Constructive Approach to the Global Markov

Constructive Approach to the Global Markov Property in Euclidean Quantum Field Theory I. Construction of transition kernels

by Sergio Albeverio Fakultat fur Mathematik, Ruhr-Universitat, Bochum, D-44780 Bochum Germany; SFB 237; BiBoS (Bielefeld, Germany); CERFIM (Locarno, Switzerland) Roman Gielerak Institute of Theoretical Physics, University of Wroclaw, 50-205 Wroclaw, Poland and BiBoS (Bielefeld, Germany) Francesco Russo Departement de Mathematiques, Institut Galilee, Universite Paris-Nord, Av. J.B. Clement, F-93430 Villetaneuse, France and BiBoS (Bielefeld, Germany)

Abstract The trace properties of the sample paths of suciently regular generalized random elds are studied. In particular, nice localisation properties are shown in the case of hyperplanes. Using techniques of Euclidean quantum eld theory a constructive description of the conditional expectation values with respect to some Gibbs measures describing Euclidean quantum eld theory models and the -algebras localised in halfspaces is given. In particular the Global Markov property with respect to hyperplanes follows from these constructions in an explicit way.

Key words Quantum eld theory, Global Markov Property, Gibbsian perturbation of the free eld

AMS Classi cation 60G60; 35Q99; 60H15

1 Introduction 1.1 Generalities

Generalized random elds play an important role in physics. Of particular interest are the elds which are homogeneous (stationary) with respect to the action of the Euclidean group. Quantum eld theory applications of generalized random elds also require that the Markov property (at least with respect to halfplanes) is ful lled together with the re ection invariance (or homogeneity with respect to the full Euclidean group). In the fundamental work of Nelson [Ne73a, Ne73b] the importance of Markovian generalized random elds for construction of quantum eld theoretical models have been demonstrated. Unfortunately it is very dicult to exhibit examples of generalized random elds ful lling simoultaneously any sort of Markov property and homogeneity with respect to the full Euclidean group. Among scalar, Gaussian generalized random elds only the Nelson free eld ful lls both requirements. Relaxing the homogeneity requirement to special Euclidean group transformations, all Gaussian generalized random elds which are somehow Markovian have been described completely by Molchan [Mo71] for the rst time (see also [Pi71; Wo68; Ko74; R82; WoH85]). However for nontrivial applications to physics examples of non-Gaussian, generalized random elds are needed. Following Nelson [Ne73c] it is possible to adopt a technique of multiplicative functionals (well known in the theory of one-dimensional Markov processes [Dy65]) to produce examples of non-Gaussian elds which are globally Markovian. However this procedure breaks down the homogeneity properties. Nevertheless this idea was crucial for many applications in constructive quantum eld theory [Si74, GJ81, AHK84, AZ92 and references therein]. The homogeneous (thermodynamical in the physical terminology) limits of these perturbations have to be investigated. Constructive eld theory provides us with several techniques for controlling those limits. In the case of 2- and 3-dimensional space time and for certain multiplicative functionals of the free eld the corresponding homogeneous limits have indeed been controlled. Similar ideas can be used to study perturbations of higher order Markov elds [Gi88]. In all cases the limiting homogeneous elds ful ll the (resp. higher order) local Markov property. i.e. the are Markovian with respect to bounded regions of IRd [New73; (resp. Gi88)]. The Global Markov Property of the free quantum eld 0 (in the sense of [AHK79a, AHK79b]) and the higher-order Gauss-Markov elds was well understood and studied. However the Global Markov Property of the non-Gaussian, homogeneous limits obtained from the multiplicative functionals perturbations is remained unproven for a long time. As for the Euclidean (Quantum) Field Theory only in 1979, Albeverio and Hegh-Krohn [AHK79a, AHK79b] succeded to present the proof in the case of weakly coupled trigonometric interactions (sine-Gordon model). In a 1983 paper of Gielerak [Gi83] some essential ideas for the veri cation of the Global Markov Property in some class of strongly coupled exponential interactions have been presented. These ideas have been improved and extended to general exponential interactions by Gielerak and Zegarlinski in [GiZ84, Ze84]. An explicit construction of the corresponding local speci cations together with many important additional results have been provided by Rockner in [Ro85, Ro86]. The most dicult case of polynomial interactions has been considered more recently [AHKZ89] and some constructive program (realized eectivelly in the case of weakly coupled '42 theory) has been proposed 1

there. See also [Gi89, Gi92b] for some related results. In this paper, Global Markov Property (GMP) will be understood as (related to germ or sharp ,algebras) Markov property with respect to hyperplanes. As for the higher order Markov elds, bounded perturbations have been studied in [Gi88] and the Global Markov Property with respect to the germ algebras and half-spaces has been checked in the weak coupling regime (modulo the problem of construction of the corresponding local speci cations solved in [Gi92c]). The problem of preservation of the Global Markov property under transformations of the elds by inverses of dierential operators has been investigated in [Iw90, Iw92, Sc96] in the case of multicomponent Euclidean covariant generalized random eld. A natural general setting for such studies on vector bundles was given in [Sc96].

1.2 Gibbsian strategies

It follows from the abstract theorem of [Kuz82] that any generalized random eld  on the space of tempered distributions S 0(IRd) can be regarded as a Gibbs eld in the sense of [Do68]. It is quite natural therefore that the problem of GMP was rst studied in the context of Gibbs perturbations theory and in fact all existing proofs of GMP follow this strategy which we call Gibbsian strategy. The lattice spin systems provided us with the simplest laboratory in which the preservation of the GMP under Gibbsian perturbations is studied. It appears that already in simple situations our understanding of this problem is very incomplete. The remarkable examples given in [Go80; GoKS90;Ke85; Is86; AFHKL86] show that the GMP can fail to hold in some cases. Some constructive strategies for the veri cation of GMP of a Gibbs eld have been worked out in the lattice context [Fo80; Go80; AHKO81] and all the existing proofs of GMP in the Euclidean quantum eld theory context followed this route. Dierent techniques in the lattice context have been developed in [Hi84; Ku88a;Ku88b]

1.3 A new constructive strategy for GMP

In this paper d is an integer greater than 1 and IRd is the d-dimensional Euclidean space-time set. Let S (IRd ) be the space of real valued smooth functions with fast decay equipped with its usual topology, see e.g. [Si74] and S 0(IRd) stands for the usual topological dual equipped with the usual inductive topology: it is the space of tempered distributions over IRd . A generic point x 2 IRd will be denoted also as x = (x0; x) where x0 stands for time coordinate and x for the space coordinate(s). S 0(IRd), equipped with the Borel , algebra F , it becomes a standard Borel space; we recall that F can also described as the , algebra d generated by random variables <
; f > where f belongs to S (IR ). If E is a topological space we will denote by F (E ) the related Borel , algebra. A (generalized) random eld will be a probability measure on (S 0(IRd); F ): The free scalar (Nelson) eld will be denoted by 0; it is a Gaussian probability measure on (S 0(IRd); F ) whose characteristic functional is given by the following well known formula: Z ,kf k2,1 i ( ;f ) i ( ;f ) e d0 ()  E0 e = e 2 (1.1) 0 d S (IR )

2

where

Z

djf^()j2(2 + 1),1 (1.2) is the classical Sobolev norm related to and f^ denotes the Fourier transform of f . d For an open subset  of IR we de ne () as the sharp , algebra generated by the random elements (
; f ) with f 2 C01() and completed by all the 0 null sets. If  is closed then () will stand for the germ , eld which is the intersection over " > 0 of (") where  is an ", neighborhood of . The indexing space for the canonical random eld 0 can be identi ed with H,1 (IRd ). For an open subset  of IRd, we de ne the local subspace H,1 () as the closure of C01() into H,1(IRd); for a closed subset  of IRd, H,1 () will be T the intersection " H,1(" ). For a given subset  of IRd we denote by e the corresponding orthogonal projector onto the subspace H,1 (c). If  is open and its boundary @  is suciently regular, the action of e is explicitely given by the following prescription. For f 2 C (IRd) \ H,1 (IRd ); g = e(f ) is the solution of the following classical Dirichlet problem: (,
+ I )(g)(x) = 0 for x 2  and g restricted with c coincides with f . The Markov property of the random eld 0 has been studied in the literature extensively from dierent point of views, see for instance [Ne73a, Ne73b, Pi71, Ro83, Ro85, R82, Wo69, WoH85]. From the general theory developed in [R82, DoM76], it follows there exists a Borel subset R of S 0(IRd) such that 0(R) = 1 and such that the action of e can be extended to R. Moreover for any  2 R the following formula related to conditional expectation holds: E0 fF j(c)g() = E@0  F (
+ e()) (1.3)

kf k,1 = 2

IRd H,1(IRd)

where @0  is the centered Gaussian measure on (S 0(IRd); F ) whose covariance is given by the "inverse" S @ of ,
@d  + I , where ,
@ is the d-dimensional Laplace operator with Dirichlet boundary conditions on @ . If f is a smooth function on IRd, then u = S @f is the unique solution of (I ,
)u = f on IRd n  and vanishing on the boundary @ . There exist several explicit descriptions of the set R and the actions of e on it [DoM76, Ro85]. We say that a random eld  has the global (germ) Markov property if for any (bounded or unbounded) open set , the , elds () and ( c ) are conditionally independent with respect to (@ ). In fact 0 has the global (germ) Markov property, see e. [Ro83,Ro85]. Moreover, considering an open set , it follows that for any f 2 C01() and for any  2 R, there exist regular versions of E0 fei(
;f )j(c)g() which are measurable with respect to the , algebra (@ ). In particular, the germ Markov property is realised with respect to halfspaces by taking  = IRd+  f(x0; x)jx0 > 0g;  c = IRd,  f(x0; x)jx0 < 0g and @   f(x0; x)jx0 = 0g. Often we will denote (@ )  (t = 0); ()  (t > 0); ( c)  (t < 0) and so on. The main inspiration for the present paper come from the work of Rockner [Ro88]. The main idea of [Ro88] was to associate to the canonical random eld (S 0(IRd ); F ; 0), a certain process (t0)t0 of Ornstein-Uhlenbeck type whose law coincides with 0. For this goal, it was shown in [Ro88] that for almost all realisations  2 S 0(IRd) or more precisely for  2 R and for any t 2 IR, there exists a trace THt () of  (de ned in a suitable sense, see [Ro88] and section 2 of the present paper) on Ht  f(x0; x) 2 IRd jx0 = tg; THt () will belong to some separable Banach space that we will denote here by T (IRd,1) and whose precise de nition is provided in section 2 below and also [Ro88]. 3

We have to remark that the (germ) , algebra (t = 0) coincides with the (sharp) , algebra generated by the random elements of the form () = (TH0 (); f ); where f 2 C01(IRd,1): This de nition has no ambiguity since the TH0 () is de ned for 0 almost all  2 S 0(IRd): Such a  can be also formally denoted by (; 0 f ). In particular we say that 0 has the Global (sharp) Markov property with respect to halfspaces and that (t = 0) is the minimal plitting , algebra of (t < 0) and (t > 0). In particular the extension P  of e to R can be expressed as the composition P~  (TH0 ) so it only depends on the traces. From the explicit formulas for the conditional expectation (1.3), and from previous considerations, it follows that, for any f 2 Cc1(IRd+ );

E0 fei(THt (
);f )j(t  0)g() = Yt;f (THt ()); 0 a:s:; where Yt;f is a measurable map from T (IRd,1) to L2(S 0(IRd ); 0). We denote 0t (ei(
;f )); !))  Yt;f (!)

