Stochastic quantization of the two-dimensional ... - Semantic Scholar

Stochastic quantization of the two-dimensional polymer measure Sergio Albeverio1) Michael Rockner4)

Yao-Zhong Hu2) 3) Xian Yin Zhou1) 5) ;

;

Abstract

We prove that there exists a diusion process whose invariant measure is the two-dimensional polymer measure g . The diusion is constructed by means of the theory of Dirichlet forms on in nite-dimensional state spaces. We prove the closability of the appropriate pre-Dirichlet form which is of gradient type, using a general closability result by two of the authors. This result does not require an integration by parts formula (which does not hold for the two-dimensional polymer measure g ) but requires the quasi-invariance of g along a basis of vectors in the classical Cameron-Martin space such that the RadonNikodym derivatives (have versions which) form a continuous process. We also show the Dirichlet form to be irreducible or equivalently that the diusion process is ergodic under time translations.

AMS Subject Classi cation Primary: 60 J 65 Secondary: 60 H 30 Key words: two-dimensional polymer measure, closability, Dirichlet forms, diusion processes, ergodicity, quasi-invariance. Running head: Two-dimensional polymer measure.

1) Institut fur Mathematik, Ruhr-Universitat Bochum, D-44780 Bochum, Germany. 2) Department of Mathematics, University of Oslo, N-0316 Oslo, Norway. 3) Institute of Mathematical Sciences, Academia Sinica, Wuhan 430071, China. 4) Fakultat fur Mathematik, Universitat Bielefeld, 33501 Bielefeld, Germany 5) Institute of Mathematics, Beijing Normal University, Beijing 100875, China

1 Introduction and main results Let  be a given probability measure on a space X . The stochastic quantization of  means the construction of a Markov process whose invariant measure is just  . Parisi and Wu [PW81] rst studied the stochastic quantization for certain gauge elds in quantum eld theory. After that, many works (see e.g. [J-LM85], [AR89a], [AR89b], [AR91]) on the stochastic quantization of Euclidean quantum elds such as the P ()2 - eld appeared. There is a close connection between the Edward's model (or say polymer measure) and the 4 - eld (see e.g. [Sy69, AFH-KL86]). This naturally directs the interest to the stochastic quantization of a polymer measure. The main aim of the present paper is to study the stochastic quantization of the two dimensional polymer measure. Let us rst introduce the (rigorous) de nition of the two dimensional polymer measure g . Let X := C ([0; 1]; 2)0 be the set of all continuous paths in 2 indexed by [0; 1] and starting at zero. Let B be the -algebra generated by all maps ! 7! !t; t  0, from X to 2 and let 0 denote the Wiener measure on (X; B). Let (x; A) be the self-intersection local time at x 2 2 of ! 2 X on the set A  [0; 1]  [0; 1], i.e. IR

IR

IR

IR

(x; A) =

Z

A

x(!s ? !t )dsdt:

(For its precise de nition, the reader is referred to [Ro86] and the references therein.) It is well-known that (0; f(s; t) : 0  s < t  1g) = 1; 0 ? a:e: Therefore, one has to use renormalization. To this end we set i;k =  0; 2(i2?k 1) ; 2i2?k 1  2i2?k 1 ; 22ki ; i = 1; : : : ; 2k?1; k = 1; 2; : : : : Let E be the expectation with respect to 0, and 





n =



n 2k?1

X X

k=1 i=1



(i;k ? Ei;k ); n  1:

Then one can prove (see [LG85], [V69]) that (n)n0 is almost surely convergent to a random variable  2 L2( ; 0) and limn!1 E jn ? j2 = 0: The random variable  is usually called the normalized self-intersection local time of planar Brownian motion. It is proved in [AHZ95] that the random variable  is not dierentiable in the sense of Meyer-Watanabe. One can prove that there is g0 2 (0; 1) (see e.g. [LG94], [PY96]) such that 1; 8g 2 (?g0; 1), E exp(?g) < = 1; 8g 2 (?1; ?g0). The two dimensional polymer measure g is de ned by g := (E exp(?g))?1 exp(?g) 0 ; g 2 (?g0; 1): 

