The zero‑mass limit problem for a relativistic spinless particle in an electromagnetic field
著者 Ichinose Takashi, Murayama Taro journal or
publication title
Proceedings of the Japan Academy Series A:
Mathematical Sciences
volume 90
number 3
page range 60‑65
year 2014‑01‑01
URL http://hdl.handle.net/2297/37854
The zero-mass limit problem for a relativistic spinless particle in an electromagnetic field
By Takashi ICHINOSE and Taro MURAYAMA
Division of Mathematical and Physical Sciences, Graduate School of Natural Science and Technology, Kanazawa University, Kakuma-machi, Kanazawa 920-1192, Japan
(Communicated by Masaki KASHIWARA,M.J.A., Feb. 12, 2014)
Abstract: It is shown that mass-parameter-dependent solutions of the imaginary-time magnetic relativistic Schro¨dinger equations converge as functionals of Le´vy processes represented by stochastic integrals of stationary Poisson point processes if mass-parameter goes to zero.
Key words: Magnetic relativistic Schro¨dinger operator; imaginary-time relativistic Schro¨dinger equation; Le´vy process; path integral formula; Feynman-Kac-Itoˆ formula.
1. Introduction and results. Kasahara- Watanabe [12] discussed limit theorems in the frame- work of semimartingales represented by stochastic integrals of point processes. In fact, they considered a sequence of point processes and their certain func- tionals represented by stochastic integrals, and proved their convergence in that context.
In this paper we treat a sequence of a slightly more general functionals of special kind of Le´vy processes, which have no Gaussian part stemming from relativistic quantum mechanics, to discuss its convergence. Naturally we have in mind the following relativistic Schro¨dinger equation which describes a spinless quantum particle of mass m >
0 (for example, pions) in Rd under the influence of the vector and scalar potentialsAðxÞ; VðxÞ:
i@
@t ðx; tÞ ¼ ½HAmmþV ðx; tÞ ðt >0Þ; ð1:1Þ
wherex2Rd. In this paper, to see the main idea, we only consider the case thatA2C01ðRd;RdÞand V 2C0ðRd;RÞ. Here thenHAm is defined by
ðHAmfÞðxÞ:¼Os- 1 ð2Þd
Z Z
RdRd
eiðxyÞ
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Aðxþy2 Þ
2þm2
q
fðyÞdyd for f 2C10 ðRdÞ, where ‘‘Os’’ means oscillatory integral. HAm is called the Weyl pseudo-differential operator with mid-point prescription, correspond- ing to the classical relativistic Hamiltonian
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi jAðxÞj2þm2
p . It is essentially selfadjoint in
L2ðRdÞon C01ðRdÞand bounded from below by m ([5], [10]). We have H0m¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
þm2
p for A0,
whereis the Laplacian inRd. The light velocity c, electric charge e and Planck’s constant h are taken to be 1, 1 and2 respectively.
The operatorHAmmþV was first studied in [9]
by one of the authors of this paper to treat thepure imaginary-timerelativistic Schro¨dinger equation
@
@tuðx; tÞ ¼ ½HAmmþVuðx; tÞ ðt >0Þ;
ð1:2Þ
where x2Rd. An imaginary-time path integral formula was given on path spaceD0to represent the solution of the Cauchy problem for (1.2). HereD0is the set of the right-continuous paths X:½0;1Þ ! Rd with left-hand limits andXð0Þ ¼0.
We use the probability spaceðD0;F; mÞtreated in [9] with the natural filtration fFðtÞgt0, where FðtÞ:¼ðXðsÞ;stÞ F. fXðtÞgt0 is Le´vy proc- ess, namely, it has stationary independent increments and is stochastically continuous (cf., [11], [15], [1]).
mðX;XðtÞ 2dyÞis equal tokm0ðy; tÞdy, wherekm0ðy; tÞ is the integral kernel of the operator etðpffiffiffiffiffiffiffiffiffiffiffiffiffiþm2mÞ and has an explicit expression
km0ðy; tÞ ð1:3Þ
¼ 2 m2 ðdþ1Þ=2temtKðdþ1Þ=2ðmðjyj
2þt2Þ1=2Þ
ðjyj2þt2Þðdþ1Þ=4 ; m >0,
ððdþ1Þ=2Þ ðdþ1Þ=2
t
ðjyj2þt2Þðdþ1Þ=2; m¼0.
