The zero-mass limit problem for a relativistic spinless particle in an electromagnetic field

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 R^d under the influence of the vector and scalar potentialsAðxÞ; VðxÞ:

i@

@t ðx; tÞ ¼ ½H_A^mmþV ðx; tÞ ðt >0Þ; ð1:1Þ

wherex2R^d. In this paper, to see the main idea, we only consider the case thatA2C₀¹ðR^d;R^dÞand V 2C0ðR^d;RÞ. Here thenH_A^m is defined by

ðH_A^mfÞðxÞ:¼Os- 1 ð2Þ^d

Z Z

R^dR^d

e^iðxyÞ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Að^xþy₂ Þ

²þm²

q

fðyÞdyd for f 2C¹₀ ðR^dÞ, where ‘‘Os’’ means oscillatory integral. H_A^m 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

L²ðR^dÞon C₀¹ðR^dÞand bounded from below by m ([5], [10]). We have H₀^m¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

þm²

p for A0,

whereis the Laplacian inR^d. The light velocity c, electric charge e and Planck’s constant h are taken to be 1, 1 and2 respectively.

The operatorH_A^mmþV was first studied in [9]

by one of the authors of this paper to treat thepure imaginary-timerelativistic Schro¨dinger equation

@

@tuðx; tÞ ¼ ½H_A^mmþVuðx; tÞ ðt >0Þ;

ð1:2Þ

where x2R^d. 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 ! R^d with left-hand limits andXð0Þ ¼0.

We use the probability spaceðD0;F; ^mÞtreated in [9] with the natural filtration fFðtÞg_t0, where FðtÞ:¼ðXðsÞ;stÞ F. fXðtÞg_t0 is Le´vy proc- ess, namely, it has stationary independent increments and is stochastically continuous (cf., [11], [15], [1]).

^mðX;XðtÞ 2dyÞis equal tok^m₀ðy; tÞdy, wherek^m₀ðy; tÞ is the integral kernel of the operator e^tðpffiffiffiffiffiffiffiffiffiffiffiffiffi_þm2mÞ and has an explicit expression

k^m₀ðy; tÞ ð1:3Þ

¼ 2 ^m₂ ^{ðdþ1Þ=2temtKðdþ1Þ=2ðmðjyj}

2þt²Þ¹⁼²Þ

ðjyj²þt²Þ^ðdþ1Þ=4 ; m >0,

ððdþ1Þ=2Þ ^ðdþ1Þ=2

t

ðjyj²þt²Þ^ðdþ1Þ=2; m¼0.

8<

:

Here K stands for the modified Bessel function of the third kind of order.

The characteristic function ofXðtÞis E^m½e^iXðtÞ ¼e^tð

ffiffiffiffiffiffiffiffiffiffiffiffi

jj²þm²

p mÞ; 2R^d; ð1:4Þ

doi: 10.3792/pjaa.90.60

#2014 The Japan Academy

2010 Mathematics Subject Classification. Primary 60G51;

Secondary 60F17, 60H05, 35S10, 81S40.

where E^m denotes the expectation over D0 with respect to^m. By theLe´vy-Khintchine formula,

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi jj²þm²

p m

ð1:5Þ

¼ Z

jyj>0

ðe^iy1iy1_jyj<1Þn^mðdyÞ:

Here n^mðdyÞ is the Le´vy measure, that is a - finite measure on R^dn f0g satisfying R

jyj>0ð1^ jyj²Þn^mðdyÞ<1, and having density

n^mðyÞ ¼n^mðjyjÞ ð1:6Þ

¼

2ð₂^mÞ^{ðdþ1Þ=2Kðdþ1Þ=2ðmðjyjÞ}

jyj^ðdþ1Þ=2 ; m >0,

ððdþ1Þ=2Þ ^ðdþ1Þ=2

1

jyj^dþ1; m¼0.