(1.4)

The Markovian and homogeneous character of 0, insures that (0t )t0 extends to a Markovian semigroup on the trace spaces T (IRd,1 ). Let us denote by 0 be the restriction of 0 of the , algebra (t = 0). r0 will stand for for its image measure on T (IRd,1) with respect to the trace map which can be in fact extended to S 0(IRd) 2  ! THt () 2 T (IRd,1 ), by setting 0 outside the carrying set of measure 0, that is to say R. It follows that r0 is a stationary measure for the Markovian semigroup (0t ). From r0, considered as initial law and the semigroup (0t ), the canonical Kolmogorov construction provides a process (t0) whose law coincides with 0. Additionally [Ro88] shows that the process t0 has a version with Holder continuous paths. The main goal of the present paper, is to extend Rockner construction in [Ro88] to the case of interacting eld theories, see e.g. [Si74, Fro74, GJ81]. For this aim, we show that the typical realisations of random elds models of self-interacting elds [Si74, GJ81] has very good properties with respect to trace operations. This will be the content of section 2 below. These results enable us to de ne trace processes t related to suciently regular (see section 2) random elds  on (S 0(IRd ); F ). For  belonging to a suitable R carrying measure , we will have t () = THt (): The random elds providing models of interacting eld theories are obtained by Gibbsian perturbations of the free eld 0, see e.g. [Ne73c, Si76, GJ81, Ro86]. Using this procedure for the nite volume, we can determine the corresponding transitions semigroups and precise the corresponding trace spaces T (IRd,1 ), for the nite volume perturbations of 0. In section 3 below, we present sucient conditions for the identi cation of the "thermodynamical limits" of the perturbed transition semigroups with the corresponding limits of the conditional expectation values of the limiting Gibbs measures; this leads to a new proof of the Global Markov property with respect to halfspaces (GMP) for some Gibbs measures. Particular perturbations of 0 are examined in detail in section 3; this shows that our conditions are ful lled in several models of quantum eld theory. Having constructed explicitely the transition kernels of the "interacting" trace processes, we hope in the future to develop some stochastic analysis tools for understanding the nature of trace processes and identify them as solutions of in nite dimensional stochastic (partial) dierential equation. In a separate paper [AGR97], we have proved the Holder continuity of 4

the trace processes corresponding to suciently regular Gibbs measures describing models of interacting quantum elds.

Remark

It is signi cant to note that a similar idea for verifying the preservation of the global Markov property in the context of lattice systems, appears also in [Ku88a,Ku88b].

1.4 The Dirichlet form approach

An \in nitesimal (or Dirichlet form) approach" has been already introduced in [AHK77a] (see also [AHK76, AHK77b]) in order to prove the GMP of classical Quantum Field Theory models; those were described for instance in [Si74, GJ81] by certain measures on (S 0(IRd); 1): Those measures are obtained by perturbations of the free eld 0 through additive functionals U , see section 3 below. Denote by  the corresponding nite volume perturbed measure, see e.g. [Si 74, GJ81] and let us assume that  converges weakly to 1 as  goes to IRd. Let J C be a \suciently big" set of cylindrical functions in L2(S 0(IRd); dr1 ) where r1 is the restriction of 1 to 1 (t = 0) and such that the formula Z D1 (F; G)  12 dr1 (')rF (')rG(') (1.5) leads to some closable Dirichlet form, see [AHK77, ARo89]. Let (Hph; ph; q0) be the corresponding quantum eld theory objects reconstructed from 1 , see e.g. [S74, GJ81]. Using the weak convergence  ! 1 , certain integrability conditions (which are in most cases ful lled) one can prove the following equality: 1 Z dr (')rF (')rG(') = DF^ (q ) ; H G^ (q ) E ; (1.6) 1 ph 0 ph 0 ph H ph 2 where F^ , G^ are transported (through the natural embedding map J C 3 F ! F^ 2 Hph) functions of the time-0 quantum eld q0 . The rst problem we encounter is that there might exist dierent closed extensions of the Dirichlet form D and a priori the equality (1.6) does not extend to the closures. Even if there exists an unique closed extension D of D which leads to some Markov process tD (concerning the construction of such a process, see [AMRo91, ARo89, ARo90a, ARo90b] ARo91] [ARZ92], [RoZh92]) it may happen that the equality (1.6) is not sucient to identify the corresponding Dirichlet form generator H  with Hph although they coincide on JdC (modulo the embedding J C ! JdC ). Only if J C is a core for H and JdC is a core for Hph we can identify H  with Hph. The only way to remedy this defect is to show that the path space measure of tD is exactly 1 . However, the problem of the uniqueness of the closed extension of D (and the related problem of the uniqueness of the Markov extension [Ta83]) is a very dicult one in the "in nite volume limit". For a solution in the bounded volume case, see [RoZh92, RoZh94]. Our new approach advocated in 1.3 avoids all these subtle and dicult problems as we are dealing exclusively with semigroups and therefore all the domains problems do not arise. 5

2 Trace properties of regular generalized random elds A given probability Borel measure  on the space of tempered distributions S 0(IRd) will be called a 2FR-regular generalized random eld i:

9c,1; c1; cp  0 : ('2(f ))  c,1kf k2,1 + c1kf ? S kL1 + cpkf ? S kpLp (2.1) 8f 2 S (IRd) where p 2 (1; 1) and S = (,
+ I ),1 is the fundamental solution (Green function) of the operator ,
+ I . Similarly we will say that  gives a 2CR-regular random eld on S 0(IRd) i: 9c,1 > 0 : ('2(f ))  c,1
kf k2,1 8f 2 S (IRd): (2.2) It is worthwhile to remark that all the so far constructed bosonic models of Euclidean Quantum eld theory (see [Si74], [GJ81], [Fro74], [AHK74]) are regular random elds in one of the above sense. To compare the above introduced notions of regularity the following lemma is useful.

Lemma 2.1 [AHKZ 89; Gi 86]. 1) If d = 2, then for any p 2 [2; 1): 9 Cp0 > 0 : 8 f 2 S (IR2) kf ? S kLp  Cp0 kf k,1 2) If d = 3, then for any p 2 [2; 6):

9 C 00p > 0 : 8 f 2 S (IR3) kf ? S kLp  C 00pkf k,1: 3) If d = 4, then for any p 2 [2; 4):

9 C 000p > 0 : 8 f 2 calS (IR4) kf ? S kLp  C 000pkf k,1 : 4) For any d = 2; 3; : : : and for any f 2 C01(IRd):

9 C,0 1 > 0 kf ? S kL1  C 0,1 jsupp (f )j 12
kf k,1; where jj means the volume of the Borel set   IRd and supp(f ) stands for the support of f . Let   IRd be a bounded region in IRd having a boundary @  of Jordan type (i.e. IRd n @  is the sum of two connected regions) and at least C 1-piecewise smooth. Then the classical Poisson kernel P  for the classical Dirichlet problem connected to the operator ,
+ I exists and we summarize some relevant properties of P  : 6

(PK1)

E P (x)jP  (y) ,1 (2.3) = (,
+ I ),1(x; y) , (
@ + I ),1(x; y)  K @(x; y) where (,
@ + I ),1  S @(x; y) is the Green function of the operator ,
@ + I ,
@ being the Laplacian with the Dirichlet boundary condition on @ . (PK2) K @(x; y) 2 C 1(IRd  IRd n (fx = yg [ (@   @ ))) and moreover for any x 62 @  there exists limy!x K @ (x; y)  K @(x; x) which has the following properties: (i) K @ 2 C 1(IRd n@ ) (ii) (C 1 for d > 2 , d dist(x;@ )d,2 + G@d  (x) @  (2.4) K (x) 'x!@ 1 , 2 ln (dist(x; @ )) + G@2  (x) for d = 2 where: G@d  2 C 1(IRd ); G@d (0) = 0, and Cd are some numerical constants depending on d. (iii) K @(x) 'dist(x;@)!1 e,jxj: (2.5) (PK3)   8x; y 2 IRdn@  : P (x) ? S (y) = K @(x; y): (2.6) The following result can easily be obtained from the methods and results of [Ro85] combined with lemma 2.1 and the listed properties of the Poisson kernels P  . For a given random eld  and open   Rd we denote by  () or simply by () the -completion of the - eld generated by h'; f i with supp f  . For  closed, we de ne the germ  eld ()  () as the intersection of () where  is an -neighborhood of . This is the so-called germ - eld associated with  and .

8x; y 2 ; x 6= y :

D

Proposition 2.2 For any domain   IRd with a boundary @  of Jordan type and C 1-piecewise smooth, and for any  2 2FR there exists a (@ ) F () measurable map ( F () standing for the Borel -algebra of subsets of  )

P  : S 0(IRd )   3 ('; x) ,! P  (')(x)

such that: (i) For .a.e ' 2 S 0(IRd) : P  (') 2 Ha (,
+ I ), where Ha (,
+ I )  ff 2 C 1()j(,
+ I ) f = 0g :

7

(ii) For f 2 C (IRd) the function P  (f ) is a solution of the classical Dirichlet boundary problem in  for the operator ,
+ I with the boundary condition fj@. (iii) For any molli er  &  as  # 0 and for  a.e. ' 2 S 0(IRd ):

P (') = lim P (' ? ) #0 locally uniformly in , where ? means convolution. The notion of trace of a distribution in the Lions-Magenes [LM72] sense will be used in the present paper. For this goal let us recall the de nition of the fractional order Sobolev spaces. For any  2 IR we de ne:   Z ,  d 0 d 2 2 ,  H (IR )   2 S (IR )j jF ()(k)j (1 + jkj ) dk < 1 with the corresponding scalar product h ; i, and the norm k
k,. Here F () denotes the Fourier transform of . For any open   IRd we de ne H,0  () as the metric completion of the space C01() with respect to the norm k k, and then we de ne H, () as the topological dual of H0 ().

