- Email: [email protected]
Markov chain approximations to symmetric diffusions
Ann.
Inst.
Henri
Poincarb, ProbabilitPs
Vol. 33, no 5, 1997, p. 619-649
er Statistiques
Markov chain approximatims to symmetric diffusions Daniel
W. STRCKXK
M.I.T., rm. 2-272, Cambridge, MA 02139-4307, USA E-mail: [email protected] and Weian ZHENG Dept. of Math., UC Irvine, Irvine, CA 92174001, E-mail: [email protected]
USA
ABSTRACT. - This paper contains an attempt to carry out for diffusions corresponding to divergence form operators the sort of approximation via Markov chains which is familiar in the non-divergence form context. At the heart of the our procedure are an extension to the discrete setting of the famous De Giorgi-Moser-Nash theory.
RI%J&. - Cet article contient une tentative pour etablir dans le cas ou le gknerateur don& sous forme de divergence, les approximations par des chaines de Markov qui sont bien connues pour des diffusions dont le gCn&ateur est dorm6 sous une forme de nondivergence. L’essentiel de notre mkhode repose sur une extension de la celbbre th&rie de De Giorgi, Moser. et Nash au cas discret.
The first author acknowledges partial support from Ann&s
de I’lnstituf
NSF
grant
9302709-DMS.
Henri Poincar~ - Probabilit& et Statistiques Vol. 33/97105/$ 7.00/O Gauthier-Villars
- 0246-0203
620
D. W. STROOCK
AND
W. ZHENG
0. INTRODUCTION
Given a continuous, symmetric matrix a : Rd + W’ @ Rd satisfying XI < a(x) 5 X-r1 for some X E (0, l] and a bounded continuous b : Rd --+ Rd, the martingale problem for the operator
(04pb=f:
on C,-(Wd; R)
i,j=l
is well-posed (cf. Theorem 7.2.1 in [l 11). As a consequence, one can use any one of a large variety of procedures to approximate the associated diffusion by Markov chains (cf. Section 11.2 in [l 11). Indeed, aside from checking a uniform infinitesimality condition, all that one has to do is make sure that the action on f E C,-(I@; R) of the approximating generators is tending, in a reasonable way, to ,Pbf. If, on the other hand, one replaces the Pb in (0.1) by the divergence form operator
(04
L” = v . (av)
= 2
-&
&+(x)&.
i=l
then, unless a is sufficiently smooth to allow one to re-write L” as fYb with d &i,j
hi(x) = c -(x), j=l d?i then the theory alluded to above does not apply. Nonetheless, the magnificent analytic theory of De Giorgi, Nash, Moser, and Aronson (cf. [9]) tells us that the operator L” not only determines a diffusion, it does so via transition probability functions which are, in general, better than those determined by C”lb. Thus, one should hope that probability theory should be able to provide a scheme for approximating these diffusions via Markov chains, and in the present article we attempt to do just that. In order to understand the issues involved, it may be helpful to keep in mind the essential difference between the theories for the non-divergence form operators C”ab and the divergence form operators L”. Namely, all the operators Pb share CcW(lRd; R) as a large subset of their domain. By contrast, when a is not smooth, finding non-trivial functions in the domain of L” becomes a highly delicate matter. In particular, as a changes by even a little bit, the domains of the corresponding La’s may undergo radical change. For this reason, one has to approach L” via the quadratic form Annales
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which it determines in L2(Rd;R). can interpret L” f as a distribution finds that
TO SYMMETRIC
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621
Namely, if f E C,-(Rd;W), then one (in the sense of L. Schwartz) and one
-($I, W) = W,S) z (RI, w),, 9 E TV@: w. (0.3) where we have introduced the notation ( . , . )a to denote the inner product on the usual Lebesgue space L2(Wd; R). For historical reasons, the quadratic form E” is called a Dirichlet form. Clearly, for any a satisfying our hypotheses, E” is well-defined on C,W(RBd;W). In fact, one finds that each E” 1 C,-(I@; R) admits a closure, which we continue to denote by E”, and that Dam(P) coincides with the Sobolev space IV; (Rd; R) of f E L2 (I@; W) having one (distributional) derivative in L2( Rd; R). Furthermore (cf. Theorem 11.3.1 in [9]), there is a unique, strongly Feller continuous, Markov semigroup {Pp : t > 0} with the property that, for each f E C,“(Rd; R), t * PFf is the unique continuous map t E [0, x) I-+ ut E W,‘(Wd; W) such that (Ut,g)O
(0.4)
=
(f,do
-
p(w) n
dT-
t E [0, co) and g ;“CT(Rd; W). On the basis of the characterization in (0.4), one might think that convergence of the semigroups should follow from convergence of the associated Dirichlet forms. That is, one might hope that P,“” f --+ Ppf for all f E Cc(Rd; W) should follow from Ean (f, J) E”(f, f) for all f E Cc” ( Rd; 53). However, this is not the case, not even when d = 1. To wit, given a measurable a : R --+ [l, 31, set
J
1 1 d[ and & = a 0 (tP)-1. 0 4l) At least when a is differentiable, it is easy to check that UpI)
=
x
L”(f
0 u”) = (cq)
0 ua,
and therefore, if, for X > 0, R; is the resolvent operator given by JFe- xtP; dt , then an elementary‘application of the Feynman-Kac formula leads to (0.5)
[R;f]
Vol. 33, no 5-1997.
(CC)= 2Ep”-“(z)
cx2a 0 (If-‘(w(t))
x exp (-2X
/f a o (u”)-’ .o
x f o (Q-l
(w(t))
1
dt ,
(W(T)) d7-)
622
D. W. STROOCK
AND
W. ZHENG
where hcLydenotes the standard Wiener measure for l-dimensional Brownian motion starting at 1~.In fact, (0.5) hoIds whether or not a is differentiable. Now suppose that {un}y is a sequence of [l ,3]-valued measurable functions. Then &“- (f, f) &“(f, f) for all f E C,-(W; R) is equivalent to saying that a, a weakly in the sense that
JR
anOP(I) dx -
J R
a(()~(<) dx
for all cp E L1(R; R).
On the other hand, (0.5) shows that it is weak convergence of the reciprocals 2 to k which determines whether the associated semigroups converge. Indeed, it is clear that 5 t implies first that uan --+ u and therefore that (zL”~)-~ @“)-I uniformly on compacts. In particular, since (?P)-l(y) = Joya, o (rP-l(~) dq, this means that ca, o (@)-l ao (u”)-l weakly. Hence, after representing ji uo(rP-’ (W(T)) dr in terms of the local time of Brownian motion, one sees from (0.5) that R;1- f RT f follows from -& weak2. Finally, take u,(x) = 2 + sin(2nrrx), and observe that a, 2 weakly while
-+ J3 1 a,
weak
-;
1
2 + sin(2sl)
dt
=
J3 1 0
1 - 4-l sin(27rt)2 d(>
;.
As the preceding discussion indicates, even in the context of diffusions, identification of limits via Dirichlet forms is a delicate matter. Thus, it should come as no surprise that the difficulties become only worse when one is trying to approximate diffusions by Markov chains. In order to handle these problems, we devote the next section to a derivation of a number of a priori estimates. These estimates are extensions to the discrete setting of estimates which are familiar in the diffusion setting, and our proof of them mimics that of their diffusion analogs. In $2 we use the results in $1 to find general criteria which tell us when our Markov chains are canverging to a diffusion, and in $3 we construct Markov chain approximations to the diffusion corresponding to a’s satisfying (O.l), first in the case when a is continuous (cf. Theorem 3.9) and then in the general (cf. Theorem 3.14). Finally, in $4 we describe a possible application.
1. THE BASIC
ESTIMATES
Our purpose in this section is to derive the a priori estimates (especiahy (1.11) and (1.32)) on which our whole program rests. The general setting in Ann&s
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we will be working is described as follows. We are given a function p : Ed x Zd --+ [O,ce) satisfying’
which
p(k,e) Cl.11
>_ 1 for all (k,e) E Zd x Zd with llej\ = 1 and yks Idk.ell < 30. d
For (Y E (O,m), we set aZd = {ak D’(cIZ~;R) of f : aZd --+ W with
: k E Z”}, introduce the spaces
and define the symmetric quadratic forms ( . , . )oi and &“J’ on L2 (aZd; R) by kQ*
(1.2)
P@(f,g)
= ad-’
c
c
kEZ*
e#
x (f(4k
p(k,e)
+ 4)
-
fbW(g(4k
f
4)
-
W41.
By the elementary theory of Markov chains, it is an easy matter to check that, for each a E (0, cc), there is a unique symmetric, Markov semigroup {P,“@ : t > 0) on L2(dd; Et) with the property that, for each f E L2(aZd;W), t E (0,cc) ++ P,*l”f E L2(dd;R) is the unique differentiable t E (0, cc) ut E L2 (aPd; W) satisfying lim+,u uf = f pointwise, and
(1.3)
$ (9, Ut), = -qg,
Q)
for all g E L2(oZd; R). Moreover, if p”@ : (0, x) x aZd x aZd ---+ [0, oc) is determined by
(1.4
.[P,“@f] (cxk) = ~8 c
f(al)pa+(t.
ak, al),
1EZd
then (1.5)
p”rP(t, ok, al) = p-‘(t,
’ for v E Wd, we use llvll the Euclidean length of v. Vol. 33. no 5-1997
01, ok)
and
@dp”fl+‘(P2t, apk, ~$1) = pa+(t, ak, ~1). E maxl
IVil,
as distinguished
from
1~1, which
reserved
for
624
D. W. STROOCK
AND
W. ZHENG
The symmetry assertion in (1 S) comes from the symmetry of the operator P,“>” in L2(oZd; W). To see the second assertion, for f E L” (aZd; R), define fp E L” (a/3Zd; R) so that f3(ct/3k) = f(ak). Next, take u&k) Then ut -+
= [P;$‘fo]
f pointwise as t \
(@Jk).
0 and
Hence, ut = P,“@f, and so we are done. Note that, as a consequence of the symmetry and Jensen’s inequality, we know that Ilptq,,,
(14
for all t E (0, co) and p E [l, co].
5 Ilfllw
We turn next to the derivation of pointwise estimates on @tP(t,x, y), and for this purpose we will need the following Nash inequality (cf. [l]). 1.7 LEMMA. the property
- There is a constant C E [l, oc), depending only on d, with that
(1.8)
IlfllT;” 5 C@If, fNll?J>
f E qc
q.
Proof. - We begin by noticing that, since ]]f]]l,2 5 ]]f]]r,r, will assume that 0 < El+(f,f) < ]]f]]~,,. Next, set
f(C)= c f(k) exp(fl2n(&k)w),
we may and
< E P,lld.
kEZd
By Parseval’s identity, for each T E (0, l]:
with f’(r) being the set of t E [0, lid with the property that, for each 1 I i 5 d, either & 5 f or & 2 1 - f. Clearly, I(r) < rd]]f]]r,r. At the same time,
rl+(f,f) 2 c ‘&(f(k + ei>- f(k)J2 (1
-
~0~(27r&))lj(t)/~
Annales de l’lnstitut
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P&car6
dt
L
~r~J(r),
Probabilitts
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APPROXIMATIONS
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625
where
Hence, we know that
Ml;,2 I 4lfll2,I In particular, can take
+
~lwJ) yr2
since we are assuming that 0 < El@(f, f) EYf,
r = and thereby obtain our result. 1.9
r E (o q > .
’
5 I]f]&,
we
f)
Ilfll~,I 0
- Assume that,
LEMMA.
(1.10) Then there exists a C E [l, o), depending only on d, such that (1.11)
p”@(i, x, y) 5 CeK(P)V, (t”) --I exp (-g,
where Proof. - Observe that aVr
(4
(1.12)
2
d
5 Va(r) 5 (a! V r)“.
Thus, we need only prove (1.11) with V, (t a ) replaced by CIVt 3. In addition, by the second part of (1.5), it suffices for us to handle the case when CI:= 1. Because of (1.8), we can apply Theorem 3.25 in [l] to see that there is a C, depending only on d, such that pll~(t,X,Y)
5 ct-~exp(-13(~;x.y))),
where D(t; x, y) 2 sup (.(&Y - x&~ i tI’(e)2 r(G2
=
SUP{~
,@,e)(exp(l(Z,e)Rd/ e#
Vol. 33. Ilo s-1997.
: E E APdj with -
1)”
: k E Zd>.
626
D. W. STROOCK
AND
W. ZHENG
Thus, when t > 1, we get (1.11) by considering c = t-;
y-x IY - Xl
andusing (elql-1)2 5 Q’ e ‘1111 . When t E (0, I], we have to argue somewhat differently. Namely, by (3.19) in [l], we know that, for any < E Rd, 1Jq5-cpt’“($tf)
111,~L etr(t)Z[Iflll,2
where &(z)
E e@)@.
