Stochastic Processes and their Applications 105 (2003) 211–255 www.elsevier.com/locate/spa
Large deviations in the Langevin dynamics of a random ÿeld Ising model GÃerard Ben Arousa, Michel Sortaisb;∗;1 aCourant Institute of Mathematical Sciences, 251 Mercer Street, New York, NY 10012, USA bFakultat II der T.U.Berlin, Institut fur Mathematik, MA 7-4, Strasse des 17ten Juni 136, D-10623 Berlin, Germany
Received 29 May2002; received in revised form 25 November 2002; accepted 27 November 2002
Abstract We consider a Langevin dynamics scheme for a d-dimensional Ising model with a disordered external magnetic ÿeld and establish that the averaged law of the empirical process obeys a large deviation principle (LDP), according to a good rate functional Ja having a unique minimiser Q∞. The asymptotic dynamics Q∞ maybe viewed as the unique weak solution associated with an inÿnite-dimensional system of interacting di7usions, as well as the unique Gibbs measure corresponding to an interaction on inÿnite dimensional path space. We then show that the quenched law of the empirical process also obeys a LDP, according to a deterministic good rate functional Jq satisfying: Jq ¿ Ja, so that (for a typical realisation of the disordered external magnetic ÿeld) the quenched law of the empirical process converges exponentiallyfast to a
Dirac mass concentrated at Q∞. c 2003 Elsevier Science B.V. All rights reserved.
Keywords: Large deviations; Statistical mechanics; Disordered systems; Interacting di7usion processes
1. Introduction and statement of the main results
In several recent papers, BrÃezin and De Dominicis have considered the statics and the dynamics of an Ising model submitted to a Gaussian random ÿeld: in BrÃezin and De Dominicis (1998b) they proceed to a “Replica Theory” analysis of the statics of this random ÿeld Ising model (RFIM), whereas in BrÃezin and De Dominicis (1998a) theystudythe Langevin dynamicsassociated with the model. In the latter note, they
∗ Corresponding author. E-mail address: [email protected] (M. Sortais). 1 Research supported bythe Swiss National Science Foundation under contract no 21-54120.98.
0304-4149/03/$ - see front matter c 2003 Elsevier Science B.V. All rights reserved. doi:10.1016/S0304-4149(02)00265-X 212 G. Ben Arous, M. Sortais / Stochastic Processes andtheir Applications 105 (2003) 211–255 examined the time correlator associated with these dynamics and made the following observation concerning the low temperature regime of low dimensional (d 6 8) RFIMs: the singularities appearing in the time correlator of the dynamics when letting the initial time t0 go to −∞ are preciselythe same as those appearing in the Replica Theoryof the statics in the n→0 limit. Theyconclude to the occurrence of an aging phenomenon in the low temperature regime of such low dimensional RFIMs: the dynamical properties of such disordered spin systems depend upon a “waiting time” t0. In the present paper, our aim is to consider the verysame Langevin dynamics framework for such Gaussian RFIM and establish some large deviation principles for the empirical process of the corresponding trajectories, both in the averaged regime (when performing an average over the realisations of the disordered external ÿeld) and in the quenched regime (i.e. for a ÿxed, typical realisation of the disorder variables). As a consequence of these large deviation results, we mayalso state the following strong law of large numbers: for a typical realisation of the disorder variables, the quenched law of the empirical process converges exponentiallyfast to a Dirac mass concentrated at some asymptotic dynamics Q∞, that maybe explicitlydescribed as the unique weak solution associated with some inÿnite-dimensional system of interacting di7usions. Such large deviations approach to the dynamics of disordered systems has alreadybeen applied in the context of the Sherrington–Kirkpatrick model (see Ben Arous and Guionnet, 1997; Ben Arous and Guionnet, 1998; Grunwald, 1996, 1998), and more recentlythe same kind of results