2

In this paper we always assume that g 2 (?g0; 1). For the construction of the stochastic quantization of g we shall use the theory of Dirichlet forms. We want to emphasize that this method only gives a way to "reconstruct" g (cf. Theorems 1.1 and 1.5 below), since it uses g and is based on some of its speci c properties. However, also in all approaches to the stochastic quantization of measures in Euclidean eld theory (such as e.g. P ()2- elds) the respective measures have been used in an essential way. A truly constructive approach is not known. It seems that the situation for measures in Euclidean eld theory as well as for polymer measures is just too singular. (E.g. all attempts using strong solutions for appropriate stochastic dierential equations to obtain the required Markov processes have failed so far.) To describe the Dirichlet form in question precisely, we need some preparations. X equipped with the supremum-norm is a separable real Banach space. Let X 0 denote its dual. Let H  X be the classical Cameron-Martin space, i.e., H := fh 2 X : h is absolutely continuous and jhj2H = 01 jh0tj2dt < 1g. Furthermore, let K := fh 2 H : sup0t1 jh0(t)j < 1g. (H; h ; iH ) is a real separable Hilbert space which is densely and continuously embedded into X , and K is a dense linear subspace of (H; h ; iH ). Identifying H with its dual we obtain that X 0 is densely embedded into H . Hence X 0  H  X , and h ; iH restricted to X 0  H coincides with the dualization between X 0 and X . De ne the set of smooth bounded cylinder functions by F Cb1 := ff (l1; : : :; lm) : m 2 ; f 2 Cb1( m); l1; : : : ; lm 2 X 0g; where Cb1( m) denotes the set of all bounded in nitely dierentiable functions on m with all partial derivatives bounded. Let k 2 X . For u 2 F Cb1; ! 2 X , de ne @u (!) := d u(! + sk)j s=0 @k ds and let ru(!) denote the unique element in H such that @u (!) for all k 2 H . hru(!); kiH = @k De ne for u; v 2 F Cb1 R

IN

IR

IR

IR

Z

Eg (u; v) := hru(!); rv(!)iH g (d!) and Eg ;1 := Eg + ( ; )L2(X ;g ). Note that since g  0 and since 0(U ) > 0 for every ; 6= U  , U open, we can identify each u 2 F Cb1 with the corresponding class in L2(X ; g ). In particular, (Eg ; F Cb1) is a symmetric positive de nite bilinear form on L2(X ; g ). Now we can formulate our rst main result. 3

Theorem 1.1 (Eg ; F Cb1) is closable on L2(X ; g ) and the closure (Eg ; D(Eg )) is a symmetric Dirichlet form (i.e., a closed positive de nite symmetric bilinear form such that u# := min(max(u; 0); 1) 2 D(Eg ) and Eg (u#; u#)  Eg (u; u) for all u 2 D(Eg )). For more details on Dirichlet forms the reader is referred e.g. to [MR92], [FOT94], [BH91], [Si74], and for the special type of Dirichlet form appearing here, namely so-called classical Dirichlet forms to [AR90]. For the convenience of the reader, however, we recall all notions necessary here in Section 3 below. Let (L; D(L)) be the generator of (Eg ; D(Eg )), i.e., the unique negative de nite self-adjoint operator on L2(X ; g ) such that

p

D( ?L) = D(Eg )

and

p

p

( ?Lu; ?Lv) = Eg (u; v) for all u; v 2 D(Eg ).

(1.1)

Let Tt := etL; t  0. Theorem 1.2 There exists a diusion process IMI = ( ; F ; (Ft)t0, (Xt)t0; (P! )!2X ) which is associated with (Eg ; D(Eg )), i.e., for all (g versions of) f 2 L2 (X ; g ) and all t  0 the function

! 7! ptf (!) :=

Z



f (Xt ) dP! ; ! 2 X;

is a g -version of Ttf . IMI is conservative and g -symmetric. In particular, g is an invariant measure for IMI .