Proposition 2.3 d Let   IR be a bounded region in IRd with smooth boundary @  of Jordan type and let  2 2FR where d = 2; 3; 4 and the corresponding p in de nition 2.1 are restricted respectively to (2; 1) (resp. [2; 6), resp. [2; 4)). Then for  a.e. ' 2 S 0(IRd): P (')(
) 2

\

> d,2 2

H,+ 12 ():

Proof

Let f be a C 1-function on  such that for all z 2 @  f (x) = const. 6= 0: lim x&z dist (x; @ ) From Fubini theorem and Lemma 2.1 we have, with s 2 (0; 1) to be choosen below: Z Z  2 d(') dx fs (x) P (')(x) Z Z  2 = dx f2s(x) d(') P  (') (x) (2.7) Z    c + c0jj 12  dx f2s (x) K @(x) for some constants c; c0 2 IR. The last integral is nite for d = 4 if we choose s 2 ( 12 ; 1) and in d = 2; 3; if we choose 8

s 2 (0; 12 ) as it follows from PK2 (ii). This says that fs P  2 L2. Theorems 11.2 and 11.3 of Chapter 1 of [LM72] say that u ! f,s u are continuous linear mapping from H s() (resp. H0s ()) into L2(). Therefore P (') 2 H,s (). Consequently P  (')(
) 2 \> d,2 2 H,+ 12 () for  a.e. ' 2 S 0(IRd) and d = 2; 3; 4. According to the general theory,1 as developed in [LM72], there exists an operator (trace operator) T@ de ned on H,+ 2 () with values in the space H,(@ ) (see the de nition below) with the following properties: (i) T@ is a linear bounded operator from H,+ 12 () to L2(@ ; do (x)) (where do is the Lebesgue surface measure on @ ) (ii) for f 2 C () : T@(f ) = fj@ (iii) Ker T@ = H,0 + 2 () 1

(iv) H, (@ ) = T@(H,+ 12 ()). For the proofs see Chapter 2 in [LM72]. We summarize our discussion in the following theorem:

Theorem 2.4 Let   IRd for d = 2; 3; 4 be a bounded domain with the boundary @  of Jordan type and C 1-smooth. For any  2 2FR with p as in lemma 2.1 there exists the trace T@(') 2 H, (@ ) for  a.e. ' 2 S 0(IRd) and any  > d,2 2 and such that 8x 2  P  (T@(')) (x) = P @(')(x): We extend this "local trace properties" to the case of half-spaces. A similar analysis for more general unbounded regions in IRd can be carried on also, however for the main purposes of the present paper the case of half-spaces is sucient. We remark that the germ - eld 0 (@ ) coincides with the sharp -algebra generated by < T@('); f > where f is a smooth function supported in @ . Let Ht = f(s; x) 2 IRdjs = tg: Denote by P + the corresponding Poisson kernel P  where  is the upper halfplane IRd+ = f(s; x) 2 IRdjs > 0g. It is well known that, for t  0;

p,
d,1 +I

Pt(x)  P + (t; x) = e,t

(x)

(2.8)

where
d,1 means the "spatial part" of the d-dimensional Laplacian (we think of (s; x) 2 IRd with s as time and x as space). For any f 2 L2(H0) \ C (H0) we have:

P +(f )(t; x) = (Pt f )(x):

(2.9)

The basic spaces of traces in the hyperplane H0 for a typical ' 2 S 0(IRd) will be introduced, 9

following Rockner [Ro88], under the assumption that the corresponding ' is distributed according to a 2FR regular eld . The novel aspect here is the remark that some of the spaces of traces introduced by Rockner [Ro88] for the free elds are proper spaces of traces also for 2FR regular generalized random elds at least in dimension d = 2, including several models of interacting Euclidean quantum elds. We start with describing the appropriate trace spaces for  2 2CR. Let % 2 L2(IRd,1) be such that (i) % 2 C 1(IRd,1); % > 0 (ii) all partial derivatives of % and %,1 are polynomially bounded at in nity (iii) %2 is (Pt)t0 supermedian i.e. Pt%2(x)  %2(x) : 8 t  0; x 2 IRd,1: The scaled Sobolev space H,%  (IRd,1) is then de ned as: n o H,%  (IRd,1)  ' 2 S 0(IRd,1)j%' 2 H, (IRd,1) equipped with the norm k'kH,%   k%'kH, . The linear subspace B%, of H,% (IRd,1) is de ned as:

B%,(IRd,1)  f' 2 H,%  (IRd,1 ) j Z1 0

dt t,1

Then the space of traces

Z

  2 (x) t2 P + (') 2 (t; x) < 1g dx % d,1

(2.10)

IR B%,(IRd,1)

is de ned by n o B%,(IRd,1) = ' 2 B%,(IRd,1)

and equipped with the norm

k'k2B%, = k'k2H,%  +

Z1 0

Z dt t,1 dx %2(x)(tP +('))2(t; x):

(2.11)

The following fundamental properties of the spaces B%, have been established by Rockner [Ro88]: (p1) for any % as above,  > 0 the space B%,(Rd,1 ) is a separable Hilbert space (p2) the semigroup (Pt)t0 acts on B%, i.e. ' 2 B%, ) Pt' 2 B%, (p3) if  > d,2 2 then H 12 (IRd,1) is continuously embedded in B%,(IRd,1). 10

Proposition 2.5 Let  2 2CR. Then for  - a.e. ' 2 S 0(IRd) there exists a trace TH0 (') of ' on the hyperplane H0 taking values in the space B%,(IRd,1) for any  > d,2 2 . Proof

By the assumed 2CR-regularity of  we have: Z Z1 Z  2 d(') dt d,1 dx t2,1 P + (')(t; x)
%2(x) IR Z 1 Z0 Z  2 2 2  , 1  0 dt IRd,1 dx % (x) t d0(') P + (')(t; x) Z1 Z = C,1
dt d,1 dx %2 (x) t2,1
K H0 (t; x) 0

(2.12)

IR

which is nite if  > d,2 2 . From the Lions-Magenes1 theorem (similarly as in the proof of Theorem 2.4) it follows that %
P + (')(t; x) 2 H,+ 2 (IRd+ ) and therefore again from the general theory of traces (see [LM72, Vol.III]) it follows that there exists a trace TH0 (') of ' that takes values in the space H,%  (IRd,1). To localise the obtained trace TH0 (') better, in the space B%,(IRd,1) we have to show that the norm k'kB%, is nite, but this has already been done.

Remark

It is clear that the above arguments can be used to show that for  a.e. ' 2 S 0(IRd) there exists a trace of ' on any hyperplane Ht = f(x0; x) 2 IRd); x0 = tg and that this trace takes values in B%,(IRd,1). Therefore it is naturally to de ne a L2(S 0(IRd); ; B%,) process Xt by Xt (')  THt ('). The detailed study of various properties of the corresponding process is planned to be subject of another paper in this series. For the main purpose of the present paper the existence of traces is sucient. The situation concerning 2FR-regular random elds is more delicate. Because any such eld  is locally 2CR-regular, we conclude that for -a.e. ' 2 S 0(IRd ) there exists a trace TH0 (') of ' which takes values in the Frechet space H,loc (IRd+ ), where H,loc (IRd+) = ff j for any bounded   IRd,1 : 
f 2 H,(IRd+)g.

Proposition 2.6

Let d = 2 and let  2 2FR. Then for  a.e. ' 2 S 0(IRd) there exists a trace TH0 (') of ' on the hyperplane H0 and it takes values in the space H,%  (IRd,1) for any  > 0.

Proof

11

Let IRdT = f(t; x) 2 IRdj 0  t  T g for some T > 0: Then we have: ZT Z Z 2 d(') d,1 dx % (x) dt t2,1 (P + (')(t; x))2 Z IR Z T0 Z 2 2  , 1  C,1 IRd,1 dx % (x) 0 dt t d0(') (P + (')(t; x))2 t ZT Z 2 2  , 1 +C1 IRd,1 dx % (x) 0 dt t kP (t; x) ? S k1 Z ZT +Cp IRd,1 dx %2(x) 0 dt t2,1 kP (t; x) ? S kpp Z ZT 2 = C,1 d,1 dx % (x) dt t2,1 K H0 (t; x) Z +1 Z 0 ZT Z IR 2  , 1 2 d K H0 (t; xj; y) +C1 d,1 dx % (x) dy dt t 0 ,1 IR Z Z ZT Z +1  p +Cp d,1 dx %2(x) dy dt t2,1 d K H0 (t; xj; y) IR

(2.13)

,1

0

From the re ection principle property we know that K H0 (t; xj; y) = S (x , y; t +  ). Therefore (2.13) equals ZT Z C,1 d,1 dx %2(x) dt t2,1 K H0 (t; x) ZIR Z0 T Z Z +1 2 +C1 d,1 dx % (x) dt t2,1 dy d S (x , y; t +  ) ZIR Z0 T Z Z,1 +1 +Cp d,1 dx %2(x) dt t2,1 dy d (S (x , y; t +  ))p IR

,1

0

The case d = 2 implies that S 2 Lp(IR2) for any p  1. Taking this into account we conclude that all integrals appearing in (2.13) are nite provided  > 0. Referring again to the Lions-Magenes theorem for weighted Sobolev spaces we conclude that for -a.e. ' 2 ,+ 12 2 0 2 + S (R ); P (')(;
) 22 H% (IRT ) for any  > 0 and by the trace theorem we conclude that 2 0 ,  for  a.e. ' 2 S (IR ) the trace TH0 (') exists and belongs to H% (IR+ ) for any  > 0. At the moment we do not know whether within the class of 2FR regular elds we can localise the traces TH0 (') in the subspace B%,(IR). However, it follows from the above proof that we can introduce the following Frechet metrizable space B%,;loc(IRd,1 ), de ned below, as a substitute for B%,(IR), at least for d = 2. ) ( Z ZT def. 2 2 2  , 1 , ;loc ,  B% (IR) = f 2 H% (IR)j 8T < 1 dx % (x) dt t (Pt (f )) (x) < 1 0

and equipped with the family of norms

kf k,% ;T = kf kH,% (R) +

ZT 0

Z dt t2,1 dx %2(x)(Pt(f ))2(x):

Then obviously the Poisson semigroup (Pt )t0 acts continuously on the space B%,;loc(IR). 12

Remark

A nice regularization of the process Xt(') = THt (') can be obtained by de ning Xt()(')(
)  P (')( + t;
) for  > 0. Then the regularized process Xt (') takes values in the space of C 1-functions. By the arguments used in the proof of lemma 3.10 in [Ro88] it can be shown that for any  > 0 there a modi cation of Xt on a set N 2 F of  measure 0 such that  fexists  >0 the modi ed process Xt t2R becomes F =F (B%,) measurable and for any t > 0 it belongs to L2(S 0(IRd); ; B%,). The study of paths Holder continuity of the limiting processes (Xt), for  going to 0, has been performed in details in [AGR97]. We complete this section by providing pointwise boundary estimates for the harmonic functions P t(')(; x). For this we need to impose more restrictive regularity conditions. We will say that a given cylindrical Borel probability  on S 0(IRd ) de nes an FR-regular random eld i: Z 9 C,1; C1; Cp  0 : 8 f 2 S (IRd) e'(f ) d(')  eC,1kf k21+C1 kf?SkL1 +Cp kf?SkLp : (2.14) We say that  de nes a completely regular , i.e. CR, random eld i: Z 2 e'(f ) d(')  eC,1kf k1 :

(2.15)

Theorem 2.7

Let d = 2 and let  be an FR regular random eld. Then there exists a constant C > 0 such that: 1 0 1 (2.16) 8s 2 (0; 6 ) jP + (')j(s; x)  C @ln A() + sup 1 (ln %,1 (y))A + ln 1s ; jx,yj< 12

where

A(') 

Z 0

1 3

ds

Z

2 (x) dx e jP +(')j(s;x) % IR

is nite for  a.e. ' 2 S 0(IR2) and 2 < 4.

