞ $ abij ðxÞDij Rl f ðxÞ+
jBRl f ðx þ hÞ $ BRl f ðxÞj ¼ "
" i;j ij
"
"
X
"
þ
li ½Bi ðx þ hÞDi Rl f ðx þ hÞ $ Bi ðxÞDi Rl f ðxÞ+"
"
i
"
"
"
"X
"
"
p"
abij ðx þ hÞðDij Rl f ðx þ hÞ $ Dij Rl f ðxÞÞ"
"
" i;j
"
"
"X
"
"
"
þ"
ðb
aij ðx þ hÞ $ abij ðxÞÞDij Rl f ðxÞ"
"
" i;j
"
"
"
"X
"
"
þ"
li Bi ðx þ hÞðDi Rl f ðx þ hÞ $ Di Rl f ðxÞÞ"
"
" i
"
"
"
"X
"
"
þ"
li ðBi ðx þ hÞ $ Bi ðxÞÞDi Rl f ðxÞ"
"
" i
¼ S1 þ S2 þ S3 þ S4 .
ð5:7Þ
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Use (3.14) to see that
X
S 1 pc5
jb
aij ðx þ hÞjjf jC a jhja
i;j
pc6 Zjf jC a jhja .
By (3.12)
S2 p
X
jaij ðx þ hÞ $ aij ðxÞj jDij Rl f ðxÞj
i;j
pc7
ð5:8Þ
X
ipj
jaij jC a jhja ðl þ lj Þ$a=2 jf jC a
pc8 ðlÞjf jC a jhja ,
ð5:9Þ
where (5.3) and dominated convergence imply liml!1 c8 ðlÞ ¼ 0. By (3.13)
X
S 3 pc9
li jBi ðx þ hÞjðl þ li Þ$1=2 jf jC a jhja pc10 ðlÞjf jC a jhja ,
(5.10)
i
where c10 ðlÞ ! 0 as l ! 1 by (5.4) and dominated convergence. By (3.11)
X
S 4 pc11
li jBi jC a ðl þ li Þ$ð1þaÞ=2 jf jC a jhja pc12 ðlÞjf jC a jhja ,
(5.11)
i
where again c12 ðlÞ ! 0 as l ! 1 by (5.5). Combining (5.8)–(5.11) yields
jBRl f jC a p½c13 ðlÞ þ c14 Z+jf jC a .
This and (5.6) complete the proof.
&
C an
a
Let
denote
S a those functions in C which only depend on the first n coordinates. Note
that T0 , n C n . Note also that S l f is a real number while Rl f is a function.
S
Proposition 5.2. If f 2 n C an , then
S l f ¼ Rl f ðy0 Þ þ S l BRl f .
Proof. Fix z0 2 H. Suppose h 2 T. Since hðX t Þ $ hðX 0 Þ $
taking expectations we have
Z t
EhðX t Þ $ hðy0 Þ ¼ E
LhðX s Þ ds.
(5.12)
Rt
0
LhðX s Þ ds is a martingale,
0
Multiplying by e$lt and integrating over t from 0 to 1, we obtain
Z t
Z 1
1
$lt
Sl h $ hðy0 Þ ¼ E
e
LhðX s Þ ds dt
l
0
0
Z 1
1
1
¼ E
e$ls LhðX s Þ ds ¼ S l Lh.
l 0
l
This can be rewritten as
lS l h $ S l L0 h ¼ hðy0 Þ þ S l Bh.
(5.13)
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397
Define
Ln0 f ðxÞ ¼
n
X
i;j¼1
aij ðz0 ÞDij f ðxÞ $
n
X
i¼1
li xi Di f ðxÞ.
Let Rnl be the corresponding resolvent. The corresponding process is an n-dimensional
Ornstein–Uhlenbeck process which starting from x at time t is Gaussian with mean vector
ðxi e$li t Þipn and covariance matrix C ij ðtÞ ¼ aij ðz0 Þð1 $ e$ðli þlj Þt Þðli þ lj Þ$1 . These parameters are independent of n and the distribution coincides with the law of the first n
coordinates (with respect to !i ) of the process with resolvent Rl .
Now take f 2 C an and let hðxÞ ¼ Rl f ðxÞ ¼ Rnl f ðx1 ; . . . ; xn Þ. (Here we abuse our notation
slightly by having f also denote its dependence on the first n variables.) By Corollary 3.5
and (3.10), h 2 T. Moreover, L0 h ¼ Ln0 Rnl f ¼ lRnl f $ f ¼ lRl f $ f . The second
equality is standard since on functions in C 2b , Ln0 coincides with the generator of the
finite-dimensional diffusion. Now substitute this into (5.13) to derive (5.12). &
To iterate (5.12) we will need to extend it to f 2 C a by an approximation argument.