Remark 1.3 (i) IMI is in fact even properly associated to (Eg ; D(Eg )) in

the sense of [MR92, Chap. IV, De nition 2.5]. (ii) in Theorem 1.2 can, of course, always be taken to be C ([0; 1); X ), since IMI has continuous sample paths and is conservative. The proofs of Theorems 1.1 and 1.2 well be given in Section 3 below. The most dicult part is to prove the closability of (Eg ; F Cb1) on L2(X ; g ). The rest follows from known results. The closability is implied by a general closability criterion (cf. [AR89b, Corollary 2.5]) and the following result on g which is certainly of its own interest. For k 2 X de ne k (!) := ! + k, ! 2 X. Theorem 1.4 Let k 2 K . Then g is k-quasi-invariant, i.e., ?1 g  sk?1 is absolutely continuous w.r.t. g for all s 2 . If ask := d(gdgsk ) , s 2 , then the process (ask )s2 has a version which has 0 ? a:e: continuous sample paths. IR

IR

4

IR

The proof of Theorem 1.4 is given (independently of the rest of this paper) in Section 2 below. We want to emphasize that [AR89b, Corollary 2.5] (which just states that whenever (ask )s2 as above has a continuous version for k in an orthonormal basis of H , then the corresponding classical Dirichlet form is closable) is a closability criterion of a totally dierent kind than the ones proved in older papers like [AH-K77a], [AH-K77b], [Ku82]. There, always an integration by parts formula (cf. e.g. [AKuR90]) was assumed to hold. In the case of the polymer measure g such an integration by parts formula does not hold (cf. [AHZ95]), but nevertheless we have closability. [AR89b, Corollary 2.5] is a consequence of the main result in [AR90, Sect. 3], which "characterizes" closability. Our nal result is the following Theorem 1.5 If g < g0, then the Dirichlet form (Eg ; D(Eg )) is irreducible (i.e., u 2 D(Eg ) with Eg (u; u) = 0 implies that u is a constant) or equivalently the associated diusion IMI when started with g is ergodic under time translations. In particular, limt!1 ptf = fdg g ? a:e: for all bounded B-measurable functions f : X ! . IR

R

IR

Remark 1.6 The last convergence in Theorem 1.5 in fact holds even quasi-

everywhere. This follows from Fukushima's quasi-everywhere ergodic theorem (c.f. [F83], which by the "transfer method" in [MR92, Chap. VI] generalizes to this "non-locally compact case").

The proof of Theorem 1.5 will be given at the end of Section 3. Finally, we would like to mention that Theorems 1.1, 1.2 and 1.4 also hold for the polymer measure in three dimensions. The proof of Theorem 1.4 is, however, much more complicated since g is no longer absolutely continuous w.r.t. 0 in this case. All this is the contents of the forthcoming paper [ARZ95].

2 Quasi-invariance of the two-dimensional polymer measure In this section we prove Theorem 1.4. To this end we will use a new approximation for the random variable . Some ideas given below are also suitable for the three-dimensional case. In a forthcoming paper [ARZ95] we shall use a similar approach to prove that the three dimensional polymer measure is also quasi-invariant. 5

We set

gn (x) := cI[0;n](jxj); n  1; x 2 2; where j
j is the Euclidean norm in 2, and the constant c 2 (0; 1) will be speci ed below. It is easy to show that there is a constant c 2 (0; 1) such that the following convergence holds in the vague sense IR

IR

Z

ihx;yig

2e

IR

n (y )dy ! 0(x);

where 0 is the Dirac function at point 0 2 x = (x1; x2) and y = (y1; y2). Let

n;" (!) :=

Z

T"

Z IR

IR

ihy; !t ?!s i g

2e

n ! 1; 2,

and hx; yi = x1y1 + x2y2 if

n (y )dy

ds dt;

where T" := f(s; t) : t ? s  "; 0  s < t  1g. Let

~n;" (u; k) := n;"(! + uk) ? n;" (!): Let us rst prove two lemmas.