8<

:

As shown in [5], H_A^m has another expression connected with the Le´vy measuren^mðdyÞ

ðH^m_AfÞðxÞ ¼mfðxÞ lim

r#0

Z

jyjr

½e^{iyAðxþ12yÞ}fðxþyÞ fðxÞn^mðdyÞ:

ForX2D0, letNXðdsdyÞbe a counting measure onð0;1Þ ðR^dn f0gÞdefined by

NXðEÞ:¼#fs >0;ðs; XðsÞ XðsÞÞ 2Eg for E2Bð0;1Þ BðR^dn f0gÞ, where Bð Þ are -algebras of Borel sets.NXðdsdyÞis the stationary Poisson random measure with intensity measure dsn^mðdyÞ with respect to ^m. Let Ng_X^mðdsdyÞ:¼ NXðdsdyÞ dsn^mðdyÞ. By the Le´vy-Itoˆ theorem,

XðtÞ ¼ Z t

0

Z

jyj1

yNXðdsdyÞ ð1:7Þ

þ Z t

0

Z

0<jyj<1

ygN_X^mðdsdyÞ:

Here and below, we should understandRt 0 :¼R

ð0;t. It can be proved that the solution of (1.2) with initial datau^mðx;0Þ ¼gðxÞis given by

u^mðx; tÞ:¼E^m½e^Smðt;x;XÞgðxþXðtÞÞ;

ð1:8Þ

S^mðÞ:¼iY^mðt; x; XÞ þ Z t

0

VðxþXðsÞÞds;

ð1:9Þ

Y^mðÞ:¼ Z t

0

Z

jyj1

AðxþXðsÞ þ¹₂yÞ yN_XðdsdyÞ þ

Z t 0

Z

0<jyj<1

AðxþXðsÞ þ¹₂yÞ ygN_X^mðdsdyÞ þ

Z t 0

ds Z

0<jyj<1

½AðxþXðsÞ þ¹₂yÞ

AðxþXðsÞÞ yn^mðdyÞ:

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, H_A^mmþV, ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

jj²þm²

p m, D0, ^m, k^m₀ðy; tÞ and XðtÞ, correspond to the nonrelativistic ones

1

2mðir AÞ²þV, ^jj_2m², C0, Wiener measure, the heat kernel ð_2t^mÞ^d=2e^m2tjyj2, Brownian motion BðtÞ, respectively. Here C0 is the space of continuous paths B:½0;1Þ !R^d 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. u^mð; tÞ converges to u⁰ð; tÞ on L²ðR^dÞasm#0, uniformly on½0; T.

Here and below, 0< T <1 can be taken ar- bitrary. Theorem 2 implies the strong resolvent convergence of H_A^mmþV to H_A⁰ 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 L²ðR^dÞasm#0, uniformly on½0; T.

We will prove Theorem 2 by using following:

Theorem 3. u^mð; tÞ converges to u⁰ð; tÞ on C₁ðR^dÞasm#0, uniformly on½0; T, whereC₁ðR^dÞ is the space of the continuous functionsg:R^d !C with jgðxÞj !0 as jxj ! 1 with norm kgk₁ :¼ sup_x2RdjgðxÞj.

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 point02R^das"#0.

(iii) There exist constants >¹₂, >0 and a non- decreasing continuous functionF on½0;1Þsuch that

E^m½jXðsÞ XðrÞj jXðtÞ XðsÞj

½FðtÞ FðrÞ²; 0< m <1; 0r < s < t:

Proof. (i) follows from (1.4), and (ii) from the stochastic continuity of fXðtÞg_t0. (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, E^m½jXðsÞ XðrÞj jXðtÞ XðsÞj

¼ Z

jyj k^m₀ðy; srÞdy Z

jyj k^m₀ðy; tsÞdy

¼Cðd; Þ²ððsrÞðtsÞÞ e^mðsrÞðmðsrÞÞ^{1 2} K1

2

ðmðsrÞÞ e^mðtsÞðmðtsÞÞ^{1 2} K1

2

ðmðtsÞÞ Cðd; Þ²2^{ð1þ2 Þ}ð¹ 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⁼ 2^{ð1þ2 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: Letg2C₀¹ðR^dÞ. For R >0, we have ku^mð; tÞ u⁰ð; tÞk2 ku^mð; tÞ u⁰ð; tÞkL²ðjxj<RÞ