Hence, since (]g](Loo I ll9l11,2 5 119ll1,1~
and so @Y~,x,Y)
L exd-(t,Y
- x)Rd + tr(t)*).
Thus, we can now complete the proof by taking 6 = e. 1.13 LEMMA. Cd (1.10)) (1.14)
0
- There is a C E [l, cm), depending only on d, such that
ad c (Jai - akl12p”@(t, ak, al) 5 CeKcP)t. IEP
Proof. - Again, by (1 S), we need only look at Q = 1. When t 2 1, (1.14) with o = 1 is an obvious consequence of (1.11) with Q = 1. When t E (0, 11, we use the description of the right continuous Markov process X(t) corresponding to { @’ : t > 0) starting at k. Namely, because this is a pure jump process, we know that
IIW - w9ll 5 c
lIW4 - X(4.
sE(%tl Thus, if r, denotes the time of the nth jump and A, X(~n-)lJlp,t~(~n)~ hen
~[IFV) - WN”]
5 e
= I]X(r,)
-
$LrA]
m,n=l
= 557
Ann&s
c
+2
m=l
E[A,A,].
llm
Henri
Poincare! - ProbabiMs
et Statistiques
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627
DIFFUSIONS
But, by the standard theory of jump Markov processes (cf. Chapter 3 of [SD, the number of jumps is a Poisson-like process with rate dominated by SUP c kad
dk>e)~
eEEd
form which it is easy to estimate the preceding by, respectively,
Hence, because t 2 1, we are done. 0 From (1.14) it is immediate that there is an R, depending only on d and K(p), with the property that ad
p”+‘(t, cuk, al) 2 f
c
for each k E Zd.
([al-ak[l
In particular, by the Chapman-Kolmogorov Schwarz’s inequality: p*@(2t, ak, “k) = ad c p”+(t, Ia* ad
c
equation,
symmetry,
and
ak, al)2 @=+(t, ak, rwl)*
I/al-crk~~~Rd
2 Va(Rt+)-‘(
c
p”@(t. ok, ~1))~
IIal-akll
Thus, there is an E E (O,l], depending only on d and K(p), (cf. (1.12)) p-‘(t,
such that
elk, ak) 2 --?-
V#).
Our next goal is to show that the on diagonal estimate in (1.15) extends to a neighborhood of the diagonal. Namely, we want to show that there is an t > 0, depending only on d and K(p), such that
(1.16)
Vol. 33. iI0 s-1997.
pQ@(t, ak, al) 2 --?--vpq for cu E (0, l] and (t, k, 1) E (0, a) x Zd x Zd with J]ok - nl]] 5 ati.
628
D. W. STROOCK
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Our derivation is based on the ideas of Nash, as interpreted in [4]. We begin with a number of observations. 1.17 LEMMA. - (1.16) will follow from the existence of an e, depending only on d and K(p), such that (1.18)
p”+‘(l,O,
al) 2 t for
a E (0,2] and I(crl(l 5 2.
Proof. - In the first place, notice that our hypotheses are translation invariant. Hence, there is no loss in generality when we take k = 0. Further, if we know (1.18) and I(al(( 5 2t3, then either 1 = 0 and (1.15) applies, or cu 5 2t 4. In the latter case, take /3 = t- 3, note that (up 5 2, and apply the second part of (1.5) and (1.12) to see that (1.16) follows from (1.18) with Q replaced by a/3. q The proof of (1.18) depends on the following Poincard inequality. 1.19 LEMMA. - Let U E C” (R; [0, w)) have the properties that J e-2U(c) dt = 1 and U(l) = Ielf or a I1 sufJi ciently large 111.For Q: E (0,2], define d
g=(l) = cx-d
Ju i=l
41%+1)
e -2u(Ft) ,-&CC, 1 E jpf.
d
Then there exists a X > 0 such that (1.34
x((f - (&J2).!-tcl
bad
i=l
where
and ei is the element of Zd whose jth component is 1 if i = j and 0 otherwise. Proof. - Because of the product structure, we may and will restrict our attention to the case when d = 1. We begin by showing that p R(f(E) - (f))2 eezu(‘) dt I l f’(E)2e-2U(f) de, J where (f) s f([)e-2U(E) d<, .I for some p > 0. To this end, note that (1.21) is equivalent to (1.21)
Annales de l’lnstirut
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where V z (V’)” - U”. Thus, if H 3 - -$ +V, then (1.21) is equivalent to (1.22)
w=(H)
C (0) U bL, co>
But, because V - 1 has compact support, H is a compact perturbation of -$ + 1, and therefore the essential spectrum of H coincides with [ 1, cc), the essential spectrum of - $ + 1. On the other hand, emu is an eigenfunction of H with eigenvalue 0. Hence, 0 must be an isolated eigenvalue of H, and so (1.22) will follow once we show that it is a simple eigenvalue. However, because it lies at the bottom of spec( H), a familiar argument, based on the variational characterization of the bottom of the spectrum, shows that 0 is indeed simple. Given (1.21), we proceed as follows. For f on &Z, extend f to R by linear interpolation. Thus,
Then (remember that d = 1) Q-l C(f(c@ ea
II-
P
2 .I’l 2 $
+ 1)) - f(cke))2g,(e)
Vol. 33, no S-1997.
f’(<)2e-2U(c) R
R2(f(v) c k,eEZ
Hence.
= /
(fG+
- f([))2eC2”(E)-2u(q) - f(4)2&)sa(~)
ci?
d<
630
D. W. STROOCK
The proof of the following
AND
W. ZHENG
lemma depends on the inequality
(1.23) for all positive numbers a, , b, c and d. To verify this, first note that, by homogeneity, it suffices to treat the case when a = 1 = c A d. In addition, by using the transformation 6 +Y i, one can reduce to the case when c 5 d. In other words, (1.23) comes down to checking that (10gx)~ 5 2(,
- $1
-x)
+ (d - 1)2
for all x E (0, oo) and d 1 1. Finally, observe that, by Schwarz’s inequality, (lo&r)2
5 @ - l)“) 2
and so it suffices to see that @ ,l)’
5 2 % - 1 (1 - x) + (d - 1)2. ( >
When 5 E (0, I], this is trivial. When II: > 1,
1.24 LEMMA. - There is an E E (0, I), depending only on d and K(p), such that
(1.25) 2 i log 6 for all LL E (0,2] and JlcukJJ5 2.
Proof. - Set ~~(1) = p”+(t,ak,al)
and
G(t) = ad c
log(ut(l))ga(l),
1EZd
and note that, by Jensen’s inequality, G’(t) = ($$),. Ann&s
G(t) 5 0. Moreover, by (1.2), = -E”.+~,~).
de I’lnsritur
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631
Hence, G’(t) = -ad-2
l&p i=l _
cid-2
c
c
1EZd eEP
ba(’
2g,(l
+
‘1
+e>
-
g~(‘)j2,q
A h(1)
e)
’
’
where, in the passage from the first Iine, we have used (1.23). Because U’ is bounded and Q! E (0,2], we can find y E (0: l] such that
minsa(1+ e”) L -m(l) l
5 K(p). 7
Hence, there is au A4 E [l, CG), depending only on d and K(p), that (cf. (1.20))
such
G’(t 1 &ad
~(hWt(l)
- G(t))2g,(l)
- M.
1EP
Next, for c > 0. set
A,(g)= (1Eifd: ~(1)> e-q}. Then, for each D > 0,
2 adC(- log- W - G(tH2gdl)
2adc (g+G(t))2ga(l) LG(t)‘; c go(l) - CT~. l-L(~) 1EAt (c) Vol. 33, R0 5.1997.
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D. W. STROOCK
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Thus, we now know that G'(t)
2 A'At(a)'ga 2M
G(t)2-(M+g2),
where IAt(o)jsa
z ad c gol(l). 1~4 (~1
Finally, we use (1.14) to find an r E [l, co) such that ad c
py
t,ak,al)
2 i
for all CLE (0,2], t E (O,l],
and (Jak(( 5 2.
IlWl~ In particular, if p is the smallest value e-2U takes on [-T, r], then
c ~(1)5 eeurd
+ ““:r’y
(At(g)lg,.
Il4l
Thus, by taking c = log4rd and using (1.1 l), we conclude first that and then that there exists a S E (0, l), depending 14(“)(ga 2 ‘$;:1p) only on d and K(p), such that G’(t) 2 ~5G(t)~ - 6-l
for all 01 E (0,2],
[ 1
(1.26)
t E i, i
, and ]Jak]J 5 2.
To complete the proof starting from (1.26), suppose that G(i)
5 - 5.
By (1.26), -G(t)
2 -(26)-l,
t E
[ 1 ;,;
.
and so, again by (1.26), G’(t) 1 ;G(Q2,
tE
1 1 4’2.
[ 1
But this means that
and therefore $ exp(-86-l).
G( f ) > -8S- i. In other words, we can take ~4 = •i Annales
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The passage from (1.25) to (1.18), and therefore (1.16), is easy. Namely, by the Chapman-Kolmogorov equation and symmetry,
Hence, by Jensen’s inequality, if 1)al/j _< 2, then (1.25) gives (1.18). In the following statement, { P,“l;” : k E Z”> denotes the Markov family of measures on R, z D ([0, 00); c&) for which transition probability function has p”@(t! Qk, al) as its density. Also, for each T > 0 and k E Z”, <;/ : % ---+ [0, co] is given by &(w)
= inf{t
> 0 : (/w(t) - akj( > r>.
Finally, for l’ C_ Rd, \I’\, denotes ad times the number of k E Zd with ak E I. 1.27 LEMMA. such that
- There is a 6’ E (0, i)> depending only on d and K(p)S
(1.28) for all Q! E (O,l], r E (0, w), t E (0, (2@r)2] : (k,l) E Zd x Zd with Jjok ;(~lj/ 5 t;.
[-ti,t* 1r-lcad.
andIF (g crk+
Proof. - First suppose that tt < (Y. Then 1 = k and either I is empty, and there is nothing to do, or Jl& = Va(ti). In the latter case, PCyi”(w(t) E I? & C,&(W) > t) 2 PK;p(w(sj = k for s E [O,t])
2 exp (-K(p)$)
2 exP (-K(P)).
Thus, we need only worry about ti > a. But then, for (lak - ajll 5 ti: Cy-dP~iP(U(t) = aj & t) = p”+(t, E 2
l/,0
al, aj) - EpZi” WYt -Cc”(P)
sup o
,$-if
- Ci$(w)T w(C&), exp
- Cf~~(~)t-$ sup fs’exp
2--f,
K@j Vol. 33, Ilo 5.1997.
U>l
(-(I
(-‘l-$H’“).
,I”“‘)
olj), G&(W) < tl
634
D. W. STROOCK
AND
W. ZHENG
Hence (cf. (1.12)) we can choose 6’ E (0, i) to achieve P,“i”(w(t)
= aj & c&(w)
> t) 1%; V&z)
and (1.28) follows immediately after summing over oj E r. Cl Our main interest in (1.28) is that it allows us to derive a Nash continuity estimate (cf. Theorem 1.31 below). However, in order to do so, we will have to replace (1.10) by the much more stringent condition: (1.29)
and
1141 > R(P) * p(k e) = 0 c p(k,e) < R(p) for all k E Zd, eEZd
where R(p) E [l,co). In the following lemma, Q”((T,
k); T->E [T - ?, T] x (all : (lo4 - ok/( 5 r} for T > 0 and (T, ok) E [r2, co) x aZd.
Also, given a function u on a set S, define Osc(u; S) = sup { lu(y) - U(Z)I : 5, y E s}. 1.30
LEMMA.
-
Set(cf.
Lemma
1.27 and (1.29)) v = (1 + R(p))-‘0
determine ~7E (0, cm) by q* = (1 - 2-d-38). bounded functions f on aZd, 0+-f?
Q”((T, k);
and
Then, for al/ Q E (0, l] and
QQ((T k);
v-)) I fO+‘T
r) >,
r > 0 and (T, k) E [r2, co) x c&,
Proof. - Without loss in generality, we will assume that k = 0. Moreover, because there is nothing to do when qr < o, we will assume that 77~_> CL For ,0 E [O,Ti], set M(P) = max{UQ(s, 4) : (s,d) and m(P) = min{zP(s,oj) : (s,oj) Then Osc(~~;Q”((2;,0);~))
= M(p)
rI=y= 0 Annales
E Q”((T, 0); P) > f Q-((T,O);p)}.
-m(P).
Next, set
r 1+ R(P) de i’lnstitut
Henri
Poincart!