Lemma 2.1 Let 2 (0; 1) and k 2 K be given. Then there is a constant C 2 (0; 1) such that E j~n;"(u1; k) ? ~n;"(u2; k)j2  C ju1 ? u2j1+ for all u1; u2 2 , " 2 (0; 1) and n  1, where E denotes expectation w.r.t. 0 . IR

Proof. By de nition we know that ~n;" (u1; k) ? ~n;"(u2; k) is equal to Z

Z

T"

2e IR

ihy;!t ?!s i (eihy;u1 (kt ?ks )i ? eihy;u2 (kt ?ks )i )g

n (y )dydsdt:

Let

T1(") := f0  s1 < t1 < s2 < t2  1g \ \2i=1 fti ? si  "g; T2(") := f0  s1 < s2 < t1 < t2  1g \ \2i=1 fti ? si  "g; T3(") := f0  s1 < s2 < t2 < t1  1g \ \2i=1 fti ? si  "g; and let Il;" be equal to Z

Z

Tl (")

IR

4 Ee

ihy1 ;!t1 ?!s1 i+ihy2 ;!t2 ?!s2 i(eiu1 hy1 ;kt1 ?ks1 i ? eiu2 hy1 ;kt1 ?ks1 i )

(eiu1hy2;kt2 ?ks2 i ? eiu2hy2 ;kt2 ?ks2 i)gn(y1)gn (y2) 6

2

Y

j =1

dyj dsj dtj ; l = 1; 2; 3:

Then it is easy to see that

E j~n;"(u1; k) ? ~m;" (u2

; k)j2

=2

3

X

l=1

Il;":

If (s1; s2; t1; t2) 2 T1("), then

Eeihy1;!t1 ?!s1 i+ihy2;!t2 ?!s2 i = e? 21 jy1 j2(t1?s1 )? 12 jy2j2 (t2?s2 ):

(2.1)

Since sin(?x) = ? sin x, I1;" is equal to Z

Z

T1 (") IR4

e? 21 jy1j2(t1?s1 )? 12 jy2 j2(t2?s2 )gn(y1)gn (y2)

2

 (cos(u1hyi; kti ? ksi i) ? cos(u2hyi ; kti ? ksi i)) dyi dsi dti : Y

i=1

We remark that for any given  2 (0; 1] there is a constant C 2 (0; 1) (from now on, C 2 (0; 1) will denote a constant which might be dierent from line to line) such that

j cos y1 ? cos y2j  C jy1 ? y2j ^ 1; y1; y2 2 :

(2.2)

IR

By (2.1) and (2.2) it follows that I1;" is less than

C ju1 ? u2

j1+

Z

Z

T1 (0)

p

p

? 21 jy1 j2 ? 12 jy2 j2 g (y = t ? s )g (y = t ? s ) n 1 1 1 n 2 2 2 4e

IR

 (ti ? si )?1hyi; (kti ? ksi )=pti ? sii(1+ )=2dyidsidti i=1 p p  C ju1 ? u2j1+ T (0) 4 e? 12 jy1j2? 12 jy2j2 gn (y1= t1 ? s1)gn(y2= t2 ? s2) 2 Y

Z

Z

1

IR

2

jy1j(1+ )=2jy2j(1+ )=2 (ti ? si)?(3? )=4 dyi dsi dti i=1 1+

 C ju1 ? u2j : We now consider I2;". Let  = (t2 ? s2)?1((t2 ? t1)(t1 ? s1)+(t1 ? s2)(s2 ? s1)). If (s1; s2; t1; t2) 2 T2("), then the left hand side of (2.1) is equal to ? s2 2(t ? s ) ; exp ? 21 jy1j2 ? 12 y2 + y1 tt1 ? 2 2 2 s2 which is denoted by f1(s; t; y). We also remark that for any given  2 (0; 1] there is a constant C 2 (0; 1) such that j sin y1 ? sin y2j  C jy1 ? y2j ^ 1; y1; y2 2 : (2.3) Y