have been derived in the context of a short range spin glass (Ben Arous and Sortais, 2002). Here the situation is certainly much simpler in the sense that we are dealing with a site disordered system, and not a bond disordered one; in such a simple situation we are able to characterise the asymptotic dynamics Q∞ as the unique Gibbs measure corresponding to some ÿnite range, translation invariant interaction on path space, and the averaged LDP obtained for the empirical process maythen be seen as a consequence of the Gibbsian nature of the averaged regime. Let us now be more precise about the context we are working in. We consider the k d-dimensional lattice disordered system whose Hamiltonian H in the ÿnite volume =[− N; N]d ∩ Zd writes: k i j i i Zd H (x)=−2 x x + k x ; x ∈{−1; 1} ; j∼i i∈ i k =(k )i∈Zd being an i.i.d. familyof centred Gaussian random variables with variance 2 (j∼i means that sites j and i are nearest neighbours in , being equipped with its periodic boundaryconditions). k In order to consider the Langevin dynamics corresponding to H , we replace the “hard” spin variables (xi ∈{−1; 1}) bylinear spin variables ( xi ∈ R), introduce the real polynomial function U given by U(x)=Cx4 − Dx2 for some positive constants C and D, and consider the process i i i d dxt =dwt − U (xt)dt; i ∈ Z ; G. Ben Arous, M. Sortais / Stochastic Processes andtheir Applications 105 (2003) 211–255 213
i as our reference process (here and in the sequel, (w )i∈Zd denotes an i.i.d. familyof standard brownian motions indexed bythe lattice); observe that the equilibrium measure −2U(x) C;D corresponding to this process is proportional to e dx, so that 1 C;D =⇒ (−1 + 1) when D =2C: C;D∞ 2 Fixing ÿ¿0 (inverse temperature parameter) and T¿0 (terminal time of the ex- periment), we let 0 be some compactlysupported probabilitymeasure on R;inthe ÿnite volume equipped with periodic boundaryconditions, the nearest neighbour interacting system of di7usions Sk given by i i i j i dxt =dwt − U (xt)dt + ÿ xt + k dt (i ∈ ; 0 6 t 6 T) j∼i
⊗ with the “deep quench” initial condition: law(x|t=0)=0 then has an invariant re- k ⊗ versible measure proportional to: exp(−ÿH (x))C;D(dx). Denoting by WT the (Polish) k space of all real valued continuous functions on the interval [0; T], we let Q be the k k law of the system S ; Q is a probabilitymeasure on WT , which we shall write as k Q ∈ M(WT ): ( ) We deÿne the empirical process x corresponding to a ÿnite-dimensional vector of di7usions x ∈ WT through the identity: 1 ( ) = ; x | | x( ); (i) i∈
( ) Zd = x ∈ WT being the inÿnite-dimensional vector of di7usions obtained from x by reproducing periodicallyon the lattice the information contained in the box and (i) Zd ∈ WT being the new conÿguration obtained from byshifting the origin of the lattice at site i:
(i) d ( )j = j+i; ∀j ∈ Z :
Now let Ms( ) be the (Polish) space of all spatiallyshift invariant probability Zd ( ) measures on = WT ; x is an Ms( )-valued variable on WT , and we shall denote k ( ) k by ! the law of x under dQ (x), so that: k ! ∈ M(Ms( )): Finally ! ∈ M(M ( )) is deÿned through the identity: s k ! (A)= d"(k)! (A) holding for anyBorel set A⊂Ms( )(! is the averaged law of the empirical process). Our main results are large deviation principles for the families (! ) ⊂⊂Zd and k (! ) ⊂⊂Zd (for a typical realisation of k), yielding a fast convergence of these fam- ilies towards a Dirac mass d Q∞ as Z ; the asymptotic dynamics Q∞ maybe described in several ways: as the unique weak solution corresponding to an inÿnite 214 G. Ben Arous, M. Sortais / Stochastic Processes andtheir Applications 105 (2003) 211–255 dimensional system of interacting di7usions, as the mean of a family of disordered, inÿnite-dimensional dynamics and also as the unique Gibbs measure corresponding to a ÿnite-range, translation-invariant interaction on path space . We refer the reader to Section 1.2 in (Dembo and Zeitouni, 1998, pp. 4,5) for precise deÿnitions concerning large deviation principles, rate functions and good rate functions.