- Probabilith
et Sthtiques
MARKOV CHAIN APPROXlMATIONS TO SYMMETRIC DIFFUSIONS
635
and
Obviously, either lI’la >_ iVa(qr) or [I’(, 5 $Vcl(7p). If lIYln > iv,(y). take w = ua - m(r). Given T - (77~)~< s 5 T, set t = (27r)” - T + s E [(q~)~, (277r)“], and note that, because w 2 0 on Qa((T,O); 7.) and
Ppae - almost surely when ~~cE~~~ < ‘r/r, (1.28) impIies that
for lIdI\ 5 *or. Thus, m(q~) - m(r) means that WV-)
- h4
F M(r) I (1 -
1 2-d-“H(M(7.)
- m(r)),
which
- m(v) 2-d-30)
(M(r)
- m(7.)) = 7f(M(r)
- 7n(r)).
When IFI, 5 iV,(q~), one takes w = M(T) - u” and proceeds as above to get first that M(r) - M(qr) 5 (1 - 2-d-3q) (M(T) - m(r)) and then the same estimate as we just arrived at. q 1.31 THEOREM. -Assume that (1.29) hotdsfor some R(p) E [l, oo), and define u as in Lemma 1.30. Then there is a B E (0, CO), depending only on d and R(p), such that, fir all ar E (0, I] and f E L”(rxZ”: R), (1.32)
I[f’?‘f] (4 - [f’?‘f] (cik)I It - ~(3 v ((ak - crl(( (T (t/Is)3 > .
Vol. 33. no 5.1997
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D. W. STROOCK
In particular, (1.33)
AND
W. ZHENG
there is a C, depending only on d and R(p), such that
lpa+(t’, Qk’, al’) - jPp (t, ctk, nl) ( 5 C(lt’ - tl+ + J(ak - ak’l) + (1~1- al’(l)=+’
it)-*.
Proof - Clearly (1.33) follows from (I .32) together with (1.11) and p-‘(t,
ak, al) = p”@(t, ~1, ok) = [ Pcylppcu+ 6 (;> +)]Wb
To prove (1.32), set T = t A s and T = dm.
If
It - s/i v IId - akll 2 T, then (1.32) is easy. Otherwise, determine n 2 0 so that
- $I . rln+1 -< It - 4 v rlb1 - akll < Then, by repeated application of (1.30), one finds that
I [fY@fl (4 - [~sOI?f] w I 5 OS+“; Q” ((T, 0); $r) > 5 $gOsc(ua;
Q” ((T, 0); r))
5 3/n011fll a,c‘=5 ~vIfll%cc
2. PRELIMINARY Throughout this For each Q E (0, assumption is that there is a R E [l,
(
(t - sp v I(ak - cdl\ u 7 >.
APPLICATIONS
section we will be dealing with the following situation. 11, we are given pa : Zd x hd [0, oo). Our basic the pa’s satisfy (1.1). In addition, we will assume that oc) with the properties that Ilell > R * pa&e)
(2.1)
q
and
c p,(k,e) eEP
< R
= 0 for all k E Zd.
2.2 LEMMA. - Set (cj 1.4)) p” E p”@- , and,for x E Rd, use [x]~ to denote the element of aZd obtained by replacing each coordinate xi with CI times Anna/es de I’lnstitur
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the integer part of CX:-~Z~.Then, for any sequence {a,}? E (0, l] which decreases to 0 there is a decreasing subsequence { QI,J} and a continuous q : (0, W) x IWdx [Wd -+ [0, KI) such that (2.3) ,!@k t;;k,
1
xs,u,p,dt~eP~y-XtlP~’ (6 [XL,, , [&,) >
44
x, y)l = 0.
where ,6 E (0, XI) depends only on d and the number R in (2.1). In particular,
(2.4)
cds,x,Wt = 1, (s.4 E (‘44 x Rd, J’R* lim sup
S~OXERds It-xl>r
Q(S+ t,x,Y) =
JRd
q(s, x, t> dt = 0,
xEIWdandrE(07co),
q(s,x, t)dt, t, Y) de, s, t E (0, W) and x,y E [Wd.
Proof - Because of (1.11) and (1.32), there is essentially nothing to do
except comment that the existence of a locally uniform limit is guaranteed by (1.32) and the Arzela-Ascoli compactness criterion and that the other statements in (2.3) and (2.4) follows from this combined with (1.11). El From (2.4), it is clear that if Qt is determined by (2.5)
.
Qt./Xx)= Ad f(t)q(t,
x, 0 dt,
(6 x) E ((h ~1 x bfd and f E Cc (Rd; R) >
then { Qt : t > 0} determines both a Markov semigroup on Ct, (Rd; R) and a strongly continuous semigroup of self-adjoint contractions on L2 (BBd;R) . Our goal now is to give criteria which will enable us to identify the semigroup { Qt : t > O}. Notice that if { Qt : t > 0} is uniquely determined, in the sense that it is independent of the subsequence selected, then there will be no need to pass to a subsequence in the statement of (2.3). In order to carry out our program, we first impose the regularity condition that (cf. (2.1)), for each T E [l, oo),
(2.6)
SUP eEZd
ll+lLT
Second, we define a, : aPd -+ (2.7) Vol. 33, no 5.1997.
aa
I~,(l,e)
- Pa&e)]
Ill-klllR
Rd @ IWd by
= c pcy(k, e)e @ e, eEZ”
k E Zd.
= 0.
638
D. W. STROOCK
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W. ZHENG
Finally, we suppose that there exists a Bore1 measurable a : Rd --+ R” @ Rd with the property that, for each r’ > 0:
(2.8)
lirn ,/ ,,1,I4xl(I) - 4x)/ dx= 0. cu~o J x 1
As a consequence of (l.l), (2.1), and (2.8), it is an easy matter to check that, without loss in generality, we may assume that 21 5 u(x) < RI for all x E Rd. In particular, the general theory (cf. either 133 or [8]) of Dirichlet forms applies and says that there is a unique strongly continuous, Markov semigroup {p: : t > 0) self-adjoint contractions on L2(Rd; R) with the properties that, for each f E L2 (Rd; R), t E (0, co) -+ p,” f is the unique continuous function t E (0, oc) I--+ ut E W,l(Rd; R) (the Sobolev space of square integrable functions with one square integrable derivative) such that
(2.9)
$(p;ut)o=-/ (VywVut),,dx,cp=T’(Rd;$ andRd ut --+ f in L2(Rd; R).
In fact (cf. [9]) there is a unique pa E C((0, co) the properties that
x
Rd
x
Rd; (0, co)) with
(2.10) Pa(%x,Y)=Pa(S,Y,x),
J
p”(s,x,t)d< Rd
P%+t,x,Y)=
(SAY)
E w4x~dx~d~
= 1, (s,x) E (0,m)
p"(s,x,~)p"(t,~,y)d~, .IWd
s,t E b4,
x IWd,
X,Y E Rd,
and, for each f E C,(W”; W), (2.11)
p:f
= PFf
a.e.
where P,“!(x)
E
s
Rd UP%
x, Y) &.
What we want to show is that, after the addition of (2.6) and (2.81, we can show that the 4 in (2.3) must be pa, and for this purpose we will need a few R, let $;, denote the function preparations. Namely, given tiO : aZd on Rd obtained by baricentric extension. That is, for each k E Zd, set (2.12)
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and define Gcy on Qa (k) to be the multilinear extension of $J~ restricted to the extreme points of se(k). The crux of our argument is contained in the following lemma. 2.13
LEMMA.
-
For each Q! E (0, I], let $‘n : aZd --+ Iw be given, and
assume that (2.14)
sup ad-2 1
nE(O,ll
kEZd
c
(b (4k
+ e)) - 4, (ak)))*
< 3~.
Ile(/=l
Zf $a converges to $J in L2(Rd; R), then 4 f VVi(@; W) and
where we have taken (c$ (1.2)) E” = Err+a. Proof - We begin by observing that, for x E Qn(k), IIWCMII 5 (2-l max
{
)+$,a(al’) - $cy(*l)l
: al,&’
E QC2(k) with 111’- 111= l}.
In particular, this means that
and so, by (2.14), the L2-norm of 1V$, ) is bounded uniformly in QI E (0, 11. Hence, from the elementary functional analysis of the space Wi(Rd; W), we know that q!~E Wi (I@; R) and that 04, tends weakly in L2(Rd; Wd) to Vrj; and, after combining this observation with (2.8), we see that it suffices for us to check that
Moreover, because cp E CLY(Rd; W), it is clear from (2.14) that the preceding can be replaced by
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D. W. STROOCK
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where ca (k, e) - pm(k, e> (e, Vv4ak))
RdI
and we have introduced the notation &$J(x) s $J(X + v) - 4(x). To prove (2.16) we define a rectilinear, nearest-neighbor path from k to k + e. Namely, for (k,e) E Zd x Zd and 1 5 j < d, let
‘0 if ej = 0, Pj(k,e)
{e : l; = ki + ei for i < j, kj 5 l?j < kj + ej, and&=kifori>j} {l : li = k; + ei for i < j, kj - ej < aj < 5, and&=k;fori>j}
=
j=l
ew,
ifej>l ifej<--1.
(k,e)
where ‘h&‘(x)
3~((21,...,Zj-l,ICj+Q,2j+l,...,Zd))
-4(x);
and so ad-l
c
c
kE2d
=
ad-l
+
add1
c,(k,e)&,&(~k)
eE2d
C
C
lEZd
eELd
c,(l,
c
c
1~2~ eE2d
e) 2
ej&+&(4
j=l
2
sgn(ej)&,ghdl)
c
j=l
(dk
e)
-
41,
e)).
{k:lEPj(k,e)}
Thus, because of (2.6), (2.14), and the fact that cp E Cr(Wd; R), we see that (2.16) reduces to
Finally, to prove (2.17), observe that
g(x) sQa(k)
dx = (;)^-l
Annales
de I’lnstitut
c
h&~&l),
lEQ(kld Hew-i PoincarP
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641
where Q(k]j)
s {k : lj = kj and li E {AY;,Ic; + 1) for %# j}.
Hence, after elementary manipulation,
j=l
1EZd e# =
c kEZ”
c e6Zd
we see that
cake)/
(e,V&(x))Rddx Qm (k)
Thus, because of (2.6) all that remains is to note that
By combining Lemma 2.2 and 2.13, we get the our main result. 2.18 THEOREM. - Let a : Rd ---+ Wd@BBdbe a Bore1 measurable, symmetric matrix valued function which satisfies (2.19)
21 I a(x) 5 RI,
x E IWd:
for some R E [l,oo), and determine pa : (0,~) x R” x Wd ---+ (O?x) accordingly, as in (2.11). Also, for each cy E (0, 11, let pa : aZd x Zd --s [0, W) be a function which satisjies the conditions in (1.1) and (2.1). Zj in addition, (2.6) and (2.8) hold, then (2.20)
lim sup sup t4,.PIY-XIIpU(t,[xi,,[Y],) aLO tE(O,m) x,yERd
-f(t~~.Y)l
=O.
where p E (0, CXJ)depends only on d and R (c$ (2.1) and (2.6)). Proof. - In view of Lemma 2.2 and the discussion which follows its proof, we need only check that (cf. (2.3)) q = p”. Moreover, starting from the characterization of {p,” : t > 0} given in (2.9) and using the continuity Vol. 33, no 5.1997.
642
D. W. STROOCK
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of $3 one can apply standard soft arguments to see that it suffices for us to prove that, for each f E Cy(W”; R) and X E (0; oc),
and
when
cc %Lxz
J0 To this end, define ‘zb+ : aZd UA,a =
.I”0
e-‘“Qtf
dt.
W by e-XtPF’P”f
dt.
From the spectral theorem, it is clear that E”( %r%)
I f
llfll:,,.
Hence, because of (1. l), we know that (2.14) holds with $, = UX,~. Moreover, from (2.3), it is an easy step to UX,~ UA in L2(Rd; R). Thus, by Lemma 2.13, UA E W,‘(Rd;R) and
3. CONSTRUCTION
OF MARKOV
CHAIN
APPROXIMATIONS
Theorem 2.18 provides us with a criterion on which to base the construction of Markov chain approximation schemes. The only ingredient which is still missing is a procedure for going from a given coefficient matrix a to a family {pal : CKE (0, l]} which satisfies the hypotheses of that theorem. Thus, Iet a Bore1 measurable a : Wd Rd @ W” symmetric matrix-valued function satisfying (3-l)
MARKOV CHAIN APPROXIMATIONS TO SYMMETRIC DIFFUSIONS
643
be given. We want to construct an associated family (pa : Q E [0, 11) which satisfies (l.l), (2.1), (2.6), (2.8), and therefore, by Theorem 2.18, (2.20). An important role in our construction will be played by the following simple application of elementary linear algebra. Here, and elsewhere, elements of Wd will be thought of as column vectors. In particular, if el....,ed are given elements of I@, then [e’. . . . . e”] will be the d x d-matrix whose jth column is ej. 3.2 LEMMA. - Given a symmetric S E ‘&!’ @ Rd, there exist linearly independent el, . . , ed E Zd such that maxlljjd [l&l/ 5 32d2(IISII,, V 1) stands for the operator norm), and (II . II”,’
when E-’
= [el,...,ed].