IR

7

Using (2.2) and (2.3) we readily see that I2;" is equal to Z

T2 (")

2



Z IR

g (y )g (y ) ? (sin(u1hyi; kti ? ksi i ? sin(u2hyi; kti ? ksi i)) 4 n 1 n 2 Y

i=1

2 Y



+ (cos(u1hyi; kti ? ksi i) ? cos(u2hyi; kti ? ksi i)) f1(s; t; y) i=1

 C ju2 ? u1j1+

Z

2

Z

Y

4

T2 (0)

2

Y

i=1

dyidsi dti

? s2 dyi dsidtie? 21 jy1 j2? 12 jy2j2 (t2?s2 )gn (y1)gn y2 ? y1 tt1 ? s 

2 2 (1+

) = 2 Y (ti ? si)(1+ )=2 jy1j(1+ )=2 y2 ? y1 tt1 ?? ss2 2 2 i=1 Z Z 2 Y  C ju1 ? u2j1+ T (0) 4 dyi dsidtie? 21 jy1 j2? 12 jy2j2 (t2?s2 ) 2 i=1 (1+ )=2  2  Y (ti ? si)(1+ )=2  jy1y2j(1+ )=2 + jy1j2 tt1 ?? ss2 2 2 i=1 Z 2 Y  C ju1 ? u2j1+ T (0) dsidti(?1?(1+ )=4(t1 ? s1)(1+ )=2(t2 ? s2)?1+(1+ )=4 2 i=1 ? 1 ? (1+

)=2(t ? s )(1+ )=2(t ? s )(1+ )=2) + 1 2 1 1 1+

 C ju1 ? u2j : IR



i=1



2



IR

We now consider I3;". If (s1; s2; t1; t2) 2 T3("), then the left hand side of (2.1) is equal to e? 21 jy1 j2(t1?s1 ?t2+s2 )? 12 jy1+y2 j2 (t2?s2 ); which is denoted by f2(s; t; y). As for I2;" we see that I3;" is equal to Z

T3 (")

2



Z IR

(sin(u1hyi; kti ? ksi i ? sin(u2hyi; kti ? ksi i)) 4 gn (y1 )gn (y2) ? Y

i=1

2 Y



+ (cos(u1hyi; kti ? ksi i) ? cos(u2hyi; kti ? ksi i)) f2(s; t; y) i=1

 C ju2 ? u1j1+

Z

T3 (0)

Z

2

Y

i=1

dyi dsi dti

? 21 jy1 j2 (t1 ?s1 ?t2 +s2 )? 12 jy2 j2 (t2 ?s2 ) g (y )g (y ? y ) e n 1 n 2 1 4

IR

2 2 Y Y (1+

) = 2 (1+

) = 2 dyi dsi dti (jy1 ) (ti ? si) 2 ? y1 j i=1 i=1 Z Z 2 Y 1+

 C ju1 ? u2j T (0) 4 dyi dsi dtie? 21 jy1 j2(t1?s1 ?t2+s2 )? 12 jy2 j2(t2?s2 ) 3 i=1 2 Y (1+

) = 2 1+

(ti ? si )(1+ )=2 (jy1y2j + jy1j i=1 Z  1+

 C ju1 ? u2j T (0) (t1 ? s1 ? t2 + s2)?1?(1+ )=4(t1 ? s1)(1+ )=2(t2 ? s2)?1+(1+ )=4 3

j(1+ )=2jy

IR

8

+(t1 ? s1 ? t2 + s2)?1?(1+ )=2(t1 ? s1)(1+ )=2(t2 ? s2)(1+ )=2

 C ju1 ? u2j1+ :

2

Y

i=1

Combining the estimates for I1;", I2;" and I3;" given before, we conclude that for any given 2 (0; 1) there is a constant C 2 (0; 1) such that

I1;" + I2;" + I3;"  C ju1 ? u2j1+ ; 8u1; u2 2 " 2 (0; 1); n  1; IR

which implies the conclusion given in Lemma 2.1.