Theorem 1.1. (i) For any inverse temperature parameter ÿ andany terminal time T, Zd the family (! ) ⊂⊂Zd obeys a large deviation principle on Ms(WT ), on the scale | | andaccordingto a goodrate function
a Zd I : Ms(WT ) → [0; +∞] having a unique minimiser Q∞. Moreover, Q∞ is the unique weak solution corresponding to the following inÿnite dimensional system of nearest neighbour interacting di;usions: i i i j dxt =dvt − U (xt)dt + ÿ xt dt j∼i ⊗Zd d law(x|t=0)=0 (i ∈ Z ; 0 6 t 6 T) i where {(vt)06t6T }i∈Zd is an i.i.d. family of time inhomogeneous Ornstein–Uhlenbeck processes satisfying: 2ÿ2 vi =0 and dvi =dwi + " vi dt for " = : 0 t t t t t 1+2ÿ2t (ii) Furthermore, almost surely in the realisations of the disorder variables k, the k family (! ) ⊂⊂Zd also obeys a large deviation principle on Ms( ), on the scale | | andaccordingto a goodrate function Iq satisfying: Iq ¿ Ia:
i Note. Each of the processes (vt)t¿0 mayalso be presented as t dwi vi =(1+2ÿ2t) s t 2 2 0 1+ ÿ s and is thus a centred Gaussian process having the covariance structure:
i i 2 2 E[vsvt]=s(1 + ÿ t); ∀0 6 s 6 t:
So the onlydi7erence between the stochastic di7erential systemcharacterising Q∞ and the inÿnite volume Langevin dynamics of a standard Ising model lies in the nature of the “thermal noise” driving the system: in the case of a standard Ising model it is an i d i.i.d. family {(wt )06t6T ; i ∈ Z } of standard brownian motions, whereas here we have an i.i.d. collection of centred gaussian processes with a verysimple time correlation i structure. Actually, each of the processes (vt)06t6T mayalso be represented as i i (z · t + wt )06t6T ; G. Ben Arous, M. Sortais / Stochastic Processes andtheir Applications 105 (2003) 211–255 215
i i where (wt )06t6T is an i.i.d. familyof standard Brownian Motions and ( z ) an i.i.d. fam- 2 2 i ilyof centred Gaussian variables with variance ÿ , independent of (wt )06t6T .Soasa consequence of the preceding theorem we maystate that the disordered, inÿnite-volume Stochastic Di7erential System i i i j i dxt =dwt − U (xt)dt + ÿ xt + k dt; k (S∞) j∼i ⊗Zd d law(x|t=0)=0 (i ∈ Z ; 0 6 t 6 T) has a unique weak solution Pk, P-a.s.(k), and that k Q∞(:)= P (:)dP(k):
Furthermore, Q∞ mayalso be presented as the unique Gibbs measure associated with a ÿnite-range interaction on inÿnite-dimensional path space (equipped with an inÿnite tensor product of Wiener measures as reference measure); such presentation is proposed in Section 2 (see in particular Sections 2.2, 2.3 and Proposition 2.2 therein). The Gibbsian nature of Q∞ plays an important role in our understanding of the level 3 large deviations occurring in the Langevin dynamics of the RFIM, hopefully this presentation of Q∞ should also prove useful in the investigations related to its space and time decorrelation properties, at least in the high temperature regime. We mayalso state as an elementaryremark that the limiting probability d Q∞(x) thus obtained does not correspond to a Markov ÿeld of interacting di7usions. This is no surprise; indeed, in the case of Sherrington–Kirkpatrick spin glass dynamics, Ben Arous and Guionnet had alreadyisolated a limiting dynamicsthat was a strongly non-Markov one (see Ben Arous and Guionnet, 1998), thus conÿrming some of the predictions made by Sompolinskyand Zippelius in ( Sompolinskyand Zippelius, 1983 ). In comparison, the limiting dynamics Q∞ obtained in the context of the RFIM maynot be considered as a stronglynon-Markov one: Q∞ is characterised through an explicit equation, and i d adding the variable (vt)06t6T at each site i ∈ Z suOces to come back to a (time inhomogeneous) Markov ÿeld. As a simple consequence of the preceding large deviations results, one mayfor example ÿx some bounded continuous functionals ’i : WT →R (1 6 i 6 n) and some bounded continuous F : Rn→R to state that for a typical realisation of the disorder variables k, the distribution of 1 1 F ’ (xi);:::; ’ (xi) | | 1 | | n i∈ i∈