Proof. - Set R = 32d2 ( ]]S]]Or, V 1). Choose an orthogonal matrix 0 so that A z OTSO is diagonal, determine ej E Zd so that, for 1 5 i < d, ei is the integer part [RO;,j] of RO;,j, and set B = R-l [el, . . , e”] . We must show that B is invertible and
For this purpose, set A = 0 - I?. Then ]]A]]op B = (1 - AOT)O. Thus, B is invertible and
L3-l = OT(I - AOT)-l = OT + H
< j$ 5
i
and
where ]]H]lop < g.
In particular, IIB-lS(B-l)T
- RI]
< 8d];f]]0p
OP -
Hence, on the one hand,
while, on the other hand,
and
I]B-l(B-‘)T
- 111 5 2. OP
R
644
D. W. STROOCK
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At this point it is useful to make a distinction between the cases when n is assumed to be continuous and when no continuity is assumed. The next statement is a more or less immediate consequence of Lemma 3.2 plus elementary point-set topology. Indeed, when a is continuous, Lemma 3.2 makes it clear that each x E Rd admits a neighborhood U and a choice of basis e’,... ,ed E Zd such that max15j5d ((ej]] 5 32d’M and E=
[el,...,ed]-'
j
cl j#i
(E(o(y)-2I)E’)
ci
/i
for all 1 5 i 5 d and y E U. Hence, the following the preceding and the fact that Rd is a-compact.
(E(dY)-qqi
statement comes from
3.5 LEMMA. - If a is continuous, then there is a countable, locally jinite, open cover {Un}r of Rd such that, for each n E N, a t U, is uniformly continuous and there exists a basis el,n, . . . , ed,n such that maxl<_j$j Ilejanll 5 32d2M and, for each 1 5 i 5 d and x E U,, (3.6)
~l(E,(a(x) j#i
- 21)E;),,jl
2 (E&(x)
- 21)EnT)i,i
C” (Rd; [0, 11) to be a partition of unity En(x) = E,-‘(a(x)
( 3.7)
and, for each n, define on : Rd x Zd -
1
- 21)E,T, [0, oo) by
when x E U,, i # j, and e = (ei+ + ej’“) when x E U,, i # j, and e = (ei)n - ej’“)
f~,(x,e) =
when x E U, and e = eiJ’ 0 when x +! U,. Finally, (3.8)
set p,(k,e)
= 111) (IHI) + c
nn(k,e). n
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3.9 THEOREM. - Zf a : Rd ---+ Rd x Rd is a continuous, symmetric matrix valued function which satisfies (3.1) and if {pu : CI E (0, 11) is determined by the prescription in (3.8), then Theorem 2.18 applies and so (2.20) holds. Proof. - There is nearly nothing to do. Indeed, since each of the gn ( . , e)’ s is continuous and vanishes identically when ]]e]] > 64d2M, it suffices to observe that, by construction, c, g,(x, e) = a(x) - 21 for all x E Rd. 0 The situation when a is .not continuous is hardly different. The only substantive change is that one must begin by doing a little smoothing. For example, choose some $ E Cy (R’; [0, M)) with total integral 1, and set (3.10)
a,(x)
= (u-4
.I
Rd $(a-?(y-x))
a(y) dy, Q E (0, l] and x E Rd.
Then
Ikbb) - aa(x)Ilop I C~-~lly- XII, x, y E BBd
(3.11)
Hence, by Lemma 3.2, one can find an aa E (0, l] such that, for each a: E (0, oO] and 1 E Zd, there is a basis {el~(cu~l),. . . ,ed)(ay’)} in Wd with (cf. (2.12)) max ]]ej1(a71)]] 5 32d2M and, for each 1 < i < d, ll(~a(~) - 2’)‘Z,,)i,jl 5 (E~~.l(a(x) - “)E,‘I)~,~. j#i x E &,a (1): when E,.t = [e1)(QJ), . . . , edl(OIJ)] -l. Set %,I(x> = &,I(a,(x) Next, choose an q E C,Y((-1,l);
- 2I)El,,,
x E Qa(l).
[O,l]) so that v z 1 on [-$, i], take
ii(t) = rl(t - 1) + v(t) rl(t) + rl(t - 1)
and
,rjQ,l(x) = fi
fj(Cax~
- lj).
j=l
and define O,,J : Wd x Zd ----+ [0, cc) so that, when x E Q,+(l), ‘ *
(T,,I(X,e)
(qx);,j) + if i # j & e = (ei,Ccy,l)+ &,Ca,l))
= ( ~(%1(4i;j)if i # j & e = (ei,Cn,I) - e.i,Cm~I)) %,1(x) (%,I(x)i,i \
Vol. 33, no 5.1997
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CI%,l(X)L,jl)
j#i if e = ei,(cxy,l)
646
D. W. STROOCK
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and gol,t(x, .) = 0 when x $ Qn+ (1). Finally, set (3.13)
pm@, e) = l{l} (Hell) + c
~a~a,,l(k
e).
3.14 THEOREM. - Given a Bore1 measurable, symmetric a : Wd ---+ Wd @ Rd, define {pa : a f (O,l]} as in (3.13). Then Theorem 2.18 applies and so (2.20) holds.
4. CONVERGENCE OF PROCESSES AND AN APPLICATION TO THE DIRICHLET PROBLEM As a more or less immediate dividend of Theorems 3.9 or Theorem 3.14, we get an accompanying statement about the associated stochastic processes. Namely, for each a! E (0, 11, let R, denote the Skorohod space D([O, 00); oZd) of right continuous, aZd-valued paths on [0, co) with left limits, and note that each R, is a closed subspace of D( [0, co); Rd). Next, let { P$ : k E Zd} be the Markov family of probability measures on R, with transition density p”. Also, let {PC : x E R”} be the Markov family of probability measures on C( [0, cc); Rd) for which p” is the transition density. Because of (1.14) and standard compactness criteria (e.g., ChenZov’s in Theorem 8.8 on page 139 of [2J) for the Skorohod topology, it is clear that, as a subset of probability measures on 00,
{p:lc: Q E (O,l],
k E Zd, and ]]ok]] 5 T} is tight for each T E (0,co).
At the same time, by (2.14), if a, \ 0 and a,k, --+ x E Wd, then the only limit { Pzzk, } T can have is P,“. Hence, we have now proved the following. 4.1 COROLLARY. - Under the conditions described in either Theorem 3.9 or Theorem 3.14, for any bounded F : Ro -+ R which is continuous and any 7- E (O,oo),
(4.2) In fact, (4.2) continues to hold for bounded F’s on Ro which, for each x E Rd, are Pz-almost surely continuous with respect to the topology of uniform convergence on finite intervals. Proof. - The only point not already covered is the final one. However, because {Px : 1x1 2 T} is compact as probability measures on Anrules
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C([O, m); Bad), it follows from standard facts about the Skorohod topology that the preceding weak convergence result self-improves to cover functions which are continuous with respect to the topology of uniform convergence on finite intervals. 0 We conclude with an application of the preceding to the Dirichler problem for L” in a bounded, connected region 6 c Rd. That is, given an f E C(d6; W) we seek a function uaf E C(a; R) with the properties that:
(4.3)
Set
E [0, c-31: w 1 [O,t] p S}
E [0, cm] : w 1 [O, t] g S}.
and Because 6 is bounded, (1.16) can be applied to check that there are T E (0, CQ) and E E (0,l) with the property that
fzk(C~ I T) 2 6 for all n E (0, l] and ok E 3. Hence, by a familiar argument, there exists another E > 0 such that sup sup EP-“k ~E(O,ll a&~
[tfC~ 1
Similarly, from the lower bound in (1.0.10) of 191, one knows that
for a suitable choice of t > 0. Obviously Ce 2 (3 always. In addition, Ce and cg are, respectively, lower and upper semicontinuous with respect to the topology of uniform convergence on finite intervals. Finally, the lower bound from (I. 1.lO) in [9] leads (cf. part (iii) of Exercise 3.2.38 in [lo]) to (4.4)
P,a(&=(z)
=l,
XERd,
whenever d6 is Lebesgue regular in SC in the sense that (4.5) Vol. 33. no S-1997
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4.6 THEOREM.- Assume that (4.4) holds. Given f E C(d6;
W), choose
f E Cb(@; W) so that f t dC5 = f. Then (cc (4.1)) q(x)
(4.7)
= ;io-,
[f(L&,),
&5(w) < cm],
x E 6.
In fact, the convergence in (4.4) takes place uniformly on compact subsets of 6. Proof. - In view of Corollary 4.1 and the preceding discussion, the derivation of (4.4) comes down to checking that qx>
(4.8)
= P,” [f(42)).
b(4
< m],
x E 6.
Indeed, (4.4) guarantees that f(~(&)) is P;-almost surely continuous in a topology with respect to which we know that Pg10 is converging weakly to P,. To prove (4.8), one needs to use three facts. First, (4.8) is essentially trivial when a is smooth. Second, the convergence result in Theorem 11.3.1 together with the tightness which comes from the upper bound in (1.1.10) of [9] and the continuity discussion just given, show that w-
[f(w(W),
42 < -1 uniformly
-
EPZ [f(w(Cd),
43 < m]
on compacts in t5
if {a,}y is a sequence of coefficients matrices satisfying (3.1) and tending to a in measure. Third, one must know that, under the same conditions, UT UT uniformly on compacts, a conclusion which can be easily drawn from the facts in [9] and is explicitly contained in results of [7] or $5 of Chapter II in [6]. 0 Remark. - For actual computation, (4.9)
lEPZk[f(4)),
it may be useful to observe that
63 < m] =
EQZk [f(4Cd),
C@< a] >
where {Qzk : k E Z”} is the discrete-time parameter Markov family on ~2’~ with transition probability III”(ak,&)
= R”(k)-i(p,(k,&
k) +/~&k-1)),
when R”(k)
= c (pm@, e - k) + ~a(4 k - e,). tad Indeed, one can construct Qzk from P$ by simply recording the size of the jumps and taking the time between successive jumps to be 1. Hence, the distribution of unparameterized trajectories is the same under P$ and QEk. In particular, the distribution of ~(~~)l(~,~~(~~) is the same under Q:: as it is under P,“. Ann&s
de l’lnstifut
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649
REFERENCES 111 E. CARLEN, S. KUSUOKA and D. STROOCK, Upper bounds for symmetric Markov transition functions, Ann. Inst. H. Poincur&, Vol. 35, Sup. au #2, 1987, pp. 245-287. [2] S. ETHIER and T. KURTZ, Murkov Processes, J. Wiley & Sons, 1986. [3] M. FIJKUSHIMA, Y. OSHIMA and M. TAKEDA, Dirichlet Forms and S.vmmetric Murkol Processes, De Gruyter Series in Math., W. De Gruyter, 1994. (41 E. FABES and D. STROOCK, The De Giorgi-Moser Harnack principle via the old ideas of Nash, Arch. Ratl. Mech. & Anal., Vol. 96, #4, 1987, pp. 327-338. 151 I. I. GIHMAN and A. SKOROHOD, The Theoy of Stochastic Processes II. Springer-Verlag, Grund. #218, 1975. 161 D. KINDERLEHRER and G. STAMPACCHIA, An Introduction to Variational Inequalities and their Applications, Academic Press, Pure & Appl. Mach., #88, 1980. [7] W. LI~TMAN, G. STAMPACCHIA and H. WEINBERGER. Regular Points for elliptic equations with discontinuous coefficients, Ann. Scuola Norm. Sup Piss, #17, 1963, pp. 43-77. [8] Z.-M. MA and M. R~CKNER, Dirichlet Forms, Springer-Verlag, Universitext Series, 1992. 191 D. STROOCK, D&ion semigroups corresponding to uniformly elliptic divergence ,form operufors, Stminaire de ProbabililitC XXII (J. Azema & M. Yor, ed.), Springer-Verlag LNMS #1321, 1988, pp. 316-348. [IO] D. STROOCK, Probability Theory, An Analytic View, Cambridge Univ. Press, 1993. [ 111 D. STROOCK and S. R. S. VARADHAN, Multidimensional D@sion Theory, Springer-Verlag. Grundlehren 233, 1979. (Munuscript received May 14, 1996: Revised December 12. 1996.)
Vol. 33. no 5.1997
Inst.