Recall b
li ¼ bi ðz0 Þli .
bp
Notation. Write f n $! f if ff n g converges to f pointwise and boundedly.
bp
Lemma 5.3. (a) If f 2 C a , then pRp f $! f as p ! 1 and
sup kpRp f kC a pkf kC a .
p40
(b) For p40 there is a c1 ðpÞ such that for any bounded measurable f : H ! R, Rp f 2 C a
and kpRp f kC a pc1 ðpÞkf kC b .
Proof. (a) Note if f 2 C a , then
Z 1
pe$pt kPt f kC b dtpkf kC b
kpRp f kC b p
0
and
pRp f ðxÞ $ f ðxÞ ¼
bp
Z
1
0
pe$pt ðPt f ðxÞ $ f ðxÞÞ dt ! 0
because Pt f ðxÞ $! f ðxÞ as t ! 0.
Let X t be the solution to (2.4) (so that X has resolvents ðRl Þ) and let X it ¼ hX t ; !i i!i .
Then X it satisfies
Z t
X it ¼ X i0 þ M it $ b
li
X is ds,
(5.14)
0
is a one-dimensional Brownian motion with CovðM it ; M is Þ ¼ aii ðs ^ tÞ. Let X xt i ;i
where
denote the solution to (5.14) when X i0 ¼ xi . Then
Z t
xi þhi ;i
xi ;i
b
Xt
$ X t ¼ h i $ li
ðX sxi þhi ;i $ X xs i ;i Þ ds,
M it
0
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and so
$b
li t
X txi þhi ;i $ X x;i
hi !i .
t ¼ e
Hence, if X xt is defined by hX xt ; !i i ¼ X xt i ;i ,
"
"
"X 2 $2bl t "1=2
xþh
x
i "
"
jX t $ X t j ¼ "
hi e
" pjhj.
Therefore
jPt f ðx þ hÞ $ Pt f ðxÞjpjf jC a EðjX xþh
$ X xt ja Þpjf jC a jhja ,
t
and so
jpRp f ðx þ hÞ $ pRp f ðxÞjp
Z
1
0
pe$pt jPt f ðx þ hÞ $ Pt f ðxÞj dtpjf jC a jhja ,
i.e., jpRp f jC a pjf jC a . This proves (a).
(b) As we mentioned above, for any bounded measurable f, kpRp f kC b pkf kC b . We also
have
Z 1
Ps pRp f $ pRp f ¼
pe$pt ½Psþt f $ Pt f + dt
0
Z s
Z 1
¼ ðeps $ 1Þ
pe$pt Pt f dt $ eps
pe$pt Pt f dt.
0
0
The right-hand side is bounded by
2ðeps $ 1Þkf kC b .
This in turn is bounded by c2 ðpÞsa=2 for 0psp1. Also,
kPs pRp f $ pRp f kC b p2kf kC b p2sa=2 kf kC b
for sX1.
li g
Hence kpRp f kSa pc3 ðpÞkf kC b . Our conclusion follows by (3.10), which holds for the fb
just as it did for fli g. &
bp
Lemma 5.4. Suppose f n $! 0 where supn kf n kC a o1. Then
bp
Dij Rl f n $! 0
and
bp
Di Rl f n $! 0 as n ! 1 for all i; j.
Proof. We focus on the second order derivatives as the proof for the first order derivatives
is simpler. We know from Corollary 3.3 that Dij Rl f n is uniformly bounded in C a norm, so
in particular, it is uniformly bounded in C b norm and we need only establish the pointwise
convergence. We have from (3.8) that
kDij Pt f n kC b pc1 kf n kC a ta=2$1 .
(5.15)
From Proposition 3.2, we have
Dij Pt f n ¼ Di Pt=2 DQt=2 !j Pt=2 f n .