Proof

First we show that A(') < 1 for  a.e. ' 2 S 0(IR2). For this we shall use (PK1), (PK3) and the equality (valid for s > 0; t 6= 0; y 6= 0; x > 0):

K H0 ((x; s)j(y; t)) = S (x , y; t + s)

13

(2.17)

which follows from the re ection principle, Z d('Z) e jP + (')j(s;x) Z + (')(s;x) P  d(') e + d(') e, P + (')(s;x)  n  C,1 2 H0  2 exp 2 K (s; x) Z  1p ! o Z p +C1 dy dt S (x , y; t + s) + Cpj j dy dt jS (x , y; t + s)j

(2.18)

Obviously, by a change of integration variables Z dy dt jS (x , y; t + s)jp < 1 for any p  1: Therefore:

Z

Z Z ds %2(x) d(') e jP +(')j(s;x) 0 Z 1 Z 3  C 0 ds IR dx %2(x) e 12 2KH0 (s;x) < 1; 1 3

(2.19)

for any 2 < 4. Having proven A(') < 1 for  a.e. ' 2 S 0(IR2), we can proceed further in an analogous way as in [Ro88], Theorem 6.22(ii). The re ection principle argument used in the above proof does not help much in higher dimensions due to the fact that S 62 Lp(IRd) if d > 2. However for a completely regular random elds we also have an analogue of Theorem 6.2 (i) of [Ro88]

Theorem 2.8

Let  be completely regular random eld on S 0(IRd). Then for any p > 1;  > 1p there exists a constant C such that ,d,1   8 s > 0 : jP +(')(x; s)j  CAd() supjx,yj 0), a sequence (n) as in the Step 0, the pointwise limit on  exists: + (3.6) nlim !1 n (F )( )  1(F )( ) : Let us remark that the main technical diculty we meet here (and in the corresponding weaker version described in Step 1' below) is the construction of +1(F )( ); uniform in the parameters of (U). This uniformity problem is solved for the models presented in the next section. For the case of polynomial interactions we have the proof of the local existence of +1(F ), however our extimates are not yet uniform in the coupling constant. Now we have to check the validity of 16

Step 2. nlim !1

Z

Z

+ n) d (')n (F )(TH0 ('))G(') = d( 1 (')1 (F )(TH0 ('))G(') :

(3.7)

Assume now that for some U , some sequence (n) as above we are able to perform the steps 0 through 2. Then: ) the limiting Gibbs measure ( 1 n has the Global Markov property with respect to the hyperplane H0 (in fact to any hyperplane Ht ) and moreover: E(1 n) (F j(t  0))( ) = +1(F )(TH0 ( )) (3.8) ) ( 1 n almost everywhere.

Proof

From the very de nition of the conditional expectation values and Step 0 ) ( 1 n (F
G) = limn!1 n (F
G) = limn!1 R n (En fF j(t  0)g
G) = lim dn ( )+n (F )(TH0 ( ))G( ) R n!1 ( ) = d1 n ( )+1(F )(TH0 ( ))G( ) The later equality is a consequence of Step 1 and Step 2. We observe that +1(F )(TH0 ( )) is (t = 0) measurable since it is a limit of (t = 0) measurable functions. Taking into account the uniqueness part of the Radon-Nikodym theorem we conclude that +1(F )(TH0 ( )) = E(1 n) fF j(t  0)g( ) ) 0 d ( 1 n almost everywhere on S (IR ). To check the validity of Step 2 the following simple argument is sucient. Assume that there exists a function f : IR+ ! IR; decreasing to zero at in nity, such that j+1(F )( ) , n (F )( )j  f (dist(0; n c))
,n ( ) ; (3.9) where the functional ,n ( ) depending on F as well ful lls: Z sup dn ('),n (TH0 (')) < 1 : (3.10) n

If (3.9) and (3.10) are ful lled then equality (3.7) also holds. The condition stated as Step 1 can be weakened. For this goal let us de ne a Borel subset reg  of  such that for any suitably regular measure  we have (reg)  1. The weakened version of Step 1 is formulated as: Step 1'. for any 2 reg, any (n ) as above, any bounded F , the limit limn!1 +n (F )  +1(F )( ) exists uniformly in the parameters of U. If moreover there exists a function f : IR+ ! IR decreasing to zero at in nity such that + ( (F ) , + (F ))( )  f (dist(0; n c))
,n ( ) 1

n

17

for any 2 reg and the functionals ,n ( ) obey: Z sup dn ('),n (TH0 (')) < 1 ; n then: ) the limiting Gibbs measure ( 1 n 2 Gr (U ) has the (sharp) Markov property with respect to H0 and moreover

E(1 n) f
j(t  0)g = +1(
) ) ( 1 n almost everywhere.

Remark

(n ) n) Let v1(n)  ( 1  TH0 be the image of the measure 1 under the trace taking the map TH0 onto the space . In all applications we have in mind we are able to de ne ) a generalized random process t belonging to  whose law is given by ( 1 n , i.e.:

0 d n) Pr(t 2 B ) = ( 1 f' 2 S (IR ); THt (') 2 B g ;

for any B 2 F (). The transition kernel of the process t is given by t t ( ; eih
;f i) = +1(ei(;t f ))(TH0 () = ) and the corresponding initial (and stationary) distribution is equal to v1(n). This will be discussed in details in the third part of the present paper. Now we come to the construction of the transition kernels + for some particular models of Euclidean Quantum Field Theory.

3.1 Gentle perturbations of the free eld (with UV{cuto) Let X 2 C01(IRd,1) be such that X  0 and let the size of the support of X be less then  where 0 < . For a bounded Borel   IRd we de ne Z Z  U (') = z d() : ei' : (x)ddx; (3.11) 

where z 2 IR; d() = d(,) is a bounded (in general complex) measure on (IR1; F (IR1)), with the compact support contained in the interval [,; ], Z '(x0; x) = ('  X )(x0; x) = dy'(x0; y)X (x , y) ; (3.12)

 2  : ei' : (x) = exp 2 S(0) ei'(x) ;

(3.13)

S(x)  ((X X )  S )(x) :

(3.14)

18

The corresponding nite volume Gibbs measure (with "empty boundary conditions") d is given by: d (') = (RZ ),1 exp U (')
d0(') ; (3.15) Z  = d (') exp U  (') : 0





The corresponding DLR{equation has been studied before in [AHK73, AHK79a, AHK79b, GiZ84, Gi86, Gi92a, Gi92b, Gi91]. We summarize the known facts about the set of in nite volume regular Gibbs measures corresponding to (3.11), which we denote by Gr (z), where the subscript r corresponds to the notion of 2CR{regularity.

Proposition 3.1 1. Let z  0, d() = d(,) be a real positive measure with the properties as above. Then for any z  0 the set GCR(z) is non void. 2. If in addition, the value of z0 > 0 is regular (i.e. the corresponding in nite{volume free energy density P1 (z) is dierentiable at the point z0) then the subset GrT (z0) of 2CR{regular Gibbs measures which are translationally invariant consists of exactly one point. 3. Let  be as in (3.11) and let JK1 be the corresponding Ruelle{Kirkwood{Salsburg operator (see below) and let B be the corresponding Banach space (see below). Then for any z 2 IR: z,1 62  (JK1 ) ; jzj <  where  (JK1) is the spectral set of JK1 in B ; the set G2CR(z) consists of exactly one point. 4. Let: jzj < C1 exp[,2S(0)) , 1] where Z C = sup je, S(x) , 1jdxd()

Then ]G2CR (z) = 1; where ] denotes the number of elements. The following expression for the conditional expectation with respect to the {algebra (t = 0) can easily be derived: n o E nei(';f ) j (t  0)o ( ) = E ei(';f ) j (t = 0) (TH0 ) = exp(i(P + (TH0 ( )); f ))
e,12 SH0 (f;f ) (3.16)  H0 )(x ) R P Q n 1 n ,  ( f  S i i 
n0 n! + d(x; )1 i=1 e ,1
+ ((x; )n1 ) ; where we have de ned: (x; )n1 = (x1; 1; : : : ; xn; n) ; (3.17) ! n n O O n d(x; )1 = dxi

d(i ) ; (3.18) i=1

i=1

19

+ ((x; )n1 )  Qni=1Q+ (xi)
zn
Qnj=1eijP + ( )(xj ) (3.19)
 nj=1 : eij ' : (xj ) ;  ,1  (d') = Z exp U+ (')dH0 0 (') ; (3.20) Z Z U+ (') = z d() + dx : ei' : (x)eiP +( )(x) ; (3.21)  Z Z+ = dH0 0 (') exp U+ ('); SH0 = (X X )  S H0 ; (3.22) where nally +   \ IRd+ and dH0 0 means the Gaussian measure on (S 0(IRd); F ) with mean zero and the covariance S H0 and nally the shorthand P + ( ) = P +(TH0 ( )) is introduced. The equation (3.16) is understood in the 0 -a.e. sense. The essential volume dependence of n o E ei(';f ) j P(t  0) is contained in the correlation functions + . The thermodynamic limits of + as + % IRd+ and for all 2 B,(IRd,1), can be controlled by the analysis of the corresponding Kirkwood{Salsburg identities that we present below. An integration by parts formula with respect to the measure d gives the following identities: P + ( )(xj )  ((x; )n1 ) = zn Qnj=1 eijP
exp[,1
ni=2 iSH0 (x1 ,h xi)] (3.23)
+ Qnj=2 : eij'(xj ) : exp z R d() R+ dx    i H e,1S 0 (x,x1) , 1 : ei'(x) : eiP +( )(x) in which we easily recognize the Kirkwood{Salsburg identities, see [AHK73], [Ru60] . Let B be the Banach space consisting of all sequences of (dx d) n measurable functions f = (fn)1n=1 and supported on n+ and such that: ,n ess sup jf ((x; )n )j < 1 kf k = sup  n 1 n n (x;)1

where  > 0 will be choosen later on. In the space B we de ne several linear operators as follows. { the in nite volume Kirkwood{Salsburg operator K1 :   H0 (x ,y ) R P Q 1 1 m n m ,  S 1 j 1 j  (K f ) ((x; ) ) = d(y; ) e ,1 1 n

P

1

2 H e,1 i=n iS 0 (x1,xi)


m=1 m!
fm((y; )m1 )

1

j =1

if n = 1 ; R d(y; )m Qm e,1 j SH0 (x1,yj ) , 1 m0 m! 1 j =1 n fn+m,1 ((x; )2 ; (y; )m1 ) if n > 1.