Henri
Poincarb, ProbabilitPs
Vol. 33, no 5, 1997, p. 619-649
er Statistiques
Markov chain approximatims to symmetric diffusions Daniel
W. STRCKXK
M.I.T., rm. 2-272, Cambridge, MA 02139-4307, USA E-mail: [email protected] and Weian ZHENG Dept. of Math., UC Irvine, Irvine, CA 92174001, E-mail: [email protected]
USA
ABSTRACT. - This paper contains an attempt to carry out for diffusions corresponding to divergence form operators the sort of approximation via Markov chains which is familiar in the non-divergence form context. At the heart of the our procedure are an extension to the discrete setting of the famous De Giorgi-Moser-Nash theory.
RI%J&. - Cet article contient une tentative pour etablir dans le cas ou le gknerateur don& sous forme de divergence, les approximations par des chaines de Markov qui sont bien connues pour des diffusions dont le gCn&ateur est dorm6 sous une forme de nondivergence. L’essentiel de notre mkhode repose sur une extension de la celbbre th&rie de De Giorgi, Moser. et Nash au cas discret.
The first author acknowledges partial support from Ann&s
de I’lnstituf
NSF
grant
9302709-DMS.
Henri Poincar~ - Probabilit& et Statistiques Vol. 33/97105/$ 7.00/O Gauthier-Villars
- 0246-0203
620
D. W. STROOCK
AND
W. ZHENG
0. INTRODUCTION
Given a continuous, symmetric matrix a : Rd + W’ @ Rd satisfying XI < a(x) 5 X-r1 for some X E (0, l] and a bounded continuous b : Rd --+ Rd, the martingale problem for the operator
(04pb=f:
on C,-(Wd; R)
i,j=l
is well-posed (cf. Theorem 7.2.1 in [l 11). As a consequence, one can use any one of a large variety of procedures to approximate the associated diffusion by Markov chains (cf. Section 11.2 in [l 11). Indeed, aside from checking a uniform infinitesimality condition, all that one has to do is make sure that the action on f E C,-(I@; R) of the approximating generators is tending, in a reasonable way, to ,Pbf. If, on the other hand, one replaces the Pb in (0.1) by the divergence form operator
(04
L” = v . (av)
= 2
-&
&+(x)&.
i=l
then, unless a is sufficiently smooth to allow one to re-write L” as fYb with d &i,j
hi(x) = c -(x), j=l d?i then the theory alluded to above does not apply. Nonetheless, the magnificent analytic theory of De Giorgi, Nash, Moser, and Aronson (cf. [9]) tells us that the operator L” not only determines a diffusion, it does so via transition probability functions which are, in general, better than those determined by C”lb. Thus, one should hope that probability theory should be able to provide a scheme for approximating these diffusions via Markov chains, and in the present article we attempt to do just that. In order to understand the issues involved, it may be helpful to keep in mind the essential difference between the theories for the non-divergence form operators C”ab and the divergence form operators L”. Namely, all the operators Pb share CcW(lRd; R) as a large subset of their domain. By contrast, when a is not smooth, finding non-trivial functions in the domain of L” becomes a highly delicate matter. In particular, as a changes by even a little bit, the domains of the corresponding La’s may undergo radical change. For this reason, one has to approach L” via the quadratic form Annales
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APPROXIMATIONS
which it determines in L2(Rd;R). can interpret L” f as a distribution finds that
TO SYMMETRIC
DIFFUSIONS
621
Namely, if f E C,-(Rd;W), then one (in the sense of L. Schwartz) and one
-($I, W) = W,S) z (RI, w),, 9 E TV@: w. (0.3) where we have introduced the notation ( . , . )a to denote the inner product on the usual Lebesgue space L2(Wd; R). For historical reasons, the quadratic form E” is called a Dirichlet form. Clearly, for any a satisfying our hypotheses, E” is well-defined on C,W(RBd;W). In fact, one finds that each E” 1 C,-(I@; R) admits a closure, which we continue to denote by E”, and that Dam(P) coincides with the Sobolev space IV; (Rd; R) of f E L2 (I@; W) having one (distributional) derivative in L2( Rd; R). Furthermore (cf. Theorem 11.3.1 in [9]), there is a unique, strongly Feller continuous, Markov semigroup {Pp : t > 0} with the property that, for each f E C,“(Rd; R), t * PFf is the unique continuous map t E [0, x) I-+ ut E W,‘(Wd; W) such that (Ut,g)O
(0.4)
=
(f,do
-
p(w) n
dT-
t E [0, co) and g ;“CT(Rd; W). On the basis of the characterization in (0.4), one might think that convergence of the semigroups should follow from convergence of the associated Dirichlet forms. That is, one might hope that P,“” f --+ Ppf for all f E Cc(Rd; W) should follow from Ean (f, J) E”(f, f) for all f E Cc” ( Rd; 53). However, this is not the case, not even when d = 1. To wit, given a measurable a : R --+ [l, 31, set
J
1 1 d[ and & = a 0 (tP)-1. 0 4l) At least when a is differentiable, it is easy to check that UpI)
=
x
L”(f
0 u”) = (cq)
0 ua,
and therefore, if, for X > 0, R; is the resolvent operator given by JFe- xtP; dt , then an elementary‘application of the Feynman-Kac formula leads to (0.5)
[R;f]
Vol. 33, no 5-1997.
(CC)= 2Ep”-“(z)
cx2a 0 (If-‘(w(t))
x exp (-2X
/f a o (u”)-’ .o
x f o (Q-l
(w(t))
1
dt ,
(W(T)) d7-)
622
D. W. STROOCK
AND
W. ZHENG
where hcLydenotes the standard Wiener measure for l-dimensional Brownian motion starting at 1~.In fact, (0.5) hoIds whether or not a is differentiable. Now suppose that {un}y is a sequence of [l ,3]-valued measurable functions. Then &“- (f, f) &“(f, f) for all f E C,-(W; R) is equivalent to saying that a, a weakly in the sense that
JR
anOP(I) dx -
J R
a(()~(<) dx
for all cp E L1(R; R).
On the other hand, (0.5) shows that it is weak convergence of the reciprocals 2 to k which determines whether the associated semigroups converge. Indeed, it is clear that 5 t implies first that uan --+ u and therefore that (zL”~)-~ @“)-I uniformly on compacts. In particular, since (?P)-l(y) = Joya, o (rP-l(~) dq, this means that ca, o (@)-l ao (u”)-l weakly. Hence, after representing ji uo(rP-’ (W(T)) dr in terms of the local time of Brownian motion, one sees from (0.5) that R;1- f RT f follows from -& weak2. Finally, take u,(x) = 2 + sin(2nrrx), and observe that a, 2 weakly while
-+ J3 1 a,
weak
-;
1
2 + sin(2sl)
dt
=
J3 1 0
1 - 4-l sin(27rt)2 d(>
;.
As the preceding discussion indicates, even in the context of diffusions, identification of limits via Dirichlet forms is a delicate matter. Thus, it should come as no surprise that the difficulties become only worse when one is trying to approximate diffusions by Markov chains. In order to handle these problems, we devote the next section to a derivation of a number of a priori estimates. These estimates are extensions to the discrete setting of estimates which are familiar in the diffusion setting, and our proof of them mimics that of their diffusion analogs. In $2 we use the results in $1 to find general criteria which tell us when our Markov chains are canverging to a diffusion, and in $3 we construct Markov chain approximations to the diffusion corresponding to a’s satisfying (O.l), first in the case when a is continuous (cf. Theorem 3.9) and then in the general (cf. Theorem 3.14). Finally, in $4 we describe a possible application.
1. THE BASIC
ESTIMATES
Our purpose in this section is to derive the a priori estimates (especiahy (1.11) and (1.32)) on which our whole program rests. The general setting in Ann&s
de l’htitut
Henri
Poincati
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CHAIN
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623
DIFFUSIONS
we will be working is described as follows. We are given a function p : Ed x Zd --+ [O,ce) satisfying’
which
p(k,e) Cl.11
>_ 1 for all (k,e) E Zd x Zd with llej\ = 1 and yks Idk.ell < 30. d
For (Y E (O,m), we set aZd = {ak D’(cIZ~;R) of f : aZd --+ W with
: k E Z”}, introduce the spaces
and define the symmetric quadratic forms ( . , . )oi and &“J’ on L2 (aZd; R) by kQ*
(1.2)
P@(f,g)
= ad-’
c
c
kEZ*
e#
x (f(4k
p(k,e)
+ 4)
-
fbW(g(4k
f
4)
-
W41.
By the elementary theory of Markov chains, it is an easy matter to check that, for each a E (0, cc), there is a unique symmetric, Markov semigroup {P,“@ : t > 0) on L2(dd; Et) with the property that, for each f E L2(aZd;W), t E (0,cc) ++ P,*l”f E L2(dd;R) is the unique differentiable t E (0, cc) ut E L2 (aPd; W) satisfying lim+,u uf = f pointwise, and
(1.3)
$ (9, Ut), = -qg,
Q)
for all g E L2(oZd; R). Moreover, if p”@ : (0, x) x aZd x aZd ---+ [0, oc) is determined by
(1.4
.[P,“@f] (cxk) = ~8 c
f(al)pa+(t.
ak, al),
1EZd
then (1.5)
p”rP(t, ok, al) = p-‘(t,
’ for v E Wd, we use llvll the Euclidean length of v. Vol. 33. no 5-1997
01, ok)
and
@dp”fl+‘(P2t, apk, ~$1) = pa+(t, ak, ~1). E maxl
IVil,
as distinguished
from
1~1, which
reserved
for
624
D. W. STROOCK
AND
W. ZHENG
The symmetry assertion in (1 S) comes from the symmetry of the operator P,“>” in L2(oZd; W). To see the second assertion, for f E L” (aZd; R), define fp E L” (a/3Zd; R) so that f3(ct/3k) = f(ak). Next, take u&k) Then ut -+
= [P;$‘fo]
f pointwise as t \
(@Jk).
0 and
Hence, ut = P,“@f, and so we are done. Note that, as a consequence of the symmetry and Jensen’s inequality, we know that Ilptq,,,
(14
for all t E (0, co) and p E [l, co].
5 Ilfllw
We turn next to the derivation of pointwise estimates on @tP(t,x, y), and for this purpose we will need the following Nash inequality (cf. [l]). 1.7 LEMMA. the property
- There is a constant C E [l, oc), depending only on d, with that
(1.8)
IlfllT;” 5 C@If, fNll?J>
f E qc
q.
Proof. - We begin by noticing that, since ]]f]]l,2 5 ]]f]]r,r, will assume that 0 < El+(f,f) < ]]f]]~,,. Next, set
f(C)= c f(k) exp(fl2n(&k)w),
we may and
< E P,lld.
kEZd
By Parseval’s identity, for each T E (0, l]:
with f’(r) being the set of t E [0, lid with the property that, for each 1 I i 5 d, either & 5 f or & 2 1 - f. Clearly, I(r) < rd]]f]]r,r. At the same time,
rl+(f,f) 2 c ‘&(f(k + ei>- f(k)J2 (1
-
~0~(27r&))lj(t)/~
Annales de l’lnstitut
Henri
P&car6
dt
L
~r~J(r),
Probabilitts
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APPROXIMATIONS
TO SYMMETRIC
DIFFUSIONS
625
where
Hence, we know that
Ml;,2 I 4lfll2,I In particular, can take
+
~lwJ) yr2
since we are assuming that 0 < El@(f, f) EYf,
r = and thereby obtain our result. 1.9
r E (o q > .
’
5 I]f]&,
we
f)
Ilfll~,I 0
- Assume that,
LEMMA.
(1.10) Then there exists a C E [l, o), depending only on d, such that (1.11)
p”@(i, x, y) 5 CeK(P)V, (t”) --I exp (-g,
where Proof. - Observe that aVr
(4
(1.12)
2
d
5 Va(r) 5 (a! V r)“.
Thus, we need only prove (1.11) with V, (t a ) replaced by CIVt 3. In addition, by the second part of (1.5), it suffices for us to handle the case when CI:= 1. Because of (1.8), we can apply Theorem 3.25 in [l] to see that there is a C, depending only on d, such that pll~(t,X,Y)
5 ct-~exp(-13(~;x.y))),
where D(t; x, y) 2 sup (.(&Y - x&~ i tI’(e)2 r(G2
=
SUP{~
,@,e)(exp(l(Z,e)Rd/ e#
Vol. 33. Ilo s-1997.
: E E APdj with -
1)”
: k E Zd>.
626
D. W. STROOCK
AND
W. ZHENG
Thus, when t > 1, we get (1.11) by considering c = t-;
y-x IY - Xl
andusing (elql-1)2 5 Q’ e ‘1111 . When t E (0, I], we have to argue somewhat differently. Namely, by (3.19) in [l], we know that, for any < E Rd, 1Jq5-cpt’“($tf)
111,~L etr(t)Z[Iflll,2
where &(z)
E e@)@.