k under dQ (x) converges exponentiallyfast to a Dirac mass concentrated at
F ’1 dQ∞;:::; ’n dQ∞ 216 G. Ben Arous, M. Sortais / Stochastic Processes andtheir Applications 105 (2003) 211–255 when Zd, with an exponential speed of convergence that maybe bounded from below uniformlyin k byusing the averaged LD rate functional Ia. Here is the plan we follow in order to establish these large deviation results: in Section 2 we perform a standard Gaussian computation in order to view the ÿnite volume, averaged probability k Q = d"(k)Q as the law of a (new) system of interacting di7usions (S ) that is spatiallyhomoge- neous. One maythen remark that Shiga and Shimizu’s theorem ( Shiga and Shimizu, 1980), asserting the existence and uniqueness of a strong solution for certain inÿ- nite dimensional systems of interacting di7usions, may be used here; we thus let Q∞ denote the probabilitylaw corresponding to an inÿnite dimensional extension of (S ) and show, using an integration byparts formula established byCattiaux, Roellyand Zessin ( Cattiaux et al., 1996), that Q∞ mayalso be characterised as the unique Gibbs measure associated with some translation invariant interaction on .
Section 3 is devoted to a derivation of the LDP for (! ) ⊂⊂Zd . The LD upper bound is established using the Gibbsian nature of the averaged regime and Varadhan’s method (cf. Olla, 1988); in the present context one has to proceed carefullysince the continuous functionals entering in the deÿnition of the interaction are not uniformly bounded on : fortunatelyone mayestablish that the conÿnement induced bythe single site potential U is strong enough to compensate for the lack of compactness in the spin variables xi. We then check the validityof the Gibbsian variational principle for the interaction and establish the LD lower bound following the method devised by FQollmer and Oreyin the context of Gibbs measures on {±1}Zd (see FQollmer and Orey, 1988). k In Section 4 we give a quick derivation of the LDP for (! ) ⊂⊂Zd following q Zd Comets (1989); the corresponding rate functional I : Ms(WT )→[0; +∞] has a rather intricate expression, and studying the set of all minimisers corresponding to Iq might seem hopeless at ÿrst. Nevertheless, according to some general considerations con- cerning Large Deviations in Disordered Systems or Random Media (see e.g. Lemma 2.2.8 in Chapter 2 of Zeitouni, 2003), the deterministic rate functionals q a I ; I : Ms( )→[0; +∞] associated with the Large Deviations of the empirical pro- cess in the quenched regime and in the annealed regime do certainly satisfy
Iq ¿ Ia;
a and we also know that the set of all minimisers of I is reduced to a singleton {Q∞} (since this set also coincides with the set of all translation invariant Gibbs measures associated with , according to the Variational Principle). Finally, in Section 5 we brieRyshow that one maychange to a certain extent the k initial and boundaryconditions entering in the deÿnition of Q and still derive such LD Principles for the empirical process. G. Ben Arous, M. Sortais / Stochastic Processes andtheir Applications 105 (2003) 211–255 217
2. Gibbsian nature of the averaged regime
2.1. Inÿnite-dimensional extension of the averaged regime probabilities
k Before stating the main result of this section, we remind that S denotes the following system of nearest neighbour interacting di7usions: i i i j i dxt =dwt − U (xt)dt + ÿ xt + k dt; j∼i ⊗ Law(x|t=0)=0 (i ∈ ; 0 6 t 6 T)
(k being some ÿxed realisation of the disordered external magnetic ÿeld, 0 some compactlysupported probabilitymeasure on the real line, and being equipped with k its periodic boundaryconditions). Q stands for the probabilitylaw corresponding to k the system S , and the probability Q corresponding to the averaged regime is then deÿned by k Q (:)= d"(k)Q (:):
The following proposition shows that this new probability Q on Wiener space WT mayalso be viewed as the law of a nearest neighbour interacting di7usions system S .