P

1

(3.24)

{ the index juggling operator J , choosing indexation of (x1; : : :; xn) such that: n X 1j SH0 (xi; xj )  ,2SH0 (0) (3.25) j =2

20

{ the projectors Q (f )n ((x; )n1 ) = ni=1+ (xi)fn((x; )n1 ) ;

(3.26)

{ the vectors:

+ = (+ (x)eiP +( )(x); 0; : : : ; 0; : : :) (3.27) with these notations the Kirkwood{Salsburg identities (3.22) can be rewritten in the following way:  = z+ JK1   + + z+ (3.28)  + where    + =  (x1; 1); : : :;  ((x; )n1 ); : : : 2 B for a suitable . We choose  = (C),1 in the analysis below. The nite volume equalities (3.28) will be compared with the following one.  = z( )JK1 1+ + z (3.29)  1+

where:

  ( ) = eiP ( )(x); 0; : : : 2 B : From now on we take to be an arbitrary element of the space of traces B,;  > d,2 2 .

Proposition 3.2

1. Let (n )n be any monotone sequence of bounded Borel subsets of IRd such that [n n = IRd and let jzj < (C),1 exp[,2SH0 (0),1],  = (C ),1 Then for any 2 B,;  > d,2 2 ; the unique thermodynamic limit limn!1  +   d exists in the space B in the  IR+  n m sense that for any m: limn!1 +n ((x; )1 ) = IRd+ ((x; )m1 ) locally uniformly. Moreover  d =  1+ .  IR+

2. Let ( n )n be another sequence as in 1 and additionally such that: 8n : n  n and limn!1 dist ( n ; cn) = 1. Then for any 2 B,(IRd,1)

 

nlim !1  n   , 1+ = 0 n



The convergence is uniform on B,(IRd,1 ).

Proof:

Part 1 follows easily by standard arguments: the contraction map principle applied to (3.28) and (3.29) and the comparison analysis of the corresponding resolvent expansions, similarly as in [Ru60]. The proof of the part 2 is based on the following lemmas, the proofs of which are placed at the end of this subsection. 21

Lemma 3.3

Let (n)n , ( n ) be as above and let 0n be another sequence of that type and such that 8n : n  0n  n and limn!1 dist ( n ; 0nc) = 1, limn!1 dist ( 0n; cn) = 1. Then: strongly on the space B :   0 lim  ( J K  ) , J K  n 1

n = 0 : n!1 n n 1

Lemma 3.4

Let (n ), ( n ), ( 0n ) be as in Lemma 3.3. Then:    0 lim   J K  1 ,  =0 c

 1  n n n

n n!1 and

   0n = 0 : lim  ( J K ) 1 , 

1

n n!1

strongly on B .

Lemma 3.5

i )i=1;:::m such For any m = 1 ; 2 ; : : : , any ( ) , (

) as in Lemma 3.3 and a family (

n n n n n n1;::: that Wn;i n  : : :  in  in+1  : : :  n and limn!1 dist( n ; 1nc) = : : : = limn!1 dist( in; in+1 c) = : : : = limn!1 dist( mn; cn ) = 1 Then:

h i lim

 ( J1K  )m , (J1K )m

= 0 n!1

n

n

n

n



Having proven the above lemmas the proof of Proposition 3.2 follows by mimicking the arguments of Ruelle [Ru60], showing:

 h i,1 

, 1

lim  n [1 , zn J1K n ] n , 1 , z J1 K 

= 0 : n!1



On the space B,(IRd,1 ) let us de ne the following functional:   R  ei'(f ) ( )  ei(P +( ) ; f )
P1n=0 n1! IRd+ d(x; )n1 Qn e,1f SH0 (xi) , 1  ((x; )n) 1+

i=1

1

(3.30)

Proposition 3.6 Let: jzj < (C ),1[exp(,2SH0 (0)) , 1]; where SH0 (0) = SH0 (0; 0). Then for any 2 B,(IRd,1) we have: n o limn!1 E~n nei(';f ) j (t  0)o ( ) = limn!1 E~n ei(';f ) j (t = 0) ( ) =  ei(';f ) ( ):

and the convergence is uniform on the space B, (IRd,1), where E~n denotes the extension of En given by (3.16) to whole space B,(IRd,1). 22

Proof.

Let and (n )n monotonic sequences of bounded Borel subsets of IRd such that Sn n = S ( n=)n IR d ; for any n :   and moreover lim n n n!1 dist( n ; cn ) = 1. For  2 n n f+n ; 1+g and 2 f +n ; (n n n )+ ; 1+g we de ne:   R n ( ) = ei(P +( );f ) Pn0 n1! d(x; )n1 Qni=1 e,1f SH0 (xi) , 1 (3.31)
 ((x; )n1 ) ; where for  = +n the allowed values of are given by +n ; (n n n )+. With this notation we have:    ei(';f ) ( ) = 1n + ( +n ) + 1n + ((n n n )+) (3.32) +1n + (cn ) ; n o Een ei(';f )j(t = 0) ( ) = n +n ( +n ) + n +n ((n n n ))+ : (3.33) From the Proposition 3.2 it follows that for any  > 0 there exists n 2 IN such that for any n > n we have: +n +  ( ) , 1+ ( + ) <  uniformly in 2 B,(IRd,1) : (3.34) n

n

n

n



+ + + + 1+ c To estimate the dierence 1 n (n n n ) , n n (n n n ) and n (n ) the decay properties of f  SH0 should be taken into account. For this observe rst that the dierence + - 1+ in the norm k k

2 jzj
e2SH0 (0)

 , 


(3.35) H H 2 0 0

 +n  1+  1 , jzj eS (0)+S (0)  

Therefore we conclude that there exists a constant C (; z) such that 1+ n h ((n n n )+) ,  n n (n n ) i  exp C (; z) Rnn n dx f  SH0 (x) R jjd() , 1 Similarly: Therefore:

1+ n h ((n)c) R i )R H 0 +  exp C(;z dx f  S ( x ) jjd() , 1  (n )c 2 limn!1 E~fei(';f ) j (t = 0)g( )  ,  + ei(';f ) ( ) = 0 1

uniformly on the space of traces B,(IRd,1). Let f 2 C01(IRd+), g 2 C01(IRd, ). Then we have R d ()ei(;g)E nei(';f ) j (t  0)o (T ())   H0 = R d (')ei(';f +g) 23

(3.36) (3.37) (3.38)

(3.39)

Let 1 be the unique (if z is suciently small) Gibbs measure constructed in Proposition 3.1; then we know that n ,! 1 weakly, therefore the right hand side of (3.39) tends to 1(ei(';f +g)) as n " IRd. Collecting (3.34), (3.35) and (3.37) together we obtain the following estimate: n o j1(ei(';f ))( ) , Een ei(';f ) j (t  0) ( ) j (n) uniformly on B,, where limn (n) = 0. Therefore we have: n o R limRn dn ( )En ei(';f ) j (t  0) (TH0 ( ))ei(';g) = d1 ( )(ei(';f ))(TH0 ( ))ei(';g) for any smooth g supported in the lower halfspace t  0. We conclude our discussion in the following theorem.

Remark.

It follows from the uniform (in 2 B,) contractivity of the corresponding Kirkwood{Sals burg operators that the maps  ,! 1(ei(
;f ))( ) are holomorphic for small  : jj < 0 and for every 2 B,. In particular the following perturbative expansions converge for jj < 0: T X  1(ei(
;f ))( ) = ei(P +( );f )
n1! ei'(f ); UIR+d (')(x 1); : : : ; UIR+d (')(x n ) SH0 n0

Theorem 3.7. Let jzj < C,1 (exp ,2S(0) , 1) : Then the unique, 2CR,regular Gibbs measure 1 2 G (z) has the Markov property with respect to any hyperplane Ht = f(x0; x) 2 IRd j x0 = tg (Global Markov property). Remark. The corresponding B,{valued Markov process tz whose initial measure r10 is the image

through TH0 of (1) and whose transition kernel is

zt (TH0 (); eih
;f i) = E1 (eih
;t f i j (t = 0)(TH0 ())

(3.40)

will be studied in a forthcoming paper. See also [Gi92] where some preliminary results on the process tz have been announced.

Proof of Lemma 3.4 Take any  2 B . Then for any k > 1 we have

 n (JK1 )nn 0n ()K (x; )K1  eh 2SH0 (0)
,1
k~k i exp K (n; n)jn n n j , 1 24

(3.41)

where:

H0 (x;y) ,  S (n; n) = sup; sup y2x2n nn 0 e t , 1 n , SH0 (x;y) sup x2 n S H0 (x; y )  sup ; j  j e y2nn 0n  x;y c  const e,dist ( n;n ) Similarly for k = 1;



 n (JK1 )nn n ()1(x; )

0c  k~k,1 je const. jn n nje, dist ( n; n ) , 1j

(3.42)

(3.43)

Proof of Lemma 3.5

We use the following decomposition: m ( n JK1n )    Q = mi=1Pn  in +Q icn JK1  in +  icn n :::m =0;1 m = n 1;:::;  in (i) JK1  in( i)n m =0;1 i=1

(3.44)

in (0) = in ; in (1) = icn

(3.45)

1

where:

Then we have: n m  ( ) , (JK )m ( )o  n (n JK  ) 1  1 n Qnm n P Q
 i J K  i J ( )o :::m =0;1  n i=1  in (i ) JK1  in ( i )n ,  n m = 11;:::;

n ( i ) i=1 n (i ) 1 m =0;1 (3.46) Therefore from the Lemma 3.3 and Lemma 3.4 it follows:

h i lim

 ( J1K  )m  ( ) , (J1 K )m( )

= 0 (3.47) n!1

n

n

n

n



3.2 2D sine{Gordon model (weak coupling regime) p

In the case of d = 2 and if  < 2  it is possible to show the existence of the limit lim#0 for the models considered in Section 3.1. Several authors discussed this model in the past [Fro75, Fro76, FroSe76, FroPa79, AHK79a, AHK79b, Gi86, Gi92b]. We summarize some existence theorems below. Let Z Z U(') = z d() : ei' : (x)d2x ; (3.48) where now :z 2 IR, supp d  (,2p; 2p) and let Gr (z) denotes the set of the corresponding regular Gibbs measures. 25

Theorem 3.8. [Fro75, FroSe76, FroPa79, Gi86, AHK79a, AHK79b, Gi92] 1) Let d(,) = d(). Then for any z  0 the set GCR(z) 6= 0. For any regular value of z  0 we have G2CR(z) = 1 and the unique Gibbs measure z1 2 G2CR(z) has the

Global Markov Property. 2) For general d it is known that the set G2FR(z) is nonempty for any z 2 IR. For simplicity of our exposition we assume: p d() = 12 ( , ) + ( + ) ; for some xed  2 (0; 2 ) and we use the shorthand: : e(') :=: ei' : Now similarly as in the previous model n o n o E ei(';f ) j (t  0) ( ) = E ei(h';f ) j (t = 0) (Ti H0 ( )) R d(x; )n Qn e,if SH0 (xi) , 1 = ei(P +( );f ) P 1 TH0 ( )

C+

where:

TH0 ( )

C +

n0 n! (+ )n ((; x)n)

1

i=1

(3.49) (3.50)

(3.51)

1

n Y n n ((; x)1 ) =  eilP +( )( l=1

)(xl)+

n Y i=1

: ei (') : (xi)

!