8<
:
Here K stands for the modified Bessel function of the third kind of order.
The characteristic function ofXðtÞis Em½eiXðtÞ ¼etð
ffiffiffiffiffiffiffiffiffiffiffiffi
jj2þm2
p mÞ; 2Rd; ð1:4Þ
doi: 10.3792/pjaa.90.60
#2014 The Japan Academy
2010 Mathematics Subject Classification. Primary 60G51;
Secondary 60F17, 60H05, 35S10, 81S40.
where Em denotes the expectation over D0 with respect tom. By theLe´vy-Khintchine formula,
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi jj2þm2
p m
ð1:5Þ
¼ Z
jyj>0
ðeiy1iy1jyj<1ÞnmðdyÞ:
Here nmðdyÞ is the Le´vy measure, that is a - finite measure on Rdn f0g satisfying R
jyj>0ð1^ jyj2ÞnmðdyÞ<1, and having density
nmðyÞ ¼nmðjyjÞ ð1:6Þ
¼
2ð2mÞðdþ1Þ=2Kðdþ1Þ=2ðmðjyjÞ
jyjðdþ1Þ=2 ; m >0,
ððdþ1Þ=2Þ ðdþ1Þ=2
1
jyjdþ1; m¼0.
8<
:
As shown in [5], HAm has another expression connected with the Le´vy measurenmðdyÞ
ðHmAfÞðxÞ ¼mfðxÞ lim
r#0
Z
jyjr
½eiyAðxþ12yÞfðxþyÞ fðxÞnmðdyÞ:
ForX2D0, letNXðdsdyÞbe a counting measure onð0;1Þ ðRdn f0gÞdefined by
NXðEÞ:¼#fs >0;ðs; XðsÞ XðsÞÞ 2Eg for E2Bð0;1Þ BðRdn f0gÞ, where Bð Þ are -algebras of Borel sets.NXðdsdyÞis the stationary Poisson random measure with intensity measure dsnmðdyÞ with respect to m. Let NgXmðdsdyÞ:¼ NXðdsdyÞ dsnmðdyÞ. By the Le´vy-Itoˆ theorem,
XðtÞ ¼ Z t
0
Z
jyj1
yNXðdsdyÞ ð1:7Þ
þ Z t
0
Z
0<jyj<1
ygNXmðdsdyÞ:
Here and below, we should understandRt 0 :¼R
ð0;t. It can be proved that the solution of (1.2) with initial dataumðx;0Þ ¼gðxÞis given by
umðx; tÞ:¼Em½eSmðt;x;XÞgðxþXðtÞÞ;
ð1:8Þ
SmðÞ:¼iYmðt; x; XÞ þ Z t
0
VðxþXðsÞÞds;
ð1:9Þ
YmðÞ:¼ Z t
0
Z
jyj1
AðxþXðsÞ þ12yÞ yNXðdsdyÞ þ
Z t 0
Z
0<jyj<1
AðxþXðsÞ þ12yÞ ygNXmðdsdyÞ þ
Z t 0
ds Z
0<jyj<1
½AðxþXðsÞ þ12yÞ
AðxþXðsÞÞ ynmðdyÞ:
In (1.7) and (1.9) above, the integration regions jyj 1 and 0<jyj<1 may be replaced by jyj and 0<jyj< respectively, for any >0.
We note that these relativistic quantities, HAmmþV, ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
jj2þm2
p m, D0, m, km0ðy; tÞ and XðtÞ, correspond to the nonrelativistic ones
1
2mðir AÞ2þV, jj2m2, C0, Wiener measure, the heat kernel ð2tmÞd=2em2tjyj2, Brownian motion BðtÞ, respectively. Here C0 is the space of continuous paths B:½0;1Þ !Rd with Bð0Þ ¼0. Furthermore, (1.8) with (1.9) is what does correspond to Feynman-Kac-Itoˆ formula([16]).