(5.16)
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399
Fix t40 and w 2 H. The proof of Proposition 5.2 in [1] shows there exist random variables
Rðt; wÞ and Y t such that
Dw Pt f ðxÞ ¼ E½f ðQt x þ Y t ÞRðt; wÞ+;
f 2 C b ðHÞ,
and
E½Rðt; wÞ2 +p
jwj2
.
gt
Therefore
bp
hn ðj; t; xÞ & DQt=2 ej Pt=2 f n ðxÞ ¼ Eðf n ðQt=2 x þ Y t=2 ÞRðt=2; Qt=2 !j ÞÞ $! 0
by dominated convergence. Moreover Cauchy–Schwarz implies
khn ðj; tÞkC b pðgtÞ$1=2 sup kf m kC b .
m
Repeating the above reasoning and using (5.16) we have
bp
Dij Pt f n ðxÞ ¼ Di Pt=2 hn ðxÞ ¼ Eðhn ðQt=2 x þ Y t=2 ÞRðt=2; !i ÞÞ $! 0
and
kDij Pt f n kC b pðgtÞ$1 sup kf m kC b .
(5.17)
m
Fix e40. Write
"Z
"
jDij Rl f n ðxÞjp""
0
e
e
$lt
" "Z
" "
Dij Pt f n ðxÞ dt"" þ ""
1
e
e
$lt
"
"
Dij Pt f n ðxÞ dt"";
by dominated convergence and (5.17) the second term tends to 0, while (5.15) shows the
first term is bounded by
&
'
Z e
c2 kf n kC a ta=2$1 dtpc3 sup kf m kC a ea=2 .
0
m
Therefore
&
'
lim sup jDij Rl f n ðxÞjpc4 sup kf m kC a ea=2 .
n!1
m
Since e is arbitrary,
lim sup jDij Rl f n ðxÞj ¼ 0:
&
n!1
Proposition 5.5. Assume (5.4). If f 2 C a , then
Sl f ¼ Rl f ðy0 Þ þ S l BRl f .
bp
(5.18)
Proof. We know f p ¼ f $ pRp f $! 0 as p ! 1 by Lemma 5.3. This lemma also shows
kf p kC a p2kf kC a , and therefore we may use Lemma 5.4, the finiteness of Z, (5.4) (in fact a
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weaker condition suffices here), and dominated convergence to conclude
X
BRl f p ðxÞ ¼
ðaij ðxÞ $ aij ðz0 ÞÞDij ðRl f p ÞðxÞ
i;j
þ
X
li xi ðbi ðxÞ $ bi ðz0 ÞÞDi ðRl f p ÞðxÞ $! 0
X
li xi ðbi ðxÞ $ bi ðz0 ÞÞDi ðRl d n ÞðxÞ $! 0
i
bp
as p ! 1.
Here we also use the bounds kDij Rl f p kC b pckf kC a and kDi Rl f p kC b pcli kf kC a from
(3.11), (3.12) and Lemma 5.3(a). By using dominated convergence it is now easy to take
limits through the resolvents to see that to prove (5.18) it suffices to fix p40 and verify it
for f ¼ pRp h where h 2 C a . Fix such an h.
P
P
Let zn ðxÞ ¼ ni¼1 xi !i þ i4n ðz0 Þi !i ! x as n ! 1 and define hn ðxÞ ¼ hðzn ðxÞÞ. Then
bp
hn $! h since h 2 C a . Recall the definition of Rnp from the proof of Proposition 5.2; by the
argument there, we see that the function pRp hn ðxÞ ¼ pRnp hn ðx1 ; . . . ; xn Þ depends only on
ðx1 ; . . . ; xn Þ. By Lemma 5.3(b) pRp hn 2 C a and therefore is in C an . Proposition 5.2 shows
bp
that (5.18) is valid with f ¼ Rp hn . Now pRp hn $! pRp h as n ! 1 and supn kpRp hn kC a p
c1 ðpÞ by Lemma 5.3(b). Therefore, if d n ¼ pRp ðhn $ hÞ we may use Lemma 5.4, Corollary 3.5,
and dominated convergence, as before, to conclude
X
BRl d n ðxÞ ¼
ðaij ðxÞ $ aij ðz0 ÞÞDij ðRl d n ÞðxÞ
$1=2
i;j
þ
i
bp
as n ! 1.
We may now let n ! 1 in (5.18) with f ¼ pRp hn to derive (5.18) with f ¼ pRp h, as
required. &
Theorem 5.6. Assume (2.2), each aij and each bi is continuous, (4.7), (5.3), (5.4), and (5.5)
hold. There exists Z0 , depending only on ða; gÞ, such that if ZpZ0 , then for any y0 2 H there is
a unique solution to the martingale problem for L started at y0 .
Proof. Existence follows from Theorem 4.2.