Z exp U (') + FdH0 0 (') ; + (F ) = Z+ Z Z U (') = d() dx : e(') : (x) : e(P +( ))(x):

;

(3.52) (3.53)

Proposition 3.9. 1) 9z0 > 0 : 8jzj < z0 : 8 2 H,  (IR), any Borel n % IRd there exists a unique limit in the sense of complex measures: 1 m C+n (fm ) n,! ,!1 C1+ (fm ) for any fm 2 L (IR )

2) Moreover for jzj < z0 we have the estimate with f  C0(IR2+); for any m = 1; 2; : : : C+n (f~m) , C1+ (f~m) (3.54)  exp (, dist (s(f ); cn ))
kf km1
C (f )m uniformly on H,  (IR); where: m h i Y ~ fm = e,if SH0 (xi) , 1 : (3.55) i=1

26

Proof.

The method of cluster expansion in the version as presented in Glimm{Jae [GJ81] and adapted to the similar context in [FroSe76, AHK79a, AHK79b] will be used to control the limit of C+n . The notation used below is that of previous references. For any fm 2 L1 (IR2+m) the cluster expansion for C+ (fm ) is summarized in the following identity: C+ (fn ) = P(X;,) R01 ds,@, R < d(x)m1 Qmi=1 : ei (') : (xi)fm(xm1 )eU(\X) >CH0_(s(,)) (3.56) Z+ (+ nX )
Z (+) : @X

where: CH0 _ (s(,)) are Pinterpolating covariances with the Dirichlet boundary condition imposed on H0 . The sum X;, is taken over the pairs (X; ,) such that each connected component of X n ,c has nonzero intersection with s(fm) = suppfm . As is well known there are two basic model dependent facts which must be checked in concrete situations. We list them below. We recall, however, that our problem is to show uniformly in the convergence of the corresponding cluster expansion.

Proposition 3.10.

For any K > 0 there exists a constant K and a norm k
k on C01(IR2n) such that for any  and suciently small jzj; the following estimate holds uniformly on H,  (IR): R Q @,< mi=1 : ei : (')(xj )fm(xm1 )eU+\X >CH0_S(,) (3.57)  e,Kj,j+Kjj
kfmk

Proof. The +proof is literally the same as in [FroSe76] using additionally the simple fact that jeiP ( )(x)j  1. Proposition 3.11. There exists a constant K2 such that for suciently small jzj we have Z ( n X )Z ()  eK2jX j (3.58) uniformly in 2 H,  (IR). Proof. We rst remark that for any unit cube
 IR2+ and any 2 H,  (IR) there exists a z0 > 0 such that for any jzj < z0 we have

1 < Z (
) < eO(z)
jj (3.59) @
2 The uniform upper bound follows by an+ expansion in powers of z and the Frohlich estimates [Fro76], where again we are using jeiP ( )j  1. 27

The lower bound in (3.59) follows from the Jensen inequality   R d()eiP + ( ) z H _ @
0 e  Z@
(
) =  eU
and: Therefore:

Z

jz ()eiP + ( )j  z
Var() : Z@
(
)  e,jzj
Var()

(3.60) (3.61)

uniformly in . The existence of z0 such that (3.59) holds for jzj < z0 is one basic step in the analysis of the corresponding Kirkwood{Salsburg identities for deriving the bound (3.58) [GJS]. The uniform contractivity (in 2 H,  (IR) ) of the corresponding Kirkwood{Salsburg operator follows from the fact that jeiP +( )(x)j < 1. As the combinatorial estimates for the cluster expansion (3.56) are model independent we can conclude that this cluster expansion converges as  " IR2+. Moreover we know that for any K > 0 there exists a constant z0 such that for any jzj < z0 there exists a continuous norm k
k on S IRd such that: R Q CH0 _S(,) P U m m + (X;,):jX j>D @, < i=1 ei (') : (xj )fm (x1 )e  \X > (3.62)
Z@XZ (()nX )  kfmke,K(D,m) The concrete form of k k was derived in [FroSe76] and will be used below. To prove 2. we introduce Borel sets n such that n  n and such that n % IR2+ and dist ( n ; cn) ! 1 as n ! 1. Then we have:      ,C+n (f~m) + C1 (f~m) = C1  n
f~m , C+n  n 
f~m    + C1+ +n n +n f~m , Cn +n n +n f~m + C1 +n c f~m 
n1 (f~m ) +
n2 (f~m) +
m3 (f~m) : (3.63) The rst dierence
n1 . The rst dierence
n1 is estimated as follows. Using the cluster expansion we have: X Z , Z n X) (3.64)
n1 (f~m ) = @ < + : : : > Z@XZ(()

n (X;,): X \ cn 6= The typical term in P(X;,) indexed by (X; ,) is estimated by: @ , < R +n : : : > Z@XZ (()nX )  o 1 m n (3.65)  j +n jm sup
2 +n R
e,f SH0 (xi) , 1 pdx p
e,Kj,j+KjX j
eK2jX j 28

 n ~ for some p > 1+ 1, (see [FroSe76]). Because jX j  j n j in all terms de ning
1 (fm ) by (3.64) we have: e,Kj,j+KjX jeK2jX j  e,K(j nj)e,Kj,jeKjX j : (3.66) Let us de ne: Z p 1p H0 (x) p , f  S O1 (f ) = sup
dx e , 1 (3.67)

Then we have the estimate:

;


j
n1 (f~m )j  j +n jm
[O1p(f )]m
e,Km
e,Kj n j (3.68) in the sense that for any K > 0 there exists zo such that for any jzj < zo the estimate (3.68)

holds. The second dierence
n2 (f~m). Using the cluster expansion again and proceeding in a similar way as before we obtain that for any K > 0 there exists zo = zo(K ) such that for any jzj < zo(K ): j
n2 (f~m )j  +n n +n m
[O1p(f )]me,Km
e,Kj+n n +n j : (3.69) The term
n3 (f~m). Here we use again the cluster expansion representation for
n3 (f~m ) but instead of using the exponentially decaying factor e,KjX j we use the decay of je,f SH0 (x) , 1j. For this goal let us consider R e,f SH0 (x) , 1 pdx
 hsup;x2
e,f SH0 (x) , 1i p (3.70)  hsup;x e,f SH0 (x) , i1 h p i p H 0 p H  sup;x2
e,pf S (x) sup ;x2
jj jf  S 0 (x)j  H  sup;x2IRd+ e,pf S 0 (x) jjpC (f )pe,dist(s(f );
) where C (f ) = kf k1. For further use let us set

Cp(f ) = sup d e,pf SH0 (x) ;x2IR+

(3.71)

Supplying the cluster expansion for C1+ ((+n )C f~m) and using     + n+ fm C1+ (+n )c fm = nlim C  + ~ ~  n n !1 n

(3.72)

where ~ +n  +n for any n  1 and ~ +n % IR2 n +n monotonously we obtain   C +  + fm  kf kmC 0(f )m
e,dist(s(f );cn)m
constant

(3.73)

1

n

1

29

Proposition 3.12. 9z0 > 0 such that for jzj < z0 and for any 2 H,  we have the convergence ~ ne(i(';f )) j (t = 0)o ( )  (ei(';f ))( ) lim E "IR2

(3.74)

for any monotonic sequence n % IR2 and any f 2 C0(IR). Moreover the convergence is uniform on the space H,  (IR).

Proof. Let

  R 1+ ei(';f ) ( ) = ei(P +(h );f )
Pn0 n1! IR2+i d(; )n1 Qn e,if SH0 (xi) , 1
C ((x; )n) 1

i=1

1

(3.75)

Then if we take n to be an arbitrary sequence of bounded subsets of IR2 as before (i.e. such that n % IR2 monotonously and by inclusion, such that for any n : n  n; limn!1 dist( n ; cn) = 1;) then using the decomposition (3.62) and the estimates (3.56), (3.57) and (3.54)) we obtain   i(';f ) n o  e , E n ei(';f ) j ( t = 0) ( ) (3.76)  Pm1 m1!
n2 (f~m) + Pm1 m1!
n2 (f~m) + Pm1
n3 (f~m) ) c   + g(dist(suppf; n) where  can be taken arbitrary small provided n to be suciently big; moreover .

nlim !1 g (n) = 0

Passing to the limit n ! 1 we obtain e nei(';f ) j (t = 0)o ( ) = 1+ ei(';f ) ( ) : lim E n!1

(3.77)

A consequence of the above proven Proposition and the estimate (3.76) is that (ei(';f ))(
) as de ned in (3.75) are versions of the conditional expectation value of E fei(';f ) j (t  0)g(
), where z1 is the weakly coupled unique Gibbs measure corresponding to (3.48).

Theorem 3.13.

There exists z0 > 0 such that for any jzj < z0 the corresponding regular in nite volume Gibbs measure z1 2 G2FR(z) has the Global Markov property.

30

Proof.

We check that the steps SO,: S 2 of our plan described in Section 3.0 are really implementable for our model. The Euclidean invariance enables us to reduce the proof to the one for a P 1 particular line H0 only. Let F; G 2 L ( (IRd)) be of the form F (') = ei(';f ); G(') = ei(';g)) with f 2 C0(IR2+ ) (and g 2 C0(IR2, )). We know already that for G2FR(z)  fz1 g for any jzj < z0 and moreover n ! z1 weakly as n % IR2 monotonously. Let n % IR2 by any such sequence. Then n o R dn ( )En nei(';g)ei(';f ) j (t  0)o ( ) (3.78) R d ( )E ei(';g)ei(';f ) j (t = 0) (T ( )) n n H0 and TH0 ( ) 2 H,  (IR) for n a.e. 2 S (IR2). R d1 ( ) ei(';f ) (TH0 ('))ei( ;g) n ) o , R dn ( )En ei(';f j (t  0) (TH0 )ei( ;g)  n o  R d1 ( ) j  ei(';f ) (T H0 ( ))n , E ei(';f ) j o(t  0) (TH0 ( ))j + R (d1 ( ) , d ( )) E ei(';f ) j (t  0) (TH ( )) n

(3.79)

0



Using Proposition 3.5, the weak convergence zn ! z1 and E~ f, j (t  0)g ! ( ) again we conclude by the application of (3.76) n i(';f ) o i( ;g) R limn!1 dn ( )E e j ( t  0) ( )e   n   i(';f (3.80) R z i ( ;g )  ) = d1 ( )e  e (TH0 ( )) By the uniqueness part of the Radon{Nikodym theorem we conclude: n o   Ez1 ei(';f ) j (t  0) ( )   ei(';f ) (TH0 ( )) ; (3.81) for z1 a.e. able.

2 S 0(IR2). But the functional (ei(';f ))(TH0 ( )) is obviously (H0 ) measur-

Remarks.