Hence, since (]g](Loo I ll9l11,2 5 119ll1,1~
and so @Y~,x,Y)
L exd-(t,Y
- x)Rd + tr(t)*).
Thus, we can now complete the proof by taking 6 = e. 1.13 LEMMA. Cd (1.10)) (1.14)
0
- There is a C E [l, cm), depending only on d, such that
ad c (Jai - akl12p”@(t, ak, al) 5 CeKcP)t. IEP
Proof. - Again, by (1 S), we need only look at Q = 1. When t 2 1, (1.14) with o = 1 is an obvious consequence of (1.11) with Q = 1. When t E (0, 11, we use the description of the right continuous Markov process X(t) corresponding to { @’ : t > 0) starting at k. Namely, because this is a pure jump process, we know that
IIW - w9ll 5 c
lIW4 - X(4.
sE(%tl Thus, if r, denotes the time of the nth jump and A, X(~n-)lJlp,t~(~n)~ hen
~[IFV) - WN”]
5 e
= I]X(r,)
-
$LrA]
m,n=l
= 557
Ann&s
c
+2
m=l
E[A,A,].
llm
Henri
Poincare! - ProbabiMs
et Statistiques
MARKOV
CHAIN
APPROXlMATlONS
TO SYMMETRIC
627
DIFFUSIONS
But, by the standard theory of jump Markov processes (cf. Chapter 3 of [SD, the number of jumps is a Poisson-like process with rate dominated by SUP c kad
dk>e)~
eEEd
form which it is easy to estimate the preceding by, respectively,
Hence, because t 2 1, we are done. 0 From (1.14) it is immediate that there is an R, depending only on d and K(p), with the property that ad
p”+‘(t, cuk, al) 2 f
c
for each k E Zd.
([al-ak[l
In particular, by the Chapman-Kolmogorov Schwarz’s inequality: p*@(2t, ak, “k) = ad c p”+(t, Ia* ad
c
equation,
symmetry,
and
ak, al)2 @=+(t, ak, rwl)*
I/al-crk~~~Rd
2 Va(Rt+)-‘(
c
p”@(t. ok, ~1))~
IIal-akll
Thus, there is an E E (O,l], depending only on d and K(p), (cf. (1.12)) p-‘(t,
such that
elk, ak) 2 --?-
V#).
Our next goal is to show that the on diagonal estimate in (1.15) extends to a neighborhood of the diagonal. Namely, we want to show that there is an t > 0, depending only on d and K(p), such that
(1.16)
Vol. 33. iI0 s-1997.
pQ@(t, ak, al) 2 --?--vpq for cu E (0, l] and (t, k, 1) E (0, a) x Zd x Zd with J]ok - nl]] 5 ati.
628
D. W. STROOCK
AND
W. ZHENG
Our derivation is based on the ideas of Nash, as interpreted in [4]. We begin with a number of observations. 1.17 LEMMA. - (1.16) will follow from the existence of an e, depending only on d and K(p), such that (1.18)
p”+‘(l,O,
al) 2 t for
a E (0,2] and I(crl(l 5 2.
Proof. - In the first place, notice that our hypotheses are translation invariant. Hence, there is no loss in generality when we take k = 0. Further, if we know (1.18) and I(al(( 5 2t3, then either 1 = 0 and (1.15) applies, or cu 5 2t 4. In the latter case, take /3 = t- 3, note that (up 5 2, and apply the second part of (1.5) and (1.12) to see that (1.16) follows from (1.18) with Q replaced by a/3. q The proof of (1.18) depends on the following Poincard inequality. 1.19 LEMMA. - Let U E C” (R; [0, w)) have the properties that J e-2U(c) dt = 1 and U(l) = Ielf or a I1 sufJi ciently large 111.For Q: E (0,2], define d
g=(l) = cx-d
Ju i=l
41%+1)
e -2u(Ft) ,-&CC, 1 E jpf.
d
Then there exists a X > 0 such that (1.34
x((f - (&J2).!-tcl
bad
i=l
where
and ei is the element of Zd whose jth component is 1 if i = j and 0 otherwise. Proof. - Because of the product structure, we may and will restrict our attention to the case when d = 1. We begin by showing that p R(f(E) - (f))2 eezu(‘) dt I l f’(E)2e-2U(f) de, J where (f) s f([)e-2U(E) d<, .I for some p > 0. To this end, note that (1.21) is equivalent to (1.21)
Annales de l’lnstirut
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where V z (V’)” - U”. Thus, if H 3 - -$ +V, then (1.21) is equivalent to (1.22)
w=(H)
C (0) U bL, co>
But, because V - 1 has compact support, H is a compact perturbation of -$ + 1, and therefore the essential spectrum of H coincides with [ 1, cc), the essential spectrum of - $ + 1. On the other hand, emu is an eigenfunction of H with eigenvalue 0. Hence, 0 must be an isolated eigenvalue of H, and so (1.22) will follow once we show that it is a simple eigenvalue. However, because it lies at the bottom of spec( H), a familiar argument, based on the variational characterization of the bottom of the spectrum, shows that 0 is indeed simple. Given (1.21), we proceed as follows. For f on &Z, extend f to R by linear interpolation. Thus,
Then (remember that d = 1) Q-l C(f(c@ ea
II-
P
2 .I’l 2 $
+ 1)) - f(cke))2g,(e)
Vol. 33, no S-1997.
f’(<)2e-2U(c) R
R2(f(v) c k,eEZ
Hence.
= /
(fG+
- f([))2eC2”(E)-2u(q) - f(4)2&)sa(~)
ci?
d<
630
D. W. STROOCK
The proof of the following
AND
W. ZHENG
lemma depends on the inequality
(1.23) for all positive numbers a, , b, c and d. To verify this, first note that, by homogeneity, it suffices to treat the case when a = 1 = c A d. In addition, by using the transformation 6 +Y i, one can reduce to the case when c 5 d. In other words, (1.23) comes down to checking that (10gx)~ 5 2(,
- $1
-x)
+ (d - 1)2
for all x E (0, oo) and d 1 1. Finally, observe that, by Schwarz’s inequality, (lo&r)2
5 @ - l)“) 2
and so it suffices to see that @ ,l)’
5 2 % - 1 (1 - x) + (d - 1)2. ( >
When 5 E (0, I], this is trivial. When II: > 1,
1.24 LEMMA. - There is an E E (0, I), depending only on d and K(p), such that
(1.25) 2 i log 6 for all LL E (0,2] and JlcukJJ5 2.
Proof. - Set ~~(1) = p”+(t,ak,al)
and
G(t) = ad c
log(ut(l))ga(l),
1EZd
and note that, by Jensen’s inequality, G’(t) = ($$),. Ann&s
G(t) 5 0. Moreover, by (1.2), = -E”.+~,~).
de I’lnsritur
Henri
PoincarP
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et Statistiques
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APPROXIMATIONS
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DIFFUSIONS
631
Hence, G’(t) = -ad-2
l&p i=l _
cid-2
c
c
1EZd eEP
ba(’
2g,(l
+
‘1
+e>
-
g~(‘)j2,q
A h(1)
e)
’
’
where, in the passage from the first Iine, we have used (1.23). Because U’ is bounded and Q! E (0,2], we can find y E (0: l] such that
minsa(1+ e”) L -m(l) l
5 K(p). 7
Hence, there is au A4 E [l, CG), depending only on d and K(p), that (cf. (1.20))
such
G’(t 1 &ad
~(hWt(l)
- G(t))2g,(l)
- M.
1EP
Next, for c > 0. set
A,(g)= (1Eifd: ~(1)> e-q}. Then, for each D > 0,
2 adC(- log- W - G(tH2gdl)
2adc (g+G(t))2ga(l) LG(t)‘; c go(l) - CT~. l-L(~) 1EAt (c) Vol. 33, R0 5.1997.
632
D. W. STROOCK
AND
W. ZHENG
Thus, we now know that G'(t)
2 A'At(a)'ga 2M
G(t)2-(M+g2),
where IAt(o)jsa
z ad c gol(l). 1~4 (~1
Finally, we use (1.14) to find an r E [l, co) such that ad c
py
t,ak,al)
2 i
for all CLE (0,2], t E (O,l],
and (Jak(( 5 2.
IlWl~ In particular, if p is the smallest value e-2U takes on [-T, r], then
c ~(1)5 eeurd
+ ““:r’y
(At(g)lg,.
Il4l
Thus, by taking c = log4rd and using (1.1 l), we conclude first that and then that there exists a S E (0, l), depending 14(“)(ga 2 ‘$;:1p) only on d and K(p), such that G’(t) 2 ~5G(t)~ - 6-l
for all 01 E (0,2],
[ 1
(1.26)
t E i, i
, and ]Jak]J 5 2.
To complete the proof starting from (1.26), suppose that G(i)
5 - 5.
By (1.26), -G(t)
2 -(26)-l,
t E
[ 1 ;,;
.
and so, again by (1.26), G’(t) 1 ;G(Q2,
tE
1 1 4’2.
[ 1
But this means that
and therefore $ exp(-86-l).
G( f ) > -8S- i. In other words, we can take ~4 = •i Annales
de I’lnsrirut
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_ Probabilit.3
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633
The passage from (1.25) to (1.18), and therefore (1.16), is easy. Namely, by the Chapman-Kolmogorov equation and symmetry,
Hence, by Jensen’s inequality, if 1)al/j _< 2, then (1.25) gives (1.18). In the following statement, { P,“l;” : k E Z”> denotes the Markov family of measures on R, z D ([0, 00); c&) for which transition probability function has p”@(t! Qk, al) as its density. Also, for each T > 0 and k E Z”, <;/ : % ---+ [0, co] is given by &(w)
= inf{t
> 0 : (/w(t) - akj( > r>.
Finally, for l’ C_ Rd, \I’\, denotes ad times the number of k E Zd with ak E I. 1.27 LEMMA. such that
- There is a 6’ E (0, i)> depending only on d and K(p)S
(1.28) for all Q! E (O,l], r E (0, w), t E (0, (2@r)2] : (k,l) E Zd x Zd with Jjok ;(~lj/ 5 t;.
[-ti,t* 1r-lcad.
andIF (g crk+
Proof. - First suppose that tt < (Y. Then 1 = k and either I is empty, and there is nothing to do, or Jl& = Va(ti). In the latter case, PCyi”(w(t) E I? & C,&(W) > t) 2 PK;p(w(sj = k for s E [O,t])
2 exp (-K(p)$)
2 exP (-K(P)).
Thus, we need only worry about ti > a. But then, for (lak - ajll 5 ti: Cy-dP~iP(U(t) = aj &
l/,0
al, aj) - EpZi” WYt -Cc”(P)
sup o
,$-if
- Ci$(w)T w(C&), exp
- Cf~~(~)t-$ sup fs’exp
2--f,
K@j Vol. 33, Ilo 5.1997.
U>l
(-(I
(-‘l-$H’“).
,I”“‘)
olj), G&(W) < tl
634
D. W. STROOCK
AND
W. ZHENG
Hence (cf. (1.12)) we can choose 6’ E (0, i) to achieve P,“i”(w(t)
= aj & c&(w)
> t) 1%; V&z)
and (1.28) follows immediately after summing over oj E r. Cl Our main interest in (1.28) is that it allows us to derive a Nash continuity estimate (cf. Theorem 1.31 below). However, in order to do so, we will have to replace (1.10) by the much more stringent condition: (1.29)
and
1141 > R(P) * p(k e) = 0 c p(k,e) < R(p) for all k E Zd, eEZd
where R(p) E [l,co). In the following lemma, Q”((T,
k); T->E [T - ?, T] x (all : (lo4 - ok/( 5 r} for T > 0 and (T, ok) E [r2, co) x aZd.
Also, given a function u on a set S, define Osc(u; S) = sup { lu(y) - U(Z)I : 5, y E s}. 1.30
LEMMA.
-
Set(cf.
Lemma
1.27 and (1.29)) v = (1 + R(p))-‘0
determine ~7E (0, cm) by q* = (1 - 2-d-38). bounded functions f on aZd, 0+-f?
Q”((T, k);
and
Then, for al/ Q E (0, l] and
QQ((T k);
v-)) I fO+‘T
r) >,
r > 0 and (T, k) E [r2, co) x c&,
Proof. - Without loss in generality, we will assume that k = 0. Moreover, because there is nothing to do when qr < o, we will assume that 77~_> CL For ,0 E [O,Ti], set M(P) = max{UQ(s, 4) : (s,d) and m(P) = min{zP(s,oj) : (s,oj) Then Osc(~~;Q”((2;,0);~))
= M(p)
rI=y= 0 Annales
E Q”((T, 0); P) > f Q-((T,O);p)}.