The purpose of this paper is to answer the following question:
(Q) When the mass m >0 of the particle becomes sufficiently small, how does its property vary?
Theorem 1. m converges weakly to 0 as m#0.
Theorem 2. umð; tÞ converges to u0ð; tÞ on L2ðRdÞasm#0, uniformly on½0; T.
Here and below, 0< T <1 can be taken ar- bitrary. Theorem 2 implies the strong resolvent convergence of HAmmþV to HA0 þV ([13, IX, Theorem 2.16]). An immediate consequence is the following result for the solution mðx; tÞ of the Cauchy problem for (1.1).
Corollary 1. mð; tÞconverges to 0ð; tÞon L2ðRdÞasm#0, uniformly on½0; T.
We will prove Theorem 2 by using following:
Theorem 3. umð; tÞ converges to u0ð; tÞ on C1ðRdÞasm#0, uniformly on½0; T, whereC1ðRdÞ is the space of the continuous functionsg:Rd !C with jgðxÞj !0 as jxj ! 1 with norm kgk1 :¼ supx2RdjgðxÞj.
The crucial idea of proof is to do a change of variable ‘‘path’’. In Sections 2, 3 and 4, these theo- rems are shown by probabilistic method, although one can more easily show Theorem 2 by operator- theoretical one [6], and also by pseudo-differential calculus [14]. In this paper, as we mentiond before, we treat the problem under a rather mild assumption on the potentials AðxÞ; VðxÞ. We will come to more general case in a forthcoming paper, together for the other two different magnetic relativistic Schro¨dinger operators ([7], [8]) corresponding to the same classical relativistic Hamiltonian. Another limit problem when the light velocitycgoes to infinity (nonrelativ- istic limit) was studied in [4].
2. Proof of Theorem 1. We observe the
No. 3] Zero-mass limit problem for a relativistic spinless particle 61
following three facts which imply Theorem 1 ([2, Theorem 13.5]): (i) The finite dimensional distribu- tions with respect to m converge weakly to those with respect to 0 as m#0. (ii) For any t >0, 0ðX;XðtÞ Xðt"Þ 2dyÞconverges weakly to Dir- ac measure concentrated at the point02Rdas"#0.
(iii) There exist constants >12, >0 and a non- decreasing continuous functionF on½0;1Þsuch that
Em½jXðsÞ XðrÞj jXðtÞ XðsÞj
½FðtÞ FðrÞ2; 0< m <1; 0r < s < t:
Proof. (i) follows from (1.4), and (ii) from the stochastic continuity of fXðtÞgt0. (iii) Since
d
dKðÞ ¼ K1ðÞ ( >0; >0) ([3, (21), p. 79]) and 7!KðÞ is strictly increasing in ð0;1Þ ([3, (21), p. 82]), we have ðd=dÞðeKðÞÞ ¼ eðKðÞ K1ðÞÞ<0 if 0< < 12. Therefore 7!eKðÞ is strictly de- creasing inð0;1Þand so [3, (41), (42), (43), p. 10]
eKðÞ lim
#0KðÞ ¼21ðÞ:
ð2:1Þ
Then we have for0r < s < t, 12 < <1, Em½jXðsÞ XðrÞj jXðtÞ XðsÞj
¼ Z
jyj km0ðy; srÞdy Z
jyj km0ðy; tsÞdy
¼Cðd; Þ2ððsrÞðtsÞÞ emðsrÞðmðsrÞÞ1 2 K1
2
ðmðsrÞÞ emðtsÞðmðtsÞÞ1 2 K1
2
ðmðtsÞÞ Cðd; Þ22ð1þ2 Þð1 2 Þ2ðtrÞ2 ;
where in the second equality we use [4, Lemma 3.3(ii)] with a constantCðd; Þdepending ondand . Therefore (iii) holds for12< <1and¼ and FðpÞ:¼Cðd; Þ1= 2ð1þ2 Þ=2 ð1 2 Þ1= p.
3. Proof of Theorem 2. We will prove Theorem 2 by assuming validity of Theorem 3. In this and the next section, we assumeV 0without loss of generality, since in the general case, we have only to replaceV in (1.8), (1.9) byV infV 0.