þ ku^mð; tÞ u⁰ð; tÞk_L2ðjxjRÞ

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 ku^mð; tÞk_L2ðjxjRÞþ ku⁰ð; tÞk_L2ðjxjRÞ

Z

jxjR

dx Z

k^m₀ðy; tÞjgðxþyÞj²dy

!¹₂

þ Z

jxjR

dx Z

k⁰₀ðy; tÞjgðxþyÞj²dy

!¹₂

¼:Jðt; m; RÞ þJðt;0; RÞ:

Let be a nonnegativeC¹₀ ðR^dÞfunction such that ðxÞ ¼1 if jxj ¹₂ and ¼0 if jxj 1. Put hðxÞ ¼ jgðxÞj². Since1_jxj<Rð_R^xÞ, we have

Jðt; m; RÞ²

Z

ð1ð_R^xÞÞdx Z

k^m₀ðy; tÞhðxþyÞdy

¼ 1 ð2Þ^d

hð0Þb Z

1exp t

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

jj² R² þm² q

m

ðÞdb þ Z

ðhð0ÞÞ b hðb_RÞÞ exp t

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

jj² R²þm² q

m

b ðÞd

;

which converges to zero as R! 1 uniformly on tT and 0m1. Here, for ’2SðR^dÞ, ’b is the Fourier transform of ’ given by ’ðÞ ¼b Re^ix’ðxÞdx(2R^d).

From (1.3) and (2.1), it follows thatk^m₀ðy; tÞ ! k⁰₀ðy; tÞ as m#0, and then Jðt;0; RÞ² lim inf_m#0Jðt; m; RÞ² by Fatou’s lemma. Therefore we have Theorem 2 for this step.

Step II: Let g2L²ðR^dÞ. There is a sequence fg_ng C₀¹ðR^dÞ such that gn!g in L²ðR^dÞ as n! 1.

Put u^m_nðx; tÞ:¼E^m½e^Smðt;x;XÞgnðxþXðtÞÞ. Then we have

ku^mð; tÞ u⁰ð; tÞk₂

ku^mð; tÞ u^m_nð; tÞk2þ ku^m_nð; tÞ u⁰_nð; tÞk2

þ ku⁰_nð; tÞ u⁰ð; tÞk₂

2kgngk₂þ ku^m_nð; tÞ u⁰_nð; tÞk₂: By Step I, we have

lim sup

m#0

sup

tT ku^mð; tÞ u⁰ð; tÞk₂ 2kg_ngk₂; which converges to zero asn! 1.

4. Proof of Theorem 3. From (1.8), we have to prove that

u^mðx; tÞ ¼E^m½e^Smðt;x;XÞgðxþXðtÞÞ

!E⁰½e^S0ðt;x;XÞgðxþXðtÞÞ ¼u⁰ðx; tÞ as m#0 in C₁ðR^dÞ. But its direct proof seems difficult since both the integrand e^Smðt;x;XÞgðxþ

XðtÞÞand the probability measure^mdepend onm.

So we changeE^m½ toE⁰½ by achange of vari- able(i.e.,change of probability measure)^m¼⁰¹_m 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 n^mðdyÞ, so that it is presumed to hold that n^mðdyÞ ¼n⁰¹_mðdyÞ for some map m:R^dn f0g ! R^dn f0g.

We will determine _m in such a way that (1) n^mðdyÞ ¼n⁰¹_mðdyÞ, (2) m2C¹ðR^dn f0g;

R^dn f0gÞ, (3) m is one to one and onto, (4) detD_mðzÞ 6¼0 for all z2R^dn f0g, where D_mðzÞ is the Jacobian matrix ofm at the pointz.