Proposition 2.1. Q is the weak solution corresponding to the interacting di;usions system S given by 2ÿ2 i i i j i j dxt =dwt − U (xt)dt + ÿ xt dt + yt − ÿ zt dt 1+2ÿ2t j∼i j∼i t t yi = xi − xi + U (xi )ds; zi = xi ds t t 0 s t s 0 0 ⊗ Law(x|t=0)=0 (i ∈ ; 0 6 t 6 T):
Proof. We let PT denote the probabilitylaw on Wiener space WT corresponding to the process:
dxt =dwt − U (xt)dt with initial condition: law(x|t=0)=0. PT is such that t xt − x0 + U (xv)dv is a standard Brownian motion under dPT (x): 0
Let also Pt denote the restriction of PT to the -algebra Ft corresponding to the time interval [0;t] ⊂ [0;T], and Q ;t denote similarlythe restriction of Q to the -algebra 218 G. Ben Arous, M. Sortais / Stochastic Processes andtheir Applications 105 (2003) 211–255
in WT corresponding to this same time interval. According to Fubini and Girsanov’s theorems:
⊗ Q PT ; and dQ M = ;t t ⊗ dPt is a positive P⊗ -martingale with mean 1, such that T t j i i Mt (x)= d"(k) exp ÿ xu + k dwu i∈ 0 j∼i 2 ÿ2 t − x j + ki du 2 u i∈ 0 j∼i 2 t ÿ2 t = expÿ x j dwi − x j du u u 2 u i∈ 0 j∼i i∈ 0 j∼i ÿ2t × d"(k) exp (k ; A (x)) − (k ; k ) ; ÿ;t 2 A (x) being the -dimensional real vector deÿned by: ÿ;t t i i 2 j Aÿ;t(x)=ÿwt (x) − ÿ xu du 0 j∼i t i i i j = ÿ xt − x0 + U (xs) − ÿ xs ds : 0 j∼i Averaging over the Gaussian vector k, we then obtain 2 1 t log M (x)= A (x); A (x) + ÿ x j dwi t mart: 2 ÿ;t 1+2ÿ2t ÿ;t u u i∈ 0 j∼i t A (x) t = 2 ÿ;u ;dA (x) + ÿ x j dwi mart: 1+2ÿ2u ÿ;u u u 0 i∈ 0 j∼i t Ai (x) t = ÿ2 ÿ;u dwi (x)+ÿ x j dwi ; mart: 1+2ÿ2u u u u i∈ 0 i∈ 0 j∼i
⊗ the sign =mart: meaning here that the two PT -semimartingales under consideration (on the left hand side and on the right hand side of the equality) have the same martingale part. G. Ben Arous, M. Sortais / Stochastic Processes andtheir Applications 105 (2003) 211–255 219
At this stage one should check that the positive martingale Mt (x) is a uniformly integrable one, so as to make sure that Girsanov’s theorem maybe applied a second time. We prove a stronger fact in Section 3 (see Proposition 3.1): (Mt (x))06t6T is ⊗ in fact bounded from above, PT -a.s. in x.
Girsanov’s theorem maythus be applied to the probability Q on path space WT corresponding to the uniformlyintegrable martingale ( Mt (x))06t6T , and the proposi- tion is proved.