Let P be any solution to the martingale problem and define Sl as above. Suppose
f 2 C a . Then by Proposition 5.5 we have
S l f ¼ Rl f ðy0 Þ þ S l BRl f .
Using Proposition 5.1 we can iterate the above and obtain
!
k
X
i
S l f ¼ Rl
ðBRl Þ f ðy0 Þ þ S l ðBRl Þkþ1 f .
i¼0
Provided Z0 ¼ Z0 ða; gÞ is small enough, our hypothesis that ZpZ0 and Proposition 5.1
imply that for l4l0 ða;P
g; ðaij Þ; ðbi ÞÞ, the operator BRl is bounded on C a with norm strictly
i
kþ1
1
less than 2. Therefore 1
f also converges to 0,
i¼kþ1 ðBRl Þ f converges to 0 and ðBRl Þ
a
both
in
C
norm,
as
k
!
1.
In
particular,
they
converge
to
0 in sup norm, so
P
i
kþ1
Rl ð 1
ðBR
Þ
Þf
ðy
Þ
and
S
ðBR
Þ
f
both
converge
to
0
as
k
!
1. It follows that
l
l
l
0
i¼kþ1
!
1
X
ðBRl Þi f ðy0 Þ.
S l f ¼ Rl
i¼0
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401
This is true for any solution to the martingale problem, so Sl is uniquely defined for large
enough l. Inverting the Laplace transform and using the continuity of t ! Ef ðX t Þ, we see
that for every f 2 C a , Ef ðX t Þ has the same value for every solution to the martingale
problem. It is not hard to see that T0 , C a is dense with respect to the topology of
bounded pointwise convergence in the set of all bounded functions. From here standard
arguments (cf. [3, Section VI.3]) allow us to conclude the uniqueness of the martingale
problem of L starting at y0 as long as we have ZpZ0 . &
Set
Qb;N ¼ fx 2 H : jxjb pNg.
Theorem 5.7. Assume ðbi Þ and ðaij Þ are as in Section 2, so that (2.2) holds. Assume also that
a; b 2 ð0; 1Þ satisfy:
(a) There exist p41 and c1 40 such that lj Xc1 j p .
P
$a=2
(b) ipj jaij jC a lj o1.
P $b
(c) j lj o1. (For example, this holds if b41=p.)
(d) For all N40, for all Z0 40, and for all x0 2 Qb;N there exists d40 such that if
jx $ x0 jod and x 2 Qb;N , then
X
jaij ðxÞ $ aij ðx0 ÞjoZ0 .
i;j
(e)
P
li jbi jC a o1.
1=2
i
Then for all y 2 H b there exists a unique solution to the martingale problem for L starting
at y.
Remark. By Theorem 4.2, any solution to the martingale problem for L starting at y 2 H
will immediately enter H b and remain there a.s. for any b 2 ð0; 1Þ. Hence the spaces H b are
natural state spaces for the martingale problem.
Proof. Fix b 2 ð0; 1Þ as in (c) and write QN for Qb;N . Let P be a solution to the martingale
problem for L. By Theorem 4.2 we only need consider uniqueness. If T N ¼
infft : X t eQN g, then by Theorem 4.2 we see that T N " 1, a.s. and it suffices to show
uniqueness for PðX (^T N 2 (Þ. (c) implies QN is compact and so as in the proof of Theorem
VI.4.2 of [3] it suffices to show:
(5.19) for all x0 2 QN there exist r40, aeij , and bei such that aij ¼ aeij and bi ¼ bei on QN \
L starting at y has a unique
fx 2 H : jx $ x0 jorg and the martingale problem for f
L is defined analogously to L but with aij and bi
solution for all y 2 QN . Here f
replaced by aeij and bei , respectively.
Fix x0 2 QN , Z0 as in Theorem 5.6. Choose d as in (d). We claim we can choose 1Xd1 40
depending on d and N such that if x 2 QN and kx $ x0 k1 od1 , then jx $ x0 jod. Here
jxj1 ¼ supi jhx; !i ij.
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To prove the claim, note that kx $ x0 k1 pd1 implies that for any K 0
X
X $b
X
2
2
ðxk $ xk0 Þ2 p
d21 ^ ð4N 2 l$b
lk .
k ÞpK 0 d1 þ 4N
k
k
k4K 0
pffiffiffiffiffiffiffiffi
So first choose K 0 such that the second term is less than d2 =2 and then set d1 ¼ d= 2K 0 .