Step I: Letg2C01ðRdÞ. For R >0, we have kumð; tÞ u0ð; tÞk2 kumð; tÞ u0ð; tÞkL2ðjxj<RÞ
þ kumð; tÞ u0ð; tÞkL2ðjxjRÞ
¼:I1ðt; m; RÞ þI2ðt; m; RÞ: From Theorem 3, I1ðt; m; RÞ converges to zero as m#0 uniformly ontT. From (1.8), we have
I2ðt; m; RÞ kumð; tÞkL2ðjxjRÞþ ku0ð; tÞkL2ðjxjRÞ
Z
jxjR
dx Z
km0ðy; tÞjgðxþyÞj2dy
!12
þ Z
jxjR
dx Z
k00ðy; tÞjgðxþyÞj2dy
!12
¼:Jðt; m; RÞ þJðt;0; RÞ:
Let be a nonnegativeC10 ðRdÞfunction such that ðxÞ ¼1 if jxj 12 and ¼0 if jxj 1. Put hðxÞ ¼ jgðxÞj2. Since1jxj<RðRxÞ, we have
Jðt; m; RÞ2
Z
ð1ðRxÞÞdx Z
km0ðy; tÞhðxþyÞdy
¼ 1 ð2Þd
hð0Þb Z
1exp t
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
jj2 R2 þm2 q
m
ðÞdb þ Z
ðhð0ÞÞ b hðbRÞÞ exp t
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
jj2 R2þm2 q
m
b ðÞd
;
which converges to zero as R! 1 uniformly on tT and 0m1. Here, for ’2SðRdÞ, ’b is the Fourier transform of ’ given by ’ðÞ ¼b Reix’ðxÞdx(2Rd).
From (1.3) and (2.1), it follows thatkm0ðy; tÞ ! k00ðy; tÞ as m#0, and then Jðt;0; RÞ2 lim infm#0Jðt; m; RÞ2 by Fatou’s lemma. Therefore we have Theorem 2 for this step.
Step II: Let g2L2ðRdÞ. There is a sequence fgng C01ðRdÞ such that gn!g in L2ðRdÞ as n! 1.
Put umnðx; tÞ:¼Em½eSmðt;x;XÞgnðxþXðtÞÞ. Then we have
kumð; tÞ u0ð; tÞk2
kumð; tÞ umnð; tÞk2þ kumnð; tÞ u0nð; tÞk2
þ ku0nð; tÞ u0ð; tÞk2
2kgngk2þ kumnð; tÞ u0nð; tÞk2: By Step I, we have
lim sup
m#0
sup
tT kumð; tÞ u0ð; tÞk2 2kgngk2; which converges to zero asn! 1.
4. Proof of Theorem 3. From (1.8), we have to prove that
umðx; tÞ ¼Em½eSmðt;x;XÞgðxþXðtÞÞ
!E0½eS0ðt;x;XÞgðxþXðtÞÞ ¼u0ðx; tÞ as m#0 in C1ðRdÞ. But its direct proof seems difficult since both the integrand eSmðt;x;XÞgðxþ
XðtÞÞand the probability measuremdepend onm.
So we changeEm½ toE0½ by achange of vari- able(i.e.,change of probability measure)m¼01m with path space transformation m:D0! D0. If there is such am, we can see by (1.4) and (1.5) that the difference between the pathXðtÞand the trans- formed pathmðXÞðtÞ is expressed in terms of the difference between the two Le´vy measures n0ðdyÞ and nmðdyÞ, so that it is presumed to hold that nmðdyÞ ¼n01mðdyÞ for some map m:Rdn f0g ! Rdn f0g.
We will determine m in such a way that (1) nmðdyÞ ¼n01mðdyÞ, (2) m2C1ðRdn f0g;
Rdn f0gÞ, (3) m is one to one and onto, (4) detDmðzÞ 6¼0 for all z2Rdn f0g, where DmðzÞ is the Jacobian matrix ofm at the pointz.
Let U:¼ fy2Rdn f0g;jyj 2U0g for U02 Bð0;1Þ. Introducing the spherical coordinates by z¼r!, r >0,!2Sd1, we have
nmðUÞ ¼ Z
U
nmðjyjÞdy¼CðdÞ Z
U0
nmðrÞrd1dr;
whereCðdÞis the surface area of thed-dimensional unit ball.
Let us assume that 1mðzÞ ¼lmðjzjÞjzjz for some non-decreasing C1 function lm:ð0;1Þ ! ð0;1Þ.
Then we have n01mðUÞ
¼ Z
U
n0ðlmðjzjÞÞjzjðd1ÞlmðjzjÞd1l0mðjzjÞdz
¼CðdÞ Z
U0
n0ðlmðrÞÞlmðrÞd1l0mðrÞdr;
wherel0mðrÞ ¼ ðd=drÞlmðrÞ:Therefore we have nmðrÞrd1¼n0ðlmðrÞÞlmðrÞd1l0mðrÞ; a.s.r >0:
Ifm >0, from (1.6), we have
d
drlmðrÞ1¼2d12 ðdþ12 Þ1mdþ12 rd32 Kdþ1 2 ðmrÞ:
We solve this differential equation under boundary conditionlmð1Þ ¼ 1to get
lmðrÞ ¼ 2d12 ðdþ12 Þ mdþ12 R1
r ud32 Kdþ1 2
ðmuÞdu ð4:1Þ :
Here we note that 0<R1
r ud32 Kdþ1
2 ðmuÞdu <1by Kdþ1
2 ðÞ>0for >0, and [3, (37), (38), p. 9]
Kdþ1 2
ðÞ ¼ 2
1=2
1=2eð1þoð1ÞÞ; " 1:
Proposition 1. (i)lmðrÞis a strictly increas- ing C1 function of r2 ð0;1Þ and lmðþ0Þ ¼0, lmð1Þ ¼ 1.
(ii) For all r >0, lmðrÞ converges to r, strictly decreasingly, as m#0.
Proof. (2.1) implies lmðþ0Þ ¼0. The other claims of (i) follow from (4.1) and the fact that Kðdþ1Þ=2ðÞ is a C1 function in ð0;1Þ. The claim (ii) can be proved by the fact that KðÞ is strictly decreasing inð0;1Þ(cf. Section 2, Proof of (ii)), (2.1) and the monotone convergence theorem.
Ifm¼0, letl0ðrÞ:¼r. Let us put0ðzÞ:¼zand form >0,
mðzÞ:¼l1mðjzjÞ z
jzj; z2Rdn f0g:
Then we have
1mðzÞ ¼lmðjzjÞ z
jzj; z2Rdn f0g:
We note that
mðzÞ !z; jmðzÞj ¼l1mðjzjÞ " jzj ð4:2Þ
as m#0by Proposition 1 (ii).
Let us define0ðXÞ:¼X and form >0, mðXÞðtÞ:¼
Z t 0
Z
jyj1
yNXðds1mðdyÞÞ ð4:3Þ
þ Z t
0
Z
0<jyj<1
ygNX0ðds1mðdyÞÞ
¼ Z t
0
Z
jzjlmð1Þ
mðzÞNXðdsdzÞ þ Z t
0
Z
0<jzj<lmð1Þ
mðzÞgNX0ðdsdzÞ
¼ Z t
0
Z
jzj1
mðzÞNXðdsdzÞ þ Z t
0
Z
0<jzj<1
mðzÞgNX0ðdsdzÞ:
Proposition 2. For every sequence fmg withm#0, there exists a subsequencefm0gsuch that
sup
tT
jm0ðXÞðtÞ XðtÞj !0as m0#0; 0-a:s: X2D0: Proof. From (1.7) and (4.3), we have
sup
tT
jmðXÞðtÞ XðtÞj Z T
0
Z
jzj1
jmðzÞ zjNXðdsdzÞ þsup
tT
Z t 0
Z
0<jzj<1
ðmðzÞ zÞgNX0ðdsdzÞ
¼:I1ðm; XÞ þsup
tT
jI2ðt; m; XÞj:
We have I1ðm; XÞ !0 as m#0 by (4.2) and RT
0
R
jzj1jzjNXðdsdzÞ<1. We note that I2ðt; m; XÞ is the L2ðD0;0Þ-limit of the right-continuous fFðtÞgt0-martingale fI2"ðt; m; XÞgt0 with
No. 3] Zero-mass limit problem for a relativistic spinless particle 63
I2"ðt; m; XÞ:¼Rt 0
R
"<jzj<1ðmðzÞ zÞgNX0ðdsdzÞ as
"#0, with convergence being uniform on tT. By taking a subsequence if necessary, I2"ðt; m; XÞ converges toI2ðt; m; XÞas"#0uniformly ontT, 0-a.s., and henceI2ðt; m; XÞis right-continuous on tT, 0-a.s. ([11, p. 73, Proof of Theorem 5.1], [15, p. 128–129, Proofs of Lemmas 20.6, 20.7]). Then we use Doob’s martingale inequality [1] to have
E0 sup
tT jI2ðt; m; XÞj2
4E0hjI2ðT ; m; XÞj2i 4T
Z
0<jzj<1
jmðzÞ zj2n0ðdzÞ;
which converges to zero as m#0 by (4.2) and R
0<jzj<1jzj2n0ðdzÞ<1.
By (1.8) andm¼01m, we have
umðx; tÞ ¼E0½eSmðt;x;mðXÞÞgðxþmðXÞðtÞÞ;
and then sup
tT
kumð; tÞ u0ð; tÞk1 ð4:4Þ
kgk1 sup
tT ;x2Rd
E0½jeSmðt;x;mðXÞÞeS0ðt;x;XÞj þE0 sup
tT
kgð þmðXÞðtÞÞ gð þXðtÞÞk1
:
Since g2C1ðRdÞ is uniformly continuous and bounded on Rd, the second term on the right of (4.4) converges to zero asm#0.
Next we consider the first term on the right of (4.4). ByNmðXÞðdsdyÞ ¼NXðds1mðdyÞÞ, we have Smðt; x;mðXÞÞ
¼i Z t
0
Z
jzj1
AðxþmðXÞðsÞ þ12mðzÞÞ mðzÞNXðdsdzÞ þ
Z t 0
Z
0<jzj<1
AðxþmðXÞðsÞ þ12mðzÞÞ mðzÞgNX0ðdsdzÞ þ
Z t 0
ds Z
0<jzj<1
½AðxþmðXÞðsÞ þ12mðzÞÞ AðxþmðXÞðsÞÞ mðzÞn0ðdzÞ
þ Z t
0
VðxþmðXÞðsÞÞds
¼:i S m1ðt; x; XÞ þS2mðt; x; XÞ þSm3ðt; x; XÞ
þS4mðt; x; XÞ:
By the inequality
jeðiaþbÞeðia0þb0Þj ebjeiaeia0j þ jbb0j for any a; a02R, b; b00, supE0½ of the first term on the right of (4.4) is less than or equal to
E0 sup
tT
keiSm1ðt;;XÞeiS01ðt;;XÞk1
ð4:5Þ
þsup
x2Rd
E0 sup
tTjSm2ðt; x; XÞ S20ðt; x; XÞj
þE0 sup
tT kS3mðt;; XÞ S03ðt;; XÞk1
þE0 sup
tT kS4mðt;; XÞ S04ðt;; XÞk1
:
Now, let fmg be a sequence with m#0 and fm0g any subsequence of fmg. By Proposition 2, there exists a subsequence fm00g of fm0g such that suptTjm00ðXÞðtÞ XðtÞj !0asm00#0,0-a.s.
To prove that each term of (4.5) converges to zero asm00#0, we first note that
S1m00ðt; x; XÞ S01ðt; x; XÞ
¼ Z t
0
Z
jzj1
ðAðxþm00ðXÞðsÞ þ12m00ðzÞÞ AðxþXðsÞ þ12zÞÞ m00ðzÞNXðdsdzÞ þ
Z t 0
Z
jzj1
AðxþXðsÞ þ12zÞ ðm00ðzÞ zÞNXðdsdzÞ:
Then the integrand of the first term of (4.5) is less than or equal to
Z T 0
Z
jzj1
sup
x2Rd
jAðxþm00ðXÞðsÞ þ12m00ðzÞÞ AðxþXðsÞ þ12zÞjjzjNXðdsdzÞ
þsup
x2Rd
jAðxÞj Z T
0
Z
jzj1
jm00ðzÞ zjNXðdsdzÞ; which converges to zero as m00#0 since A2 C10 ðRd;RdÞis uniformly continuous onRd.
Next, since Sm2ðt; x; XÞ is seen to be right- continuous, by Schwarz’s inequality and Doob’s martingale inequality, E0½ of the second term of (4.5) is less than or equal to
2E0 Z T
0
ds Z
0<jzj<1
jAðxþm00ðXÞðsÞ þ12m00ðzÞÞ m00ðzÞ AðxþXðsÞ þ12zÞ zj2n0ðdzÞ
12 :
By the inequality ðaþbÞ2 2ða2þb2Þ for any a; b2R,E0½ above is less than or equal to
2
E0 Z T
0
ds Z
0<jzj<1
sup
x2Rd
jAðxþm00ðXÞðsÞ þ12m00ðzÞÞ AðxþXðsÞ þ12zÞj2jzj2n0ðdzÞ
þTsup
x2Rd
jAðxÞj2 Z
0<jzj<1
jm00ðzÞ zj2n0ðdzÞ ; which converges to zero asm00#0. As for the third term of (4.5), by the mean value theorem, we have
S3m00ðt; x; XÞ S30ðt; x; XÞ ¼1 2
Z t 0
ds Z
0<jzj<1
n0ðdzÞ
Z 1 0
½ðWx;Xm00ðs; Þm00ðzÞÞ m00ðzÞ ðWx;X0 ðs; ÞzÞ zd:
Here Wx;Xm00ðs; Þ and Wx;X0 ðs; Þ are dd matrices defined by
Wx;Xm00ðs; Þ ¼DAðxþm00ðXÞðsÞ þ12m00ðzÞÞ;
Wx;X0 ðs; Þ ¼DAðxþXðsÞ þ12zÞ;
whereDAðÞis the Jacobian matrix ofA. Since ðWx;Xm00ðs; Þm00ðzÞÞ m00ðzÞ ðWx;X0 ðs; ÞzÞ z
¼ ðWx;Xm00ðs; Þm00ðzÞÞ ðm00ðzÞ zÞ þ ððWx;Xm00ðs; Þ Wx;X0 ðs; ÞÞm00ðzÞÞ z þ ðWx;X0 ðs; Þðm00ðzÞ zÞÞ z;
the integrand of the third term of (4.5) is less than or equal to
T sup
x2Rd
kDAðxÞk Z
0<jzj<1
jm00ðzÞ zjjzjn0ðdzÞ þ1
2 Z T
0
ds Z
0<jzj<1
jzj2n0ðdzÞ
Z 1 0
sup
x2Rd
kWx;Xm00ðs; Þ Wx;X0 ðs; Þkd;
wherek kis the norm of matrices. This is less than or equal to 3Tsupx2RdkDAðxÞkR
0<jzj<1jzj2n0ðdzÞ<
1, and converges to zero as m00#0 because each component ofDAis uniformly continuous onRd.
Finally, the fourth term of (4.5) is less than or equal to E0½RT
0 kVð þm00ðXÞðsÞÞ Vð þ XðsÞÞk1ds, which converges to zero asm00#0since V 2C0ðRd;RÞis uniformly continuous onRd. Thus we have suptTkum00ð; tÞ u0ð; tÞk1!0 as m00#0, and hencesuptTkumð; tÞ u0ð; tÞk1!0asm#0.
Acknowledgements. The author (T.I.) is grateful to Prof. Yuji Kasahara for a number of helpful discussions with suggestion on the subject at a very early stage of this work and Prof. Masaaki Tsuchiya for frequent valuable and helpful discus- sions from the beginning. The other author (T.M.) would like to thank Prof. Hidekazu Ito for his kind guidance, many helpful advices and warm encour- agement during the preparation of this work. The authors are indebted to the anonymous referee for valuable comments and suggestions.
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