For real and symmetric d and for z > 0 we know from the correlation inequalities proven in [FrPa79], (see also [Gi86]) that the measure z1 is a CR{regular random eld. Therefore, instead of the space B,;loc(IR) we can use the space B,(IR) as space of traces and this enables us to associate with the Gibbsz measure z1 (at least for suciently small z > 0) some B,(IR) valued Markov processes (1 ) with the initial (stationary) measure z1 j (H0) and the corresponding transition kernel given by:    z  (t1) TH0 ( ) ; ei   ei 0 the set of 2FR regular Gibbs measures G2FR(V ) corresponding to (3.84) consists of an exactly one element 1 . The random eld 1 has the Global Markov property with respect to any hyper-surfaces , of Jordan type and at least C 1-piecewise. A construction of the transition kernels for the class of exponential models will be presented below. It is based on exponential clustering of the conditionned Gibbs measure uniformly in the boundary data and on Theorem 3.7. Let n be an arbitrary sequence of bounded Borel subsets of IR2 such that n  n+1 for any n 2 IN and Sn n = IR2 with further properties speci ed below. The nite volume transition kernel En fe'(f )j(t  0)g() can be written as  nX ,1  n '(f ) o + ();f )   , ( P ' ( f ) ' ( f ) e En e j(t  0) () = +k+1 (e ) , +k (e ) + +0 (e'(f )) (3.85) k=0

where we have assumed that supp f  s(f )  +0 and we have used the following abreviations  H0 0 (e'(f ) eV ('))  ' ( f ) +n (e ) = H0 (eV (')) ; 0 (3.86) V (') = V (' + P + ()) and +n  IR2+ \ n : Let us introduce, for s 2 [0; 1] the following interpolation

Vk+1jk (')(s)  V+k (') + sV+k+1 n+k : 32

(3.87)

Then we can write



+ k+1

(e'(f ))

Z1

d  (e'(f ))(s) ds k+1jk !T Z 1  0 ds Z + () ' ( f ) ' P = , ds k+1jk e ; + + : e : e dx (s)

,

+ k

(e'(f )) =

k+1 nk

0

where

(3.88)

H0 (,eVk+1jk (')(s))   k+1jk (e'(f ))(s) = 0H0 Vk+1jk (')(s)  (e ) 0

(3.89)

and

(F; G)T = (F; G) , (F )(G) (truncated expectation value or covariance).

Proposition 3.15. p p 1. Let supp dr  (,2 ; 2 ). Then for any f , g 2 D(IR) of constant sign the following estimate holds





k+1jk (sign fe'(f ); sign ge'(g))T (s)  kjk+1 e 21 '(f +g))2 12 @0 (2 sh 'p(f ) ; 2 sh 'p(g) ) 2 2

;

2. If supp dr  [0; ) and f  0 then:

kjk+1 (e'(f ); : e' : (x)T (s)  H0 0 (e'(f ); : e' : (x)T (s)

Proof

The proof is obtained in complete analogy to the proof of lemma 1.5.1 from [Ze84] for the point 1 and from [Gi84] in the case of 2. Now let (k) be a sequence of C 1 functions such that

(x) 0  (x) (x) sup j
j x sup jrj x

= 1 for x : dist(x; H0)  16   for x : 0  dist(x; H0)  16 = 0 for x : dist(x; H0)  121 < 1 and < 1 (uniformly in k):

where  is such that

p

j1 , j < 2 Let us perform the shift transformation ' ! ' , P + () 33

(3.90)

in the formula (3.89). Calculating the corresponding Radon-Nikodym derivative we obtain   H0 e'(f )e,'((,
+I ) P + ()) eVk+1jk ('+(1, )P + ())(s)  0 (3.91) k+1jk (e'(f ))(s) = H0  ,'((,
+1) P +()) Vk+1jk ('+(1,)P +()(s)) 0 e e Then the formula (3.88) becomes Z1 d    ' ( f ) ' ( f ) k+1 (e ) , k (e ) = ds ds k+1jk (e'(f ))(s) 0 Z Z = e,(P +();f )(,) dx e(1,)P + ()(x) dr() Z 1   k+1 nk T ds k+1jk e'(f ); : e' : (x) (s)

(3.92)

0

Therefore using Proposition 3.15 and explicit calculations we have Z Z  '(f )  (e ) ,  (e'(f ))  e,(P ();f )
jj
dx dr() O(f )e, dist(k+1 nk ;s(f ))e(1,)P +()(x) ; k+1

k

k+1 nk

where O(f ) is some constant. 1 Now let k+1 n k = [k+1 n k ]06 + [k+1 n k ]161 , where:

(3.93)

1 [k+1 n k ]06 = f(t; x) 2 k+1 n k j0 < t < 16 g ;

[k+1 n k ]161 = f(t; x) 2 k+1 n k j 16  t < 1g : 1

For (t; x) 2 [k+1 n k ]06 we have (by Theorem 2.7) the estimate:

e(1,)P + ()(t;x)  e(1,) ln A()e(1,) supjx,yj ln ,1 (y)e(1,) ln( 1t ) : (3.94) From the assumptions made on  it follows that there exists  > 0, such that: e(1,) ln( 1t ) < t,1+ : (3.95) From the properties of  (see the considerations after Theorem 2.4) it follows that there exists a natural p 2 IN , such that ,1 (y))

esupjx,yj< 121 (ln 

 (1 + jxj)p :

Collecting all results we get: e(1,)P +()(t;x)  max(A(); 1)(1 + jxj)pt,1+ ; for (t; x) 2 [0; 16 ). Now we are prepared to prove the following result 34

(3.96) (3.97)

Proposition 3.16

Let (n ) be any monotonic sequence of bounded Borel subsets of IR2, such that (i) jn+1 n n j grows at most polynomially in n; (ii) dist(f; n+1 n n) grows at least linearly in n. If the measure is supported in [0; 2p) then for -a.e.  2 B,(IR),  > 0, where  is arbitrary CR-regular measure, there exists a subsequence (n0)  (n) such that lim n0 !1

n0 X

(k0 +1 (e'(f )) , k0 (e'(f ))  e,(P + ();f )+(e'(f ))() + e,(P +();f )0 (e'(f )) (3.98)

k0 =0

exists and moreover if 1 is the corresponding Gibbs measure (1  limn n ) then: +(e'(f ))(
) = E1 fe'(f )j(t  0)g(
) (1-a.e.)

(3.99)

for all f 2 C01(IR2+).

Proof

Using the estimate (3.93) we obtain X N f + (e'(f )) ,  + (e'(f ))g k k=0 k+1 N X ,O(1)k Z , (  P (  ) ;f )  e
jj e k=0

dt dx O(f ) t,1+
max(A(); 1)(1 + jxj)p

1

[k+1 nk ]06

Z N X , (  P (  ) ;f ) , O (1) k  +e
jj e k=0 [k+1 nk ]1 1

Z

dt dx dr()
e(1,)P +()(x) :

6

(3.100) From the assumption (ii) and the fact that  > 0 it follows that the rst series (concentrated 1 on [k+1 n k ]06 ) is absolutely convergent for 1 -a.e.  2 B,(IR). The in nite volume exponential measure 1 is obviously CR- regular. The remainder RN of the second series appearing in the decomposition (3.100) is estimated in the L1(1 ) norm where 1 is an arbitrary completely regular measure. We get Z Z 1 X RN  jj e,O(1)k dt dx dr() sup 1 e 12 2KH0 (1,j1,) t;x2[k+1 nk ] 1 k=N +1 [k+1 nk ]1 6 1 (3.101) 6 Z 1 X 1  jj( dr())
const e,O(1)k jk+1 n k j 61 ; k=N +1

which could be made arbitrary small provided N is taken to be suciently big. Therefore, there exists a subsequence (n0) such that n0  X  (e'(f )) ,  (e'(f ))  lim  (e'(f )) ,  (e'(f ))) lim  (3.102) k0 +1 k0 0 n0 !1 n0 !1 n0 k0 =0

35

exists for -a.e.  2 B,, provided  is completely regular. To identify the constructed +(e'(f )) with the transition kernel of the exponential Gibbs measure 1 we use the weakened version Step 1' of section 2, nding easily the estimate: Z n X  ,  (e'(f )) < 1 sup d (3.103) n ( ) k+1 k n k=0

Remarks

The construction of the transition kernel by passing to a subsequence as well as the p restriction supp dr  [0; 2 ) are no doubt only artefacts of our construction. However, the bene t of our construction is its simplicity.

Acknowledgements We are very grateful to Prof. M. Rockner for very instructive and stimulating discussions. The second (R.G.) and third authors (F.R.) thank Profs. P. Blanchard and L. Streit for making their stay in BiBoS pleasant and fruitful. The second author is grateful to Institut Galilee, Universite Paris-Nord for the kind hospitality. We are very grateful to the referee for his remarks helping us to clarify the text. Partial nancial supports obtained from EC Mobility Grant No 722 and Polish National Committee for Scienti c Research, Grants Nr. 200609101 and 2P03B12211, are gratefully acknowledged by the second author. The third author was partially supported by the Human Capital and Mobility fellowship, contract No ERBCHBGCT 920016 and by SFB 237 (EssenBochum-Dusseldorf)

References [AFHKL86] Albeverio, S., Fenstad, J.E., Hegh-Krohn, R., Lindstrm, Non standard methods in stochastic analysis and mathematical physics, Academic Press (1986) [AGR97] Albeverio, S., Gielerak, R., Russo, F.: On the paths Holder continuity in the models of Euclidean Quantum Field Theory. BiBoS Preprint (1997). [AHK73] Albeverio, S., Hegh-Krohn, R., Uniqueness of the physical vacuum and the Wightman functions in the in nite volume limit for some non-polynomial interactions, Comm. Math Physics, 30, 171-200 (1973) [AHK74] Albeverio, S., Hegh-Krohn, R., The Wightman axioms and the mass gap for strong interactions of exponential type in two-dimensional space-time, J. Funct. Anal. 16, 39-82 (1974) [AHK75] Albeverio, S., Hegh-Krohn, R., Homogeneous random elds and statistical mechanics, Journal of Functional Analysis, 19, 242-272 (1975) 36

[AHK76] Albeverio, S., Hegh-Krohn, R., Quasi invariant measures, symmetric diusion processes and quantum elds, pp 11-59 in: \Proceedings of the International Colloquium on Mathematical Methods of Quantum Field Theory", Editions de CNRS. 1976, (Colloques Internationaux de Centre National de la Recherche Scienti que, No. 248) [AHK77a] Albeverio, S., Hegh-Krohn, R., Hunt processes and analytic potential theory on rigged Hilbert spaces, Ann. Inst. H. Poincare (Probability Theory) B13, 269-291 (1977) [AHK77b] Albeverio, S., Hegh-Krohn, R., Dirichlet forms and diusion processes on rigged Hilbert spaces, Zeitschrift fur Wahrscheinlichkeitstheorie und verwandte Gebiete 40, 1-57 (1977) [AHK79a] Albeverio, S., Hegh-Krohn, R., Uniqueness and the Global Markov property for Euclidean elds. The case of trigonometric interactions, Comm. Math. Phys., 68, 95-128 (1979) [AHK79b] Albeverio, S., Hegh-Krohn, R. The Global Markov property for Euclidean and lattice elds, Phys. Letters, 84B, 89-90 ( 1979) [AHK84] Albeverio, S., Hegh-Krohn, R. Diusion elds, Quantum elds and elds with values in groups, 1-98, in \Stochastic Analysis and Applications", Adv. in Probability, Edt. M. Pinsky, M. Dekker, New York (1984) [AHKO81] Albeverio, S., Hegh-Krohn, R., Olsen, G., The global Markov property for lattice elds J. Multiv. Anal. 11, 599-607 (1981) [AHKZ89] Albeverio, S., Hegh-Krohn, R., Zegarlinski, B., Uniqueness and Global Markov Property for Euclidean Fields: The Case of General Polynomial Interactions, Comm. Math. Phys. 123, 377-424 (1989) [AMRo91] Albeverio, S., Ma, Z.M., Rockner, M., Dirichlet forms and Markov elds | A report on recent developments, in M. Pinsky Ed. \Diusion Processes and related Problems in Analysis", Birkhauser, New York (1991) [ARo89] Albeverio, S., Rockner, M., Classical Dirichlet forms on topological vector spaces | the construction of the associated diusion process, Prob. Theory and Rel. Fields 83, 405-438 (1989) [ARo90a] Albeverio, S., Rockner, M., New developments in theory and applications of Dirichlet forms, in: \Stochastic Processes, Physics and Geometry", (edts. S. Albeverio, G. Casati, U. Cattaneo, D. Merlini, R. Moresi), World Scienti c, Singapore, 1990, pp 27-76 [ARo90b] Albeverio, S., Rockner, M., Classical Dirichlet forms on topological vector spaces | closability and a Cameron-Martin formula, J. Func. Anal. 88, 395-436 (1990) 37