Note. Adding the variables t t i i i i i i yt = xt − x0 + U (xs)ds; zt = xs ds 0 0 2 2 2 2 at each site i ∈ and letting: "t = ÿ =(1 + ÿ t), one maynotice that the interacting di7usions system S obtained in the averaged regime is a Markov one, since the di7erentials of the new variables yi and zi are i i j i j i i dyt =dwt + ÿ xt dt + "t yt − ÿ zt dt and dzt = xt dt: j∼i j∼i
Classical results of Ito calculus assert that S , viewed as a Markov system of interacting three dimensional di7usions, has a unique strong solution for anyinverse temperature parameter ÿ and anyterminal time T¿0, that maybe constructed through Euler’s method (cf. Krylov, 1990). Actually, the monotonicity properties of the drift term appearing in the di7erentials i i i dxt ,dyt and dzt are such that one mayassert, following Shiga and Shimizu, a strong existence and uniqueness result for the corresponding inÿnite dimensional system S∞, given by i i i j i j dxt =dwt − U (xt)dt + ÿ xt dt + "t yt − ÿ zt dt; j∼i j∼i i i j i j dyt =dwt + ÿ xt dt + "t yt − ÿ zt dt; j∼i j∼i dzi = xi dt; t t ⊗Zd d Law(x|t=0)=0 ; y|t=0 = z|t=0 = 0 (i ∈ Z ; 0 6 t 6 T): To be more precise, we next introduce some notations and give a strong existence and uniqueness theorem for inÿnite systems of n-dimensional di7usions that is convenient for our purpose.
n Zd Deÿnition 2.1. Fix an integer n ¿ 1. For each p ∈ Z and X =(Xi)i∈Zd ∈ (R ) , let 1 X 2 = |X |2 (with: |i| = |i | + ···+ |i |) p (1 + |i|)2p i 1 d i∈Zd 220 G. Ben Arous, M. Sortais / Stochastic Processes andtheir Applications 105 (2003) 211–255
d n and deÿne the Hilbert space Sp(Z ; R )as d n n Zd Sp(Z ; R )={X ∈ (R ) =X p ¡ + ∞}: The space of Rn-valued rapidlydecreasing sequences on Zd is then deÿned as d n d n S(Z ; R )= Sp(Z ; R ): p∈Z
d n S(Z ; R ) is a nuclear space for the sequence of norms :p, and its dual space is the space S(Zd; Rn) of all tempered Rn-valued sequences on Zd: d n d n S (Z ; R )= Sp(Z ; R ): p∈Z
d d d d We also let: S(Z )=S(Z ; R) and S+(Z )=S(Z ; R+), and we recall that any d n n Zd S (Z ; R )-valued path X: :[0;T] → (R ) that is weaklycontinuous is also strongly continuous.
Following Shiga and Shimizu, we maynow state a strong existence and uniqueness theorem for inÿnite systems of Rn-valued di7usions whose coeOcients satisfysome monotonicitycondition.
Theorem 2.1. Consider the inÿnite dimensional system of Rn-valueddi;usions (E) given by:
i i i d (E): dXt = · dwt + fi(t; Xt ; Xt)dt; i ∈ Z ;t¿ 0 where:
i n Zd • Xt =(Xt )i∈Zd ∈ (R ) . i d •{(wt )t¿0; i ∈ Z } is an i.i.d. family of standard n-dimensional brownian motions. • is ÿxedin Rn⊗n. n n Zd n • fi : R+ × R × (R ) →R is a family of continuous mappings such that:
d ◦ fi(t;0;0)=0, ∀i ∈ Z , ∀t ¿ 0 d d n n Zd n ◦∀i ∈ Z , ∀ ⊂⊂ Z , fi : R+×R ×(R ) →R is locally Lipschitz in the variables i (t; 3;(X )i∈ ) ◦ Monotonicity Condition:
d d n n Zd ∃K; L¿0; ∃c=(ci)i∈Zd ∈S+(Z ); ∀i∈Z ; ∀t¿0; ∀3; Á∈R ; ∀X; Y ∈(R ) ; 2 j j 2 ((3−Á); fi(t; 3; X )−fi(t; Á; Y ))6K|3−Á| +L|3−Á|· ci−j|X −Y | : j∈Zd
d n Then: for each X ∈ S (Z ; R ), (E) has a unique strong solution (Xt)t¿0 with ini- d n tial condition X0 = X , moreover (Xt)t¿0 has a.s. S (Z ; R )-valuedpaths that are continuous with respect to the strong topology in S(Zd; Rn). G. Ben Arous, M. Sortais / Stochastic Processes andtheir Applications 105 (2003) 211–255 221
Proof. One merelyneeds to adapt the demonstration of Theorem 4.1 in ( Shiga and Shimizu, 1980) (for a complete proof see Sortais (2001, Theorem 2.2.1 in Chapter 2)).