-m(P).
Next, set
r 1+ R(P) de i’lnstitut
Henri
Poincart!
- Probabilith
et Sthtiques
MARKOV CHAIN APPROXlMATIONS TO SYMMETRIC DIFFUSIONS
635
and
Obviously, either lI’la >_ iVa(qr) or [I’(, 5 $Vcl(7p). If lIYln > iv,(y). take w = ua - m(r). Given T - (77~)~< s 5 T, set t = (27r)” - T + s E [(q~)~, (277r)“], and note that, because w 2 0 on Qa((T,O); 7.) and
Ppae - almost surely when ~~cE~~~ < ‘r/r, (1.28) impIies that
for lIdI\ 5 *or. Thus, m(q~) - m(r) means that WV-)
- h4
F M(r) I (1 -
1 2-d-“H(M(7.)
- m(r)),
which
- m(v) 2-d-30)
(M(r)
- m(7.)) = 7f(M(r)
- 7n(r)).
When IFI, 5 iV,(q~), one takes w = M(T) - u” and proceeds as above to get first that M(r) - M(qr) 5 (1 - 2-d-3q) (M(T) - m(r)) and then the same estimate as we just arrived at. q 1.31 THEOREM. -Assume that (1.29) hotdsfor some R(p) E [l, oo), and define u as in Lemma 1.30. Then there is a B E (0, CO), depending only on d and R(p), such that, fir all ar E (0, I] and f E L”(rxZ”: R), (1.32)
I[f’?‘f] (4 - [f’?‘f] (cik)I It - ~(3 v ((ak - crl(( (T (t/Is)3 > .
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636
D. W. STROOCK
In particular, (1.33)
AND
W. ZHENG
there is a C, depending only on d and R(p), such that
lpa+(t’, Qk’, al’) - jPp (t, ctk, nl) ( 5 C(lt’ - tl+ + J(ak - ak’l) + (1~1- al’(l)=+’
it)-*.
Proof - Clearly (1.33) follows from (I .32) together with (1.11) and p-‘(t,
ak, al) = p”@(t, ~1, ok) = [ Pcylppcu+ 6 (;> +)]Wb
To prove (1.32), set T = t A s and T = dm.
If
It - s/i v IId - akll 2 T, then (1.32) is easy. Otherwise, determine n 2 0 so that
- $I . rln+1 -< It - 4 v rlb1 - akll < Then, by repeated application of (1.30), one finds that
I [fY@fl (4 - [~sOI?f] w I 5 OS+“; Q” ((T, 0); $r) > 5 $gOsc(ua;
Q” ((T, 0); r))
5 3/n011fll a,c‘=5 ~vIfll%cc
2. PRELIMINARY Throughout this For each Q E (0, assumption is that there is a R E [l,
(
(t - sp v I(ak - cdl\ u 7 >.
APPLICATIONS
section we will be dealing with the following situation. 11, we are given pa : Zd x hd [0, oo). Our basic the pa’s satisfy (1.1). In addition, we will assume that oc) with the properties that Ilell > R * pa&e)
(2.1)
q
and
c p,(k,e) eEP
< R
= 0 for all k E Zd.
2.2 LEMMA. - Set (cj 1.4)) p” E p”@- , and,for x E Rd, use [x]~ to denote the element of aZd obtained by replacing each coordinate xi with CI times Anna/es de I’lnstitur
Henri
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637
DIFFUSIONS
the integer part of CX:-~Z~.Then, for any sequence {a,}? E (0, l] which decreases to 0 there is a decreasing subsequence { QI,J} and a continuous q : (0, W) x IWdx [Wd -+ [0, KI) such that (2.3) ,!@k t;;k,
1
xs,u,p,dt~eP~y-XtlP~’ (6 [XL,, , [&,) >
44
x, y)l = 0.
where ,6 E (0, XI) depends only on d and the number R in (2.1). In particular,
(2.4)
cds,x,Wt = 1, (s.4 E (‘44 x Rd, J’R* lim sup
S~OXERds It-xl>r
Q(S+ t,x,Y) =
JRd
q(s, x, t> dt = 0,
xEIWdandrE(07co),
q(s,x, t)dt, t, Y) de, s, t E (0, W) and x,y E [Wd.
Proof - Because of (1.11) and (1.32), there is essentially nothing to do
except comment that the existence of a locally uniform limit is guaranteed by (1.32) and the Arzela-Ascoli compactness criterion and that the other statements in (2.3) and (2.4) follows from this combined with (1.11). El From (2.4), it is clear that if Qt is determined by (2.5)
.
Qt./Xx)= Ad f(t)q(t,
x, 0 dt,
(6 x) E ((h ~1 x bfd and f E Cc (Rd; R) >
then { Qt : t > 0} determines both a Markov semigroup on Ct, (Rd; R) and a strongly continuous semigroup of self-adjoint contractions on L2 (BBd;R) . Our goal now is to give criteria which will enable us to identify the semigroup { Qt : t > O}. Notice that if { Qt : t > 0} is uniquely determined, in the sense that it is independent of the subsequence selected, then there will be no need to pass to a subsequence in the statement of (2.3). In order to carry out our program, we first impose the regularity condition that (cf. (2.1)), for each T E [l, oo),
(2.6)
SUP eEZd
ll+lLT
Second, we define a, : aPd -+ (2.7) Vol. 33, no 5.1997.
aa
I~,(l,e)
- Pa&e)]
Ill-klllR
Rd @ IWd by
= c pcy(k, e)e @ e, eEZ”
k E Zd.
= 0.
638
D. W. STROOCK
AND
W. ZHENG
Finally, we suppose that there exists a Bore1 measurable a : Rd --+ R” @ Rd with the property that, for each r’ > 0:
(2.8)
lirn ,/ ,,1,I4xl(I) - 4x)/ dx= 0. cu~o J x 1
As a consequence of (l.l), (2.1), and (2.8), it is an easy matter to check that, without loss in generality, we may assume that 21 5 u(x) < RI for all x E Rd. In particular, the general theory (cf. either 133 or [8]) of Dirichlet forms applies and says that there is a unique strongly continuous, Markov semigroup {p: : t > 0) self-adjoint contractions on L2(Rd; R) with the properties that, for each f E L2 (Rd; R), t E (0, co) -+ p,” f is the unique continuous function t E (0, oc) I--+ ut E W,l(Rd; R) (the Sobolev space of square integrable functions with one square integrable derivative) such that
(2.9)
$(p;ut)o=-/ (VywVut),,dx,cp=T’(Rd;$ andRd ut --+ f in L2(Rd; R).
In fact (cf. [9]) there is a unique pa E C((0, co) the properties that
x
Rd
x
Rd; (0, co)) with
(2.10) Pa(%x,Y)=Pa(S,Y,x),
J
p”(s,x,t)d< Rd
P%+t,x,Y)=
(SAY)
E w4x~dx~d~
= 1, (s,x) E (0,m)
p"(s,x,~)p"(t,~,y)d~, .IWd
s,t E b4,
x IWd,
X,Y E Rd,
and, for each f E C,(W”; W), (2.11)
p:f
= PFf
a.e.
where P,“!(x)
E
s
Rd UP%
x, Y) &.
What we want to show is that, after the addition of (2.6) and (2.81, we can show that the 4 in (2.3) must be pa, and for this purpose we will need a few R, let $;, denote the function preparations. Namely, given tiO : aZd on Rd obtained by baricentric extension. That is, for each k E Zd, set (2.12)
MARKOY
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To
SYMMETRIC
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DIFFUSIONS
and define Gcy on Qa (k) to be the multilinear extension of $J~ restricted to the extreme points of se(k). The crux of our argument is contained in the following lemma. 2.13
LEMMA.
-
For each Q! E (0, I], let $‘n : aZd --+ Iw be given, and
assume that (2.14)
sup ad-2 1
nE(O,ll
kEZd
c
(b (4k
+ e)) - 4, (ak)))*
< 3~.
Ile(/=l
Zf $a converges to $J in L2(Rd; R), then 4 f VVi(@; W) and
where we have taken (c$ (1.2)) E” = Err+a. Proof - We begin by observing that, for x E Qn(k), IIWCMII 5 (2-l max
{
)+$,a(al’) - $cy(*l)l
: al,&’
E QC2(k) with 111’- 111= l}.
In particular, this means that
and so, by (2.14), the L2-norm of 1V$, ) is bounded uniformly in QI E (0, 11. Hence, from the elementary functional analysis of the space Wi(Rd; W), we know that q!~E Wi (I@; R) and that 04, tends weakly in L2(Rd; Wd) to Vrj; and, after combining this observation with (2.8), we see that it suffices for us to check that
Moreover, because cp E CLY(Rd; W), it is clear from (2.14) that the preceding can be replaced by
Vol. 33, no s-1997
640
D. W. STROOCK
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where ca (k, e) - pm(k, e> (e, Vv4ak))
RdI
and we have introduced the notation &$J(x) s $J(X + v) - 4(x). To prove (2.16) we define a rectilinear, nearest-neighbor path from k to k + e. Namely, for (k,e) E Zd x Zd and 1 5 j < d, let
‘0 if ej = 0, Pj(k,e)
{e : l; = ki + ei for i < j, kj 5 l?j < kj + ej, and&=kifori>j} {l : li = k; + ei for i < j, kj - ej < aj < 5, and&=k;fori>j}
=
j=l
ew,
ifej>l ifej<--1.
(k,e)
where ‘h&‘(x)
3~((21,...,Zj-l,ICj+Q,2j+l,...,Zd))
-4(x);
and so ad-l
c
c
kE2d
=
ad-l
+
add1
c,(k,e)&,&(~k)
eE2d
C
C
lEZd
eELd
c,(l,
c
c
1~2~ eE2d
e) 2
ej&+&(4
j=l
2
sgn(ej)&,ghdl)
c
j=l
(dk
e)
-
41,
e)).
{k:lEPj(k,e)}
Thus, because of (2.6), (2.14), and the fact that cp E Cr(Wd; R), we see that (2.16) reduces to
Finally, to prove (2.17), observe that
g(x) sQa(k)
dx = (;)^-l
Annales
de I’lnstitut
c
h&~&l),
lEQ(kld Hew-i PoincarP
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et Statistiques
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TO SYMMETRIC
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641
where Q(k]j)
s {k : lj = kj and li E {AY;,Ic; + 1) for %# j}.
Hence, after elementary manipulation,
j=l
1EZd e# =
c kEZ”
c e6Zd
we see that
cake)/
(e,V&(x))Rddx Qm (k)
Thus, because of (2.6) all that remains is to note that
By combining Lemma 2.2 and 2.13, we get the our main result. 2.18 THEOREM. - Let a : Rd ---+ Wd@BBdbe a Bore1 measurable, symmetric matrix valued function which satisfies (2.19)
21 I a(x) 5 RI,
x E IWd:
for some R E [l,oo), and determine pa : (0,~) x R” x Wd ---+ (O?x) accordingly, as in (2.11). Also, for each cy E (0, 11, let pa : aZd x Zd --s [0, W) be a function which satisjies the conditions in (1.1) and (2.1). Zj in addition, (2.6) and (2.8) hold, then (2.20)
lim sup sup t4,.PIY-XIIpU(t,[xi,,[Y],) aLO tE(O,m) x,yERd
-f(t~~.Y)l
=O.
where p E (0, CXJ)depends only on d and R (c$ (2.1) and (2.6)). Proof. - In view of Lemma 2.2 and the discussion which follows its proof, we need only check that (cf. (2.3)) q = p”. Moreover, starting from the characterization of {p,” : t > 0} given in (2.9) and using the continuity Vol. 33, no 5.1997.
642
D. W. STROOCK
AND
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of $3 one can apply standard soft arguments to see that it suffices for us to prove that, for each f E Cy(W”; R) and X E (0; oc),
and
when
cc %Lxz
J0 To this end, define ‘zb+ : aZd UA,a =
.I”0
e-‘“Qtf
dt.
W by e-XtPF’P”f
dt.
From the spectral theorem, it is clear that E”( %r%)
I f
llfll:,,.