[ARo91] Albeverio, S., Rockner, M., Stochastic dierential equations in in nite dimensions: solutions via Dirichlet forms, Prob. Th. Rel. Fields 91, 347-386 (1991). [ARo95] Albeverio, S., Rockner, M., Dirichlet form methods for uniqueness of martingale problems, 513-528 in\Stochastic Analysis", Proc. AMS Summer Inst. Stochastic Analysis, Cornell, 1993, Proc. Symposia in Pure Mathematics Vol. 57, Edts. M.C. Cranston, M.A. Pinsky, AMS, Providence (1995) [ARZ93] Albeverio, S., Rockner, M., Zhang, T.S., Markov uniqueness and its application to martingale problems, stochastic dierential equations and stochastic quantization, C. R. Math. Rep. Acad. Sci. Canada XV 1-6 (1993) [AZ92] Albeverio, S. Zegarlinski, B., Global Markov property in quantum eld theory and statistical mechanics: a review on results and problems, 331-369 in R. Hegh-Krohn's memorial Volume, \Ideas and Methods in Quantum and Statistical Physics", Vol 2, Edts S. Albeverio, J.E. Fenstad, H. Holden, T. Lindstrm, Cambridge University Press (1992) [Bo46] Bogolubov, N.N, Problems of dynamical theory in statistical mechanics, GostInzdat, Moscow, 1946 (in Russian) [Do68] Dobrushin, R.,L., The description of a Random Field by means of Conditional Probabilities and Conditions of its Regularity, Theor. Prob. Appl. 13, 201-229 (1968) (in Russian) (transl. 13, 197-224 (1968)) [DoM76] Dobrushin, R.L., Minlos, R.A., The theory of Markov random elds and its application to Quantum Field Theory, Acta Univ. Wratisl., 368, 23-49 (1976) [Dy65] Dynkin, E.B., Markov Processes, Springer-Verlag, Berlin, 1965 [Fo80] Follmer, H., on the Global Markov Property, in \Quantum Fields, Algebras, Proesses", Ed. Streit L., p. 293-302, Springer-Verlag, 1980. [Fro74] Frohlich, J., Schwinger functions and their generating functionals I, Helv. Phys. Acta 47, 265-300 (1974) [Fro76] Frohlich, J., Classical and quantum statistical mechanics in one and two dimensions: Two component Yukawa and Coulomb systems, Comm. Math. Phys. 47, 233 (1976) [FroPA77] Frohlich, J., Park, Y.M., Remarks on exponential interactions and the quantum sine-Gordon equation in two space-time dimension, Helv. Phys. Acta 50, 315-329, 1977 [FroSe76] Frohlich, J., Seiler, E., The massive Thirring Schwinger model (QED)2: convergence of perturbation theory and particle structure, Helv. Phys. Acta 49, 889-924 (1976) [Gi83] Gielerak, R., Veri cation of the Global Markov property in some class of strongly coupled exponential interactions, J. Math. Phys. 24, 347-355 (1983) 38

[Gi85]

Gielerak, R., in Critical phenomena. Theoretical aspects, Progress in Physics, 12, Eds. A. Jae, D. Ruelle, Birkauser (1985) [Gi86] Gielerak, R., On the DLR equation for the two dimensional sine-Gordon model, Journ. Math. Phys. 27, 2892, 1986 [Gi88] Gielerak, R., Trigonometric perturbation of the Gaussian generalized random elds. Weak coupling results, Ann. Henri-Poincare 48, 205-228 (1988) [Gi89] Gielerak, R. On the DLR equation for (4 + b2 + ;  6= 0)2 Quantum Field Theory, Ann. of Physics 189, 1-28 (1989) [Gi91] Gielerak, R., Bounded perturbations of the Gaussian generalized random elds, Journ. Math. Phys. 32 (9), 2329-36, 1991 [Gi92a] Gielerak, R., On the phase diagram for a class of continuous systems, Journ. Math. Phys. 33, 68-84, 1992 [Gi92b] Gielerak, R., Veri cation of the Global Markov property for the strongly coupled trigonometric interactions, in the book: \Ideas and methods in Math. Physics", Vol. 2, Cambridge University Press, 1992, Edts.: Albeverio, S., Holden, H., Fenstad, M. and Lindstrm, T. [Gi92c] Gielerak, R., Gibbsian perturbations of the generalized random elds, Proc. of the 3rd Spring School on Stochastic Evolution and Dynamical Systems, Katisiveli, Crimea May 1992, also BiBoS Preprint No 553/92 [GiZ84] Gielerak, R. Zegarlinski, B., Uniqueness and the Global (almost) Markov property for a class of regularized Yukawa like gases, Fortschr. der Physik, 1, 1-24 (1984) [GJ81] Glimm, J., Jae, A., Quantum Physics: A functional integral point of view, Springer 1981, New York, Heidelberg, Berlin [Go80] Goldstein, S., Remark on the Global Markov Property, Commun. Math. Phys. 74, 223-234 (1980) [GoKS90] Goldstein, S., Kuik, R., Schlijper, A.G., Entropy and Global Markov Property, Commun. Math. Phys. 126, 469-482 (1990) [GuRS75] Guerra, F. Rosen, L, Simon, B., The P ()2 Euclidean Quantum Field Theory as classical statistical mechanics, Ann. Math. 101, 111-259 (1975) [Hi84] Higuchi, Y. A remark on the global Markov property for d-dimensional Ising Model, Proc. Japan Acad., 60, Ser. A, 49-52 (1984) [Is86] Israel, R.B., Some examples concerning the global Markov property, Commun. Math. Phys. 105, 669-673 (1986) [Iw90] Iwata, K., On linear maps preserving Markov properties and applications to multicomponent generalized random elds, Ph. D. Thesis, Bochum (1990) 39

[Iw92] [Ke85] [Ko74] [Ku88a] [Ku88b] [Kuz82] [LM72] [Mo71] [Ne73a] [Ne73b] [Ne73c] [New73] [Pi71] [Ro83] [Ro85] [Ro86] [Ro88]

Iwata, K., The inverse od a local operator preserves the Markov property, Ann. Sc. Norm. Sup. Pisa, Cl. Sci. IV See XIX, 223-253 (1992) Kessler, C., Examples of extremal lattice elds without the Global Markov Property, Publ. RIMS 21, 877-888 (1985) Kotani, S., On a Markov property for stationary Gaussian processes with a multidimensional parameter, pp 239-250, in: Proc. 2nd Japan-USSR Symp., Lec. Notes Math. 330, Springer 1974 Kuik, R., Markov and stability property for equilibrium states which are determined by correlations in a strip, Comm. Math. Phys. 115, 529-552(1988) Kuik, R., The global Markov Property for equilibrium states which are determined by correlations in a strip, Comm. Math. Phys. 116, 177-189 (1988) Kuznetzov, C.E., Construction of Consistent Conditional Distributions, Sec. I, 3.4 in: Rozanov Y.A., \Markov random elds", Springer 1982 Lions, J.L., Magenes, E., Non-homogeneous boundary value problems and applications, vol. I-III, Springer-Verlag 1972 Molchan, G.M., Characterization of Gaussian elds with Markovian property, Dokl. Akad. Nauk. SSSR 197, 784-787, transl.: Sov. Math. Dokl. 12, 563-567 (1971) Nelson, E., The construction of quantum elds from Markov elds, J. Funct. Anal. 12, 97-112 (1973) Nelson, E., The free Markov eld, J. Funct. Anal. 12, 211-217 (1973) Nelson, E., Quantum elds and Markov elds, in \Partial Dierential Equations", Ed. D.C. Spencer, Providence (1973) Newman, C., The construction of stationary two-dimensional Marko elds with an application to quantum eld theory, J. Funct. Anal. 14, 44-61 (1973) Pitt, L.D., A Markov property for Gaussian processes with multidimensional parameter, Arch. Rat. Mech. Anal. 43, 367-391 (1971) Rockner, M., Markov property of generalized elds and axiomatic potential theory, Math. Anal. 264, 153-177 (1983) Rockner, M., A Dirichlet problem for distributions and speci cations for random elds, Memoirs of the Am. Math. Soc. 54, 324, 1985 Rockner, M., Speci cations and Martin boundary for P ()2 random elds, Comm. Math. Phys. 106 105-135 (1986) Rockner, M., Traces of harmonic functions and a new path space for the free quantum eld, J. Funct. Anal. 79, 211-249 (1988) 40

[RoZh92] Rockner, M., Zhang, T.S., Uniqueness of generalized Schrodinger operators and applications, J. Funct. Anal. 105, 187-221 (1992) [RoZh94] Rockner, M., Zhang, T.S., Uniqueness of generalized Schrodinger operators. II, J. Funct. Anal. 119, 455-467 (1994) [R82] Rozanov, Y.A., Markov random elds, Springer, New York (1982) [Ru60] Ruelle, D., Statistical Mechanics: Rigorous results, W.A. New York 1960 [Sc96] Schafer, J., Abstract Markov Property and local operators, J. Funct. Anal. 138, 137-169 (1996). [Si74] Simon, B., The P ()2 Euclidean (quantum) eld theory, Princeton, N.J. Princeton University Press 1974 [Ta83] Takeda, M., On the uniqueness of Markov self-adjoint extensions of diusion operators on in nite dimensional spaces, Osaka J. Math. 22 733-742 (1983) [Wo69] Wong, E., Homogeneous Gauss-Markov random elds, Ann. Math. Stat. 40, 16251634 (1969) [WoH85] Wong, E., Hajek, B. Stochastic processes in engineering systems, Springer, Berlin (1985) [Ze84] Zegarlinski, B., Uniqueness and the Global Markov property for Euclidean Fields: The case of general exponential interactions, Comm. Math. Phys. 96, 195-221 (1984)

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