Corollary 2.1. For any compactly supportedprobability 0 on the real line, the inÿnite system of three-dimensional interacting di;usions S given by ∞ i i i j i j dxt =dwt − U (xt)dt + ÿ xt dt + "t yt − ÿ zt dt j∼i j∼i i i j i j dyt =dwt + ÿ xt dt + "t yt − ÿ zt dt j∼i j∼i dzi = xi dt t t ⊗Zd d Law(x|t=0)=0 ; y|t=0 = z|t=0 = 0 (i ∈ Z ; 0 6 t 6 T) d 3 has a unique strong solution (Xt)t¿0, and (Xt)t¿0 has a.s. Sd(Z ; R )-valued, continuous paths.
Proof. Applysimplythe preceding theorem to the situation where 100 n =3;= 100 000
3 3 Zd 3 and fi : R+ × R × (R ) → R is deÿned through the identity: 1 2 3 3 i i i 3 Zd ∀t ¿ 0; ∀3 =(3 ;3 ;3 ) ∈ R ; ∀X =(x ;y;z)i∈Zd ∈ (R ) ; j i j −U (31)+ÿ x + "t y − ÿ z j∼i j∼i fi(t; 3; X )= ; ÿ x j + " yi − ÿ z j t j∼i j∼i xi
⊗Zd ⊗Zd d observe also that our “deep quench” initial condition 0 satisÿes: 0 (Sd(Z ; R))=1.
i i i Note. (a) (Xt)06t6T =(xt;yt ;zt )06t6T being the unique strong solution corresponding i to the system S∞,weletQ∞ denote the probabilitylaw of ( xt)06t6T ; Q∞ is a spatiallyshift invariant probabilitymeasure on inÿnite dimensional path space = Zd WT : Q∞ ∈ Ms( ). In the next sections we show that Q∞ maybe viewed as a Gibbs measure on and give the corresponding interaction on path space explicitly. 222 G. Ben Arous, M. Sortais / Stochastic Processes andtheir Applications 105 (2003) 211–255
(b) As an intermediate step in the proof of Theorem 2.1, one mayshow that Q∞ satisÿes: 2 dQ∞ sup xtd ¡ ∞: 06t6T Actually, still using Ito’s formula and an appropriate Gromwall inequality, one may also prove that the quantity 2p sup dQ∞xtd 06t6T is ÿnite for all p ¿ 1 (see the proof of Theorem 4.6 in FQollmer and Wakolbinger, 1986). As an easyconsequence, we maystate that integrals such as
T i p i p dQ∞(x)|xT | or dQ∞(x) |xt| dt 0 are bounded from above uniformlyin i ∈ Zd, for all p ¿ 1. (c) Last but not least, the dynamics Q∞ maybe veryconvenientlycompared to the dynamics of a standard Ising model by introducing the new variables: i i j vt = yt − ÿ zt j∼i
d at each site i ∈ Z . Indeed, one obtains that Q∞ mayalso be viewed as the probability law corresponding to the system i i i j dxt =dvt − U (xt)dt + ÿ xt dt; j∼i
⊗Zd Law(x|t=0)=0 ;
i {vt;06 t 6 T}i∈Zd being an i.i.d. familyof time inhomogeneous Ornstein–Uhlenbeck processes under Q∞