Lemma 2.2 Let ~(u; k) = (! + uk) ? (!) for u 2 and k 2 K . Then, for any given 2 (0; 1) and k 2 K there is a constant C 2 (0; 1) such that E j~(u1; k) ? ~(u2; k)j2  C ju1 ? u2j1+ ; u1; u2 2 : Proof. From the choice of (gn )n1 we see that n;" is convergent in probability (with respect to 0) to T" (!t ? !s )dsdt as n ! 1, if " 2 (0; 1) is xed. IR

IR

R

Moreover, we have that

nlim !1 En;"

=E

Z

T"

(!t ? !s )dsdt; " 2 (0; 1):

Therefore, (n; ") ? E(n; ") is convergent in probability with respect to 0 to (!t ? !s )dsdt ? E (!t ? !s)dsdt Z

Z

T"

T"

as n ! 1, if " 2 (0; 1) is xed. However, it is easy to check that lim

"!0+

Z

T"

(!s ? !t)dsdt ? E

Z

T"



(!s ? !t )dsdt = ; in L2(0):

Then there are sequences (mn)n2 and ("n)n2 with limn!1 mn = 1 and limn!1 "n = 0 such that mn ;"n ? Emn ;"n is convergent to  in probability with respect to 0, as n ! 1. Hence, mn ;"n (
+ uk) ? Emn ;"n is also convergent to (
+ uk) in probability with respect to 0, as n ! 1. This proves that ~mn ;"n (u; k) is convergent to ~(u; k) in probability with respect to 0, as n ! 1. Thus we get the desired result immediately from Lemma 2.1 IN

IN

Having Lemma 2.2 we can easily prove Theorem 1.4 9

dsi dti

Proof of Theorem 1.4 By Lemma 2.2 we know that for any given k 2 K there is f~0(u; k); u 2 g such that 0(~(u; k) = ~0(u; k)) = 1; u 2 ; (2.4) 0(~0(u; k) is continuous with respect to u 2 ) = 1: (2.5) By the de nition of the two dimensional polymer measure g for g 2 (?g0; 1), i.e., g := (E exp(?g))?1 exp(?g) 0; we can easily see that g  uk?1 is indeed absolutely continuous with respect to g for all u 2 and k 2 K . Let ?1 a (!) := g  uk (!): IR

IR

IR

IR

uk

dg

By de nition of g we know that

1

1

1

1

1

auk (!) = e?g~(?u;k)
eu 0 kt0 d!t? 2 u2 0 (kt0 )2dt; with 01 kt0 d!t understood as an It^o stochastic integral. Let R

R

R

R

R

1

a0uk (!) := e?g~0(?u;k)
eu 0 kt0 d!t? 2 u2 0 (kt0 )2dt: By (2.4) and (2.5) we know that 0(auk = a0uk ) = 1; u 2 ; k 2 K; (2.6) 0 0(auk is continuous with respect to u 2 ) = 1; k 2 K: (2.7) Since g is absolutely continuous with respect to 0, both (2.6) and (2.7) hold with g replacing 0. Hence the proof is complete. 2 IR

IR

3 Proofs of Theorems 1.1, 1.2 and 1.5

Before we prove Theorem 1.1 we recall the necessary de nitions. (Eg ; F Cb1) is closable on L2(X ; g ) means that the unique continuous extension of the continuous embedding F Cb1  L2(X ; g ) (where F Cb1 is equipped with norm (Eg ;1)1=2 and L2(X ; g ) with ( ; )1L=22(X ;g) ) to the completion of F Cb1 (w.r.t. (Eg ;1)1=2) is still one-to-one. The closure (Eg ; D(Eg )) is then the smallest closed extension of (Eg ; F Cb1) on L2(X ; g ). Proof of Theorem 1.1. Since K is dense in H we can nd an orthonormal basis fkn : n 2 g of H in K . Obviously, we have for u; v 2 F Cb1 1 @u @v d : (3.1) Eg (u; v) = g n=1 @kn @kn IN