$b=2
$b=2
Now let ½pj ; qj + ¼ ½xj0 $ d1 ; xj0 þ d1 + \ ½$Nlj ; Nlj + and note pj oqj as x0 2 QN . Let
cj : R ! R be defined by
8
x if pj pxpqj ;
>
<
cj ðxÞ ¼ pj if xopj ;
>
: q if x4q :
j
j
Define c : H ! QN \ fx 2 H : kx $ x0 k1 od1 g by
cðxÞ ¼
1
X
j¼1
cj ðhx; ej iÞej .
As kcj k21 pN 2 l$b
j , c is well defined by (c).
Take r ¼ d1 2 ð0; 1+ and set aeij ðxÞ ¼ aij ðcðxÞÞ. If jx $ x0 jor and x 2 QN , then
kx $ x0 k1 or and therefore cðxÞ ¼ x, which says that aeij ðxÞ ¼ aij ðxÞ for all i; j.
Define
8
if jujor;
>
<u
rðuÞ ¼ ð2r $ jujÞu=r if rpjujo2r;
>
:0
if 2rpjuj;
and set bei ðxÞ ¼ bi ðx0 þ rðx $ x0 ÞÞ. If jx $ x0 jor, then rðx $ x0 Þ ¼ x $ x0 and so
bei ðxÞ ¼ bi ðxÞ. Also bei is clearly continuous as (e) implies that bi is.
We now show that aeij satisfies the hypotheses of Theorem 5.6. For any x
X
X
je
aij ðxÞ $ aeij ðx0 Þj ¼
jaij ðcðxÞÞ $ aij ðx0 Þj.
(5.20)
i;j
i;j
Since kcðxÞ $ x0 k1 pr and cðxÞ 2 QN , it follows that jcðxÞ $ x0 jod. (d) now implies that
the right-hand side of (5.20) is less than Z0 . It remains only to check (5.3) for aeij . But
jcj ðxÞ $ cj ðx þ hj Þjpjhj j,
and so
jcðxÞ $ cðx þ hÞjpjhj.
Therefore
je
aij ðx þ hÞ $ aeij ðxÞj ¼ jaij ðcðx þ hÞÞ $ aij ðcðxÞÞj
pjaij jC a jcðx þ hÞ $ cðxÞja
pjaij jC a jhja ,
and so
je
aij jC a pjaij jC a .
Hence aeij satisfies (5.3) because aij does.
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403
If we set Bi ðxÞ ¼ xi ðbei ðxÞ $ bei ðx0 ÞÞ, it is easy to check that Bi ðxÞ is 0 for jx $ x0 jX2r,
kBi k1 pc2 jbei jC a pc2 jbi jC a , and jBi jC a pc2 jbei jC a pc2 jbi jC a , where c1 may depend on x0 .
Therefore (e) implies ðbei Þ satisfies (5.4) and (5.5).
We see then that Theorem 5.6 applies to aeij and bei and so (5.19) holds. &
Example 5.8. We discuss a class of examples where the bi ¼ 1 and the aij are zero unless i
and j are sufficiently close together. Let M 2 N, a 2 ð0; 1Þ and SM ði; jÞ be the subspace of H
generated by f!k : jk $ ij _ jk $ jjpMg. Also let PSM ði;jÞ be the projection operator onto
S M ði; jÞ. Assume that aji ðxÞ ¼ aij ðxÞ ¼ h!i ; aðxÞ!j i satisfies (2.2) and depends only on
coordinates corresponding to S M ði; jÞ, that is,
aij ðxÞ ¼ aij ðPSM ði;jÞ xÞ for all x 2 H; i; j 2 N.
(5.21)
In particular, (5.21) implies aij is constant if ji $ jj42M. Also suppose that
sup jaij jC a ¼ c1 o1.
(5.22)
i;j
Set bi ðxÞ ¼ 1 for all i, x and also assume
lj Xc2 j p
for all j for some p41,
(5.23)
and b 2 ð0; 1Þ satisfies
1
X
$ba
lj 2
þd
o1
for some d40.
(5.24)
j¼1
For example, (5.24) will hold if p42 and ba42=p. We then claim that the hypotheses of
Theorem P
5.7 hold and so P
there is a unique solution to the martingale problem for
Lf ðxÞ ¼ i;j aij ðxÞDij f ðxÞ $ i li xi Di f ðxÞ, starting at any y 2 H b .