Hence, because of (1. l), we know that (2.14) holds with $, = UX,~. Moreover, from (2.3), it is an easy step to UX,~ UA in L2(Rd; R). Thus, by Lemma 2.13, UA E W,‘(Rd;R) and
3. CONSTRUCTION
OF MARKOV
CHAIN
APPROXIMATIONS
Theorem 2.18 provides us with a criterion on which to base the construction of Markov chain approximation schemes. The only ingredient which is still missing is a procedure for going from a given coefficient matrix a to a family {pal : CKE (0, l]} which satisfies the hypotheses of that theorem. Thus, Iet a Bore1 measurable a : Wd Rd @ W” symmetric matrix-valued function satisfying (3-l)
MARKOV CHAIN APPROXIMATIONS TO SYMMETRIC DIFFUSIONS
643
be given. We want to construct an associated family (pa : Q E [0, 11) which satisfies (l.l), (2.1), (2.6), (2.8), and therefore, by Theorem 2.18, (2.20). An important role in our construction will be played by the following simple application of elementary linear algebra. Here, and elsewhere, elements of Wd will be thought of as column vectors. In particular, if el....,ed are given elements of I@, then [e’. . . . . e”] will be the d x d-matrix whose jth column is ej. 3.2 LEMMA. - Given a symmetric S E ‘&!’ @ Rd, there exist linearly independent el, . . , ed E Zd such that maxlljjd [l&l/ 5 32d2(IISII,, V 1) stands for the operator norm), and (II . II”,’
when E-’
= [el,...,ed].
Proof. - Set R = 32d2 ( ]]S]]Or, V 1). Choose an orthogonal matrix 0 so that A z OTSO is diagonal, determine ej E Zd so that, for 1 5 i < d, ei is the integer part [RO;,j] of RO;,j, and set B = R-l [el, . . , e”] . We must show that B is invertible and
For this purpose, set A = 0 - I?. Then ]]A]]op B = (1 - AOT)O. Thus, B is invertible and
L3-l = OT(I - AOT)-l = OT + H
< j$ 5
i
and
where ]]H]lop < g.
In particular, IIB-lS(B-l)T
- RI]
< 8d];f]]0p
OP -
Hence, on the one hand,
while, on the other hand,
and
I]B-l(B-‘)T
- 111 5 2. OP
R
644
D. W. STROOCK
AND
W. ZHENG
At this point it is useful to make a distinction between the cases when n is assumed to be continuous and when no continuity is assumed. The next statement is a more or less immediate consequence of Lemma 3.2 plus elementary point-set topology. Indeed, when a is continuous, Lemma 3.2 makes it clear that each x E Rd admits a neighborhood U and a choice of basis e’,... ,ed E Zd such that max15j5d ((ej]] 5 32d’M and E=
[el,...,ed]-'
j
cl j#i
(E(o(y)-2I)E’)
ci
/i
for all 1 5 i 5 d and y E U. Hence, the following the preceding and the fact that Rd is a-compact.
(E(dY)-qqi
statement comes from
3.5 LEMMA. - If a is continuous, then there is a countable, locally jinite, open cover {Un}r of Rd such that, for each n E N, a t U, is uniformly continuous and there exists a basis el,n, . . . , ed,n such that maxl<_j$j Ilejanll 5 32d2M and, for each 1 5 i 5 d and x E U,, (3.6)
~l(E,(a(x) j#i
- 21)E;),,jl
2 (E&(x)
- 21)EnT)i,i
C” (Rd; [0, 11) to be a partition of unity En(x) = E,-‘(a(x)
( 3.7)
and, for each n, define on : Rd x Zd -
1
- 21)E,T, [0, oo) by
when x E U,, i # j, and e = (ei+ + ej’“) when x E U,, i # j, and e = (ei)n - ej’“)
f~,(x,e) =
when x E U, and e = eiJ’ 0 when x +! U,. Finally, (3.8)
set p,(k,e)
= 111) (IHI) + c
nn(k,e). n
Ann&s
de l’hstiruf
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3.9 THEOREM. - Zf a : Rd ---+ Rd x Rd is a continuous, symmetric matrix valued function which satisfies (3.1) and if {pu : CI E (0, 11) is determined by the prescription in (3.8), then Theorem 2.18 applies and so (2.20) holds. Proof. - There is nearly nothing to do. Indeed, since each of the gn ( . , e)’ s is continuous and vanishes identically when ]]e]] > 64d2M, it suffices to observe that, by construction, c, g,(x, e) = a(x) - 21 for all x E Rd. 0 The situation when a is .not continuous is hardly different. The only substantive change is that one must begin by doing a little smoothing. For example, choose some $ E Cy (R’; [0, M)) with total integral 1, and set (3.10)
a,(x)
= (u-4
.I
Rd $(a-?(y-x))
a(y) dy, Q E (0, l] and x E Rd.
Then
Ikbb) - aa(x)Ilop I C~-~lly- XII, x, y E BBd
(3.11)
Hence, by Lemma 3.2, one can find an aa E (0, l] such that, for each a: E (0, oO] and 1 E Zd, there is a basis {el~(cu~l),. . . ,ed)(ay’)} in Wd with (cf. (2.12)) max ]]ej1(a71)]] 5 32d2M and, for each 1 < i < d, l
- 2I)El,,,
x E Qa(l).
[O,l]) so that v z 1 on [-$, i], take
ii(t) = rl(t - 1) + v(t) rl(t) + rl(t - 1)
and
,rjQ,l(x) = fi
fj(Cax~
- lj).
j=l
and define O,,J : Wd x Zd ----+ [0, cc) so that, when x E Q,+(l), ‘ *
(T,,I(X,e)
(qx);,j) + if i # j & e = (ei,Ccy,l)+ &,Ca,l))
= ( ~(%1(4i;j)if i # j & e = (ei,Cn,I) - e.i,Cm~I)) %,1(x) (%,I(x)i,i \
Vol. 33, no 5.1997
-
CI%,l(X)L,jl)
j#i if e = ei,(cxy,l)
646
D. W. STROOCK
AND
W. ZHENG
and gol,t(x, .) = 0 when x $ Qn+ (1). Finally, set (3.13)
pm@, e) = l{l} (Hell) + c
~a~a,,l(k
e).
3.14 THEOREM. - Given a Bore1 measurable, symmetric a : Wd ---+ Wd @ Rd, define {pa : a f (O,l]} as in (3.13). Then Theorem 2.18 applies and so (2.20) holds.
4. CONVERGENCE OF PROCESSES AND AN APPLICATION TO THE DIRICHLET PROBLEM As a more or less immediate dividend of Theorems 3.9 or Theorem 3.14, we get an accompanying statement about the associated stochastic processes. Namely, for each a! E (0, 11, let R, denote the Skorohod space D([O, 00); oZd) of right continuous, aZd-valued paths on [0, co) with left limits, and note that each R, is a closed subspace of D( [0, co); Rd). Next, let { P$ : k E Zd} be the Markov family of probability measures on R, with transition density p”. Also, let {PC : x E R”} be the Markov family of probability measures on C( [0, cc); Rd) for which p” is the transition density. Because of (1.14) and standard compactness criteria (e.g., ChenZov’s in Theorem 8.8 on page 139 of [2J) for the Skorohod topology, it is clear that, as a subset of probability measures on 00,
{p:lc: Q E (O,l],
k E Zd, and ]]ok]] 5 T} is tight for each T E (0,co).
At the same time, by (2.14), if a, \ 0 and a,k, --+ x E Wd, then the only limit { Pzzk, } T can have is P,“. Hence, we have now proved the following. 4.1 COROLLARY. - Under the conditions described in either Theorem 3.9 or Theorem 3.14, for any bounded F : Ro -+ R which is continuous and any 7- E (O,oo),
(4.2) In fact, (4.2) continues to hold for bounded F’s on Ro which, for each x E Rd, are Pz-almost surely continuous with respect to the topology of uniform convergence on finite intervals. Proof. - The only point not already covered is the final one. However, because {Px : 1x1 2 T} is compact as probability measures on Anrules
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C([O, m); Bad), it follows from standard facts about the Skorohod topology that the preceding weak convergence result self-improves to cover functions which are continuous with respect to the topology of uniform convergence on finite intervals. 0 We conclude with an application of the preceding to the Dirichler problem for L” in a bounded, connected region 6 c Rd. That is, given an f E C(d6; W) we seek a function uaf E C(a; R) with the properties that:
(4.3)
Set
E [0, c-31: w 1 [O,t] p S}
E [0, cm] : w 1 [O, t] g S}.
and Because 6 is bounded, (1.16) can be applied to check that there are T E (0, CQ) and E E (0,l) with the property that
fzk(C~ I T) 2 6 for all n E (0, l] and ok E 3. Hence, by a familiar argument, there exists another E > 0 such that sup sup EP-“k ~E(O,ll a&~
[tfC~ 1
Similarly, from the lower bound in (1.0.10) of 191, one knows that
for a suitable choice of t > 0. Obviously Ce 2 (3 always. In addition, Ce and cg are, respectively, lower and upper semicontinuous with respect to the topology of uniform convergence on finite intervals. Finally, the lower bound from (I. 1.lO) in [9] leads (cf. part (iii) of Exercise 3.2.38 in [lo]) to (4.4)
P,a(&=(z)
=l,
XERd,
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4.6 THEOREM.- Assume that (4.4) holds. Given f E C(d6;
W), choose
f E Cb(@; W) so that f t dC5 = f. Then (cc (4.1)) q(x)
(4.7)
= ;io-,
[f(L&,),
&5(w) < cm],
x E 6.
In fact, the convergence in (4.4) takes place uniformly on compact subsets of 6. Proof. - In view of Corollary 4.1 and the preceding discussion, the derivation of (4.4) comes down to checking that qx>
(4.8)
= P,” [f(42)).
b(4
< m],
x E 6.
Indeed, (4.4) guarantees that f(~(&)) is P;-almost surely continuous in a topology with respect to which we know that Pg10 is converging weakly to P,. To prove (4.8), one needs to use three facts. First, (4.8) is essentially trivial when a is smooth. Second, the convergence result in Theorem 11.3.1 together with the tightness which comes from the upper bound in (1.1.10) of [9] and the continuity discussion just given, show that w-
[f(w(W),
42 < -1 uniformly
-
EPZ [f(w(Cd),
43 < m]
on compacts in t5
if {a,}y is a sequence of coefficients matrices satisfying (3.1) and tending to a in measure. Third, one must know that, under the same conditions, UT UT uniformly on compacts, a conclusion which can be easily drawn from the facts in [9] and is explicitly contained in results of [7] or $5 of Chapter II in [6]. 0 Remark. - For actual computation, (4.9)
lEPZk[f(4)),
it may be useful to observe that
63 < m] =
EQZk [f(4Cd),
C@< a] >
where {Qzk : k E Z”} is the discrete-time parameter Markov family on ~2’~ with transition probability III”(ak,&)
= R”(k)-i(p,(k,&
k) +/~&k-1)),
when R”(k)
= c (pm@, e - k) + ~a(4 k - e,). tad Indeed, one can construct Qzk from P$ by simply recording the size of the jumps and taking the time between successive jumps to be 1. Hence, the distribution of unparameterized trajectories is the same under P$ and QEk. In particular, the distribution of ~(~~)l(~,~~(~~) is the same under Q:: as it is under P,“. Ann&s
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REFERENCES 111 E. CARLEN, S. KUSUOKA and D. STROOCK, Upper bounds for symmetric Markov transition functions, Ann. Inst. H. Poincur&, Vol. 35, Sup. au #2, 1987, pp. 245-287. [2] S. ETHIER and T. KURTZ, Murkov Processes, J. Wiley & Sons, 1986. [3] M. FIJKUSHIMA, Y. OSHIMA and M. TAKEDA, Dirichlet Forms and S.vmmetric Murkol Processes, De Gruyter Series in Math., W. De Gruyter, 1994. (41 E. FABES and D. STROOCK, The De Giorgi-Moser Harnack principle via the old ideas of Nash, Arch. Ratl. Mech. & Anal., Vol. 96, #4, 1987, pp. 327-338. 151 I. I. GIHMAN and A. SKOROHOD, The Theoy of Stochastic Processes II. Springer-Verlag, Grund. #218, 1975. 161 D. KINDERLEHRER and G. STAMPACCHIA, An Introduction to Variational Inequalities and their Applications, Academic Press, Pure & Appl. Mach., #88, 1980. [7] W. LI~TMAN, G. STAMPACCHIA and H. WEINBERGER. Regular Points for elliptic equations with discontinuous coefficients, Ann. Scuola Norm. Sup Piss, #17, 1963, pp. 43-77. [8] Z.-M. MA and M. R~CKNER, Dirichlet Forms, Springer-Verlag, Universitext Series, 1992. 191 D. STROOCK, D&ion semigroups corresponding to uniformly elliptic divergence ,form operufors, Stminaire de ProbabililitC XXII (J. Azema & M. Yor, ed.), Springer-Verlag LNMS #1321, 1988, pp. 316-348. [IO] D. STROOCK, Probability Theory, An Analytic View, Cambridge Univ. Press, 1993. [ 111 D. STROOCK and S. R. S. VARADHAN, Multidimensional D@sion Theory, Springer-Verlag. Grundlehren 233, 1979. (Munuscript received May 14, 1996: Revised December 12. 1996.)
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