X

Z

10

Hence e.g. by [MR92, Chap.I, Proposition 3.7(i)] it suces to prove the closability of each (Ekn ; F Cb1) on L2(X ; g ) where @u @v d ; u; v 2 F C 1 : Ekn (u; v) := @k g b n @kn But this is a direct consequence of Theorem 1.4 and [AR89b, Corollary 2.5]. The fact that the closure (Eg ; D(Eg )) of (Eg ; F Cb1) on L2(X ; g ) is then a Dirichlet form, follows by [AR90, Theorem 3.8] or [MR92, Chap.I, Proposition 3.5]. Z

2

Proof of Theorem 1.2 By virtue of the representation (3.1) the existence

of a g -symmetric IMI is a consequence of [MR92, Chap.IV, Subsection 4b and Theorem 3.5] or [Sch90]. The continuity of the sample paths follows from [MR92, Chap.V, Example 1.12(ii) and Theorem 1.11]. The conservativity of IMI is obvious since 1 2 D(L) and L1 = 0, hence Tt1 = 1 for all t  0. The fact that g is an invariant measure for IMI follows then immediately.

2

Proof of Theorem 1.5. Suppose u 2 D(Eg ) such that Eg (u; u) = 0:

We may assume that u is bounded (cf. [AKR95, Proof of Proposition 2.3(ii)) (i)]). Since g 2 (?g0; g0) we can nd  2 ( 21 ; 1) such that E exp 1 ?  g < 1 (i.e., ? 1 ?  g 2 (?g0; 1)): By Holder's inequality we have that for all f 2 L2(X ; g ) 



Z

jf j2d0 = 

Z

jf j2 exp(?g) exp(g)d0

Z

Z

Hence

Z

jf j2d

jf j2 exp(?g)d





0

exp 1 ?  g d0 !



1 2

1?

!

:

 c jf jL2(X ;g)

(3.2) for some constant c 2 (0; 1) independent of f . Applying (3.2) to jrvjH for v 2 F Cb1 we obtain that Z



1

jrvj2Hd0 2

0

 c Eg (v; v)1=2 for all v 2 F Cb1 . 11

(3.3)

By de nition of the closure (Eg ; D(Eg )), we can nd vn 2 F Cb1, n 2 , 2 such that vn n?! !1 u and jrvnjH n?! !1 0 in L (X ; g ). Applying (3.2), (3.3) we see that the same convergence holds in L2(X ; 0). Hence u 2 D1;2 and ru = 0 (cf. [N95, Chap. 1, Sect. 1.5] for the notation). Since 2 > 1 it follows by [N95, Proposition 1.5.5 and Remark 1 on p. 35] that u is constant 0-a.e., hence g -a.e. The statement that Pg := X P! g d(!) is ergodic unter time shifts on (= C ([0; 1); X ) (cf. Remark 1.3(ii)) is well-known to be equivalent with the irreducibility of (Eg ; D(Eg )) (cf. [F82] which by the "transfer method" in [MR92, Chap.VI] generalizes immediately to this "nonlocally compact case"). The last statement follows similarly from [F82, F83]. IN

R

2

Acknowledgements. When this paper was nished, Professor M.Yor told us that one can also use the Tanaka-Rosen formula of  as de ned in Sect. 1 (see e.g. [Ro86] or [Y85]): u 1 1  = ? hd!u ; ds j!!u ??!!sj2 i + ds log j!1 ? !s j 0 0 0 u s to prove Theorem 1.4. We thank him very much for the above suggestion and other interesting comments. The nancial support of SFB 343 Bielefeld and SFB 237 Bochum-Dusseldorf-Essen is gratefully acknowledged. Z

Z

Z

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