We must check conditions (b)–(d) of Theorem 5.7. Note first that
jaij ðx þ hÞ $ aij ðxÞjp1ðji$jjp2MÞ jaij jC a jhja ,
so that jaij jC a p1ðji$jjp2MÞ c3 and hence by (5.24),
X
ipj
$a=2
jaij jC a lj
pð2M þ 1Þc5
X
$a=2
lj
o1.
j
This proves (b), and (c) is immediate from (5.24). If N40, x; x0 2 Qb;N , then for small
enough e40,
X
i;j
jaij ðxÞ $ aij ðx0 Þj
p2
X
ipj
jaij jC a
"
X
k
2
1ðjk$ij_jk$jjpMÞ ðxðkÞ $ x0 ðkÞÞ
#a=2
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p2jx $ x0 j
e
"
X iþ2M
X X
i
j¼1
pjx $ x0 je c4 ðMÞ
pc5 ðMÞjx $ x0 je
k
1
X
k¼1
#a=2
jxðkÞ $ x0 ðkÞja$e
k¼1
1
X
1ðjk$ijpMÞ jxðkÞ $ x0 ðkÞj
2$ð2e=aÞ
$b
ð2NÞa$e lk2
ða$eÞ
pcðM; NÞjx $ x0 je .
We have used (5.22), x; x0 2 Qb;N and (5.24) in the above. This proves (d), as required.
Example 5.9. We give a more specific realization of the previous example. Continue to
assume bi ¼ 1 for all i, (5.23), and (5.24). Let L; NX1 (we can take N ¼ 1, for example)
and for kX1 let I k ¼ fðk $ 1ÞN þ 1; . . . ; kNg. For each k assume aðkÞ : R2LþN ! Sþ
N , the
space of symmetric positive definite N - N matrices. Assume for all k, for all x 2 R2LþN ,
and for all z 2 RN ,
N X
N
X
i¼1 j¼1
2 $1
2
aðkÞ
ij ðxÞzi zj 2 ½gjzj ; g jzj +
(5.25)
and
a
sup max jaðkÞ
ij jC o1.
k
(5.26)
1pi;jpN
Now for x 2 H, let pk x ¼ ðhx; !ðð‘þk$1ÞN$LÞ_1 iÞ‘¼1;...;2LþN 2 R2LþN and define a : H !
LðH; HÞ by
haðxÞ!i ; !j i ¼ aij ðxÞ ¼ aji ðxÞ
8
< aðkÞ
i$ðk$1ÞN;j$ðk$1ÞN ðpk xÞ if i; j 2 I k ; kX1;
¼
S
:0
if ði; jÞe 1
k¼1 I k - I k :
Then for all x; z 2 H,
1 X
XX
X
aij ðxÞzi zj ¼
aij ðxÞzi zj
i
j
k¼1 i;j2I k
¼
1 X
N
X
k¼1 i;j¼1
aðkÞ
ij ðpk xÞzðk$1ÞNþi zðk$1ÞNþj
2 ½gjzj2 ; g$1 jzj2 +
by (5.25), and so (2.2) holds. Note that if i; j 2 I k , then (using the notation of Example 5.8)
SLþN ði; jÞ . fðk $ 1ÞN $ L þ 1; . . . ; kN þ Lg, and so (5.21) with M ¼ L þ N is immediate
from the above definitions. Also (5.22) is implied by (5.26). The conditions of Example 5.8
therefore hold and so weak existence and uniqueness of solutions hold for the martingale
problem for L with initial conditions in H b .
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Remark 5.10. The above examples demonstrate the novel features of our results. The
fact that our perturbation need not be nonnegative facilitates the localization argument
$a=2
(see Remark 9 in [23] for comparison) and the presence of flj g in condition (b) of
Theorem 5.7 means that the perturbation need not be Hölder in the trace class norm. The
latter allows for the possibility of locally dependent Hölder coefficients with just bounded
Hölder norms, something that seems not to be possible using other results in the literature.
On the other hand [23] includes an SPDE example which our approach cannot handle in
general unless, for example, the orthonormal basis in the equation diagonalizes the second
derivative operator. This is because he has decoupled the conditions on the drift operator
and noise term, while ours are interconnected. The latter leads to the double summation in
conditions (b) and (d) of Theorem 5.7, as opposed to the trace class conditions in [23]. All
of these approaches seem to still be a long way from resolving the weak uniqueness
problem for the one-dimensional SPDE described in the introduction which leads to much
larger perturbations.
Acknowledgement
We would like to thank L. Zambotti for helpful conversations on the subject of this
paper.
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