Let U:¼ fy2R^dn f0g;jyj 2U⁰g for U⁰2 Bð0;1Þ. Introducing the spherical coordinates by z¼r!, r >0,!2S^d1, we have

n^mðUÞ ¼ Z

U

n^mðjyjÞdy¼CðdÞ Z

U⁰

n^mðrÞr^d1dr;

whereCðdÞis the surface area of thed-dimensional unit ball.

Let us assume that ¹_mðzÞ ¼l_mðjzjÞ_jzj^z for some non-decreasing C¹ function lm:ð0;1Þ ! ð0;1Þ.

Then we have n⁰¹_mðUÞ

¼ Z

U

n⁰ðl_mðjzjÞÞjzj^ðd1Þl_mðjzjÞ^d1l⁰_mðjzjÞdz

¼CðdÞ Z

U⁰

n⁰ðl_mðrÞÞl_mðrÞ^d1l⁰_mðrÞdr;

wherel⁰_mðrÞ ¼ ðd=drÞlmðrÞ:Therefore we have n^mðrÞr^d1¼n⁰ðlmðrÞÞlmðrÞ^d1l⁰_mðrÞ; a.s.r >0:

Ifm >0, from (1.6), we have

d

drlmðrÞ¹¼2^d12 ð^dþ1₂ Þ¹m^dþ12 r^d32 Kdþ1 2 ðmrÞ:

We solve this differential equation under boundary conditionl_mð1Þ ¼ 1to get

lmðrÞ ¼ 2^d12 ð^dþ1₂ Þ m^dþ12 R1

r u^d32 Kdþ1 2

ðmuÞdu ð4:1Þ :

Here we note that 0<R1

r u^d32 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 l_mðþ0Þ ¼0, l_mð1Þ ¼ 1.

(ii) For all r >0, lmðrÞ converges to r, strictly decreasingly, as m#0.

Proof. (2.1) implies l_mðþ0Þ ¼0. The other claims of (i) follow from (4.1) and the fact that K_ðdþ1Þ=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:¼l¹_mðjzjÞ z

jzj; z2R^dn f0g:

Then we have

¹_mðzÞ ¼lmðjzjÞ z

jzj; z2R^dn f0g:

We note that

mðzÞ !z; jmðzÞj ¼l¹_mðjzjÞ " jzj ð4:2Þ

as m#0by Proposition 1 (ii).

Let us define₀ðXÞ:¼X and form >0, mðXÞðtÞ:¼

Z t 0

Z

jyj1

yNXðds¹_mðdyÞÞ ð4:3Þ

þ Z t

0

Z

0<jyj<1

ygN_X⁰ðds¹_mðdyÞÞ

¼ Z t

0

Z

jzjl_mð1Þ

mðzÞNXðdsdzÞ þ Z t

0

Z

0<jzj<l_mð1Þ

mðzÞgN_X⁰ðdsdzÞ

¼ Z t

0

Z

jzj1

mðzÞNXðdsdzÞ þ Z t

0

Z

0<jzj<1

mðzÞgN_X⁰ðdsdzÞ:

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 zÞgN_X⁰ðdsdzÞ

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 fFðtÞg_t0-martingale fI₂^"ðt; m; XÞg_t0 with

No. 3] Zero-mass limit problem for a relativistic spinless particle 63

I₂^"ðt; m; XÞ:¼Rt 0

R

"<jzj<1ðmðzÞ zÞgN_X⁰ðdsdzÞ 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) and^m¼⁰¹_m, we have

u^mðx; tÞ ¼E⁰½e^{Smðt;x;mðXÞÞ}gðxþmðXÞðtÞÞ;

and then sup

tT

ku^mð; tÞ u⁰ð; tÞk₁ ð4:4Þ

kgk₁ sup

tT ;x2R^d

E⁰½je^{Smðt;x;mðXÞÞ}e^S0ðt;x;XÞj þE⁰ sup

tT

kgð þ_mðXÞðtÞÞ gð þXðtÞÞk₁

:

Since g2C₁ðR^dÞ is uniformly continuous and bounded on R^d, 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). ByN_mðXÞðdsdyÞ ¼NXðds¹_mðdyÞÞ, we have S^mðt; x;mðXÞÞ

¼i Z t

0

Z

jzj1

AðxþmðXÞðsÞ þ¹₂mðzÞÞ mðzÞNXðdsdzÞ þ

Z t 0

Z

0<jzj<1

AðxþmðXÞðsÞ þ¹₂mðzÞÞ mðzÞgN_X⁰ðdsdzÞ þ

Z t 0

ds Z

0<jzj<1

½AðxþmðXÞðsÞ þ¹₂mðzÞÞ AðxþmðXÞðsÞÞ mðzÞn⁰ðdzÞ

þ Z t

0

VðxþmðXÞðsÞÞds

¼:i S ^m₁ðt; x; XÞ þS₂^mðt; x; XÞ þS^m₃ðt; x; XÞ

þS₄^mðt; x; XÞ:

By the inequality

je^ðiaþbÞe^ðia0þb0Þj e^bje^iae^ia0j 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

ke^iSm1ðt;;XÞe^iS01ðt;;XÞk₁

ð4:5Þ

þsup

x2R^d

E⁰ sup

tTjS^m₂ðt; x; XÞ S₂⁰ðt; x; XÞj

þE⁰ sup

tT kS₃^mðt;; XÞ S⁰₃ðt;; XÞk₁

þE⁰ sup

tT kS₄^mðt;; 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 sup_tTj_m⁰⁰ðXÞðtÞ XðtÞj !0asm⁰⁰#0,⁰-a.s.

To prove that each term of (4.5) converges to zero asm⁰⁰#0, we first note that

S₁^m00ðt; 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

x2R^d

jAðxþm⁰⁰ðXÞðsÞ þ¹₂m⁰⁰ðzÞÞ AðxþXðsÞ þ¹₂zÞjjzjNXðdsdzÞ

þsup

x2R^d

jAðxÞj Z T

0

Z

jzj1

j_m⁰⁰ðzÞ zjN_XðdsdzÞ; which converges to zero as m⁰⁰#0 since A2 C¹₀ ðR^d;R^dÞis uniformly continuous onR^d.

Next, since S^m₂ðt; 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

x2R^d

jAðxþ_m⁰⁰ðXÞðsÞ þ¹₂m⁰⁰ðzÞÞ AðxþXðsÞ þ¹₂zÞj²jzj²n⁰ðdzÞ

þTsup

x2R^d

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

S₃^m00ðt; x; XÞ S₃⁰ðt; x; XÞ ¼1 2

Z t 0

ds Z

0<jzj<1

n⁰ðdzÞ

Z 1 0

½ðW_x;X^m00ðs; Þ_m⁰⁰ðzÞÞ _m⁰⁰ðzÞ ðW_x;X⁰ ðs; ÞzÞ zd:

Here W_x;X^m00ðs; Þ and W_x;X⁰ ðs; Þ are dd matrices defined by

W_x;X^m00ðs; Þ ¼DAðxþm⁰⁰ðXÞðsÞ þ¹₂m⁰⁰ðzÞÞ;

W_x;X⁰ ðs; Þ ¼DAðxþXðsÞ þ¹₂zÞ;

whereDAðÞis the Jacobian matrix ofA. Since ðW_x;X^m00ðs; Þm⁰⁰ðzÞÞ m⁰⁰ðzÞ ðW_x;X⁰ ðs; ÞzÞ z

¼ ðW_x;X^m00ðs; Þm⁰⁰ðzÞÞ ðm⁰⁰ðzÞ zÞ þ ððW_x;X^m00ðs; Þ W_x;X⁰ ðs; ÞÞm⁰⁰ðzÞÞ z þ ðW_x;X⁰ ðs; Þðm⁰⁰ðzÞ zÞÞ z;

the integrand of the third term of (4.5) is less than or equal to

T sup

x2R^d

kDAðxÞk Z

0<jzj<1

jm⁰⁰ðzÞ zjjzjn⁰ðdzÞ þ1

2 Z T

0

ds Z

0<jzj<1

jzj²n⁰ðdzÞ

Z 1 0

sup

x2R^d

kW_x;X^m00ðs; Þ W_x;X⁰ ðs; Þkd;

wherek kis the norm of matrices. This is less than or equal to 3Tsup_x2RdkDAðxÞkR

0<jzj<1jzj²n⁰ðdzÞ<

1, and converges to zero as m⁰⁰#0 because each component ofDAis uniformly continuous onR^d.

Finally, the fourth term of (4.5) is less than or equal to E⁰½RT

0 kVð þ_m⁰⁰ðXÞðsÞÞ Vð þ XðsÞÞk₁ds, which converges to zero asm⁰⁰#0since V 2C0ðR^d;RÞis uniformly continuous onR^d. Thus we have sup_tTku^m00ð; tÞ u⁰ð; tÞk₁!0 as m⁰⁰#0, and hencesup_tTku^mð; 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.

References

[ 1 ] D. Applebaum, Le´vy processes and stochastic calculus, 2nd ed., Cambridge Studies in Ad- vanced Mathematics, 116, Cambridge Univ.

Press, Cambridge, 2009.

[ 2 ] P. Billingsley, Convergence of probability meas- ures, 2nd ed., Wiley Series in Probability and Statistics: Probability and Statistics, Wiley, New York, 1999.

[ 3 ] A. Erde´lyi, W. Magnus, F. Oberhettinger and F. G. Tricomi,Higher transcendental functions.

Vol. II, McGraw-Hill, New York, 1953.

[ 4 ] T. Ichinose, The nonrelativistic limit problem for a relativistic spinless particle in an electromag- netic field, J. Funct. Anal.73(1987), no. 2, 233–

257.

[ 5 ] T. Ichinose, Essential selfadjointness of the Weyl quantized relativistic Hamiltonian, Ann. Inst.

H. Poincare´ Phys. The´or.51(1989), no. 3, 265–

297.

[ 6 ] T. Ichinose, Remarks on the Weyl quantized relativistic Hamiltonian, Note Mat. 12 (1992), 49–67.

[ 7 ] T. Ichinose, On three magnetic relativistic Schro¨dinger operators and imaginary-time path integrals, Lett. Math. Phys. 101 (2012), 323–

339.

[ 8 ] T. Ichinose, Magnetic relativistic Schro¨dinger operators and imaginary-time path integrals, in Mathematical physics, spectral theory and stochastic analysis, Oper. Theory Adv. Appl., 232, Birkha¨user/Springer Basel AG, Basel, 2013, pp. 247–297.

[ 9 ] T. Ichinose and H. Tamura, Imaginary-time path integral for a relativistic spinless particle in an electromagnetic field, Comm. Math. Phys.105 (1986), no. 2, 239–257.

[ 10 ] T. Ichinose and T. Tsuchida, On essential self- adjointness of the Weyl quantized relativistic Hamiltonian, Forum Math.5(1993), no. 6, 539–

559.

[ 11 ] N. Ikeda and S. Watanabe,Stochastic differential equations and diffusion processes, North-Hol- land Mathematical Library, 24, North-Holland, Amsterdam, 1981.

[ 12 ] Y. Kasahara and S. Watanabe, Limit theorems for point processes and their functionals, J. Math.

Soc. Japan38(1986), no. 3, 543–574.

[ 13 ] T. Kato,Perturbation theory for linear operators, 2nd ed., Springer, Berlin, 1976.

[ 14 ] M. Nagase and T. Umeda, Weyl quantized Hamiltonians of relativistic spinless particles in magnetic fields, J. Funct. Anal.92(1990), no. 1, 136–154.

[ 15 ] K. Sato, Le´vy processes and infinitely divisible distributions, translated from the 1990 Japanese original, Cambridge Studies in Advanced Mathematics, 68, Cambridge Univ. Press, Cam- bridge, 1999.

[ 16 ] B. Simon, Functional integration and quantum physics, Pure and Applied Mathematics, 86, Academic Press, New York, 1979.

No. 3] Zero-mass limit problem for a relativistic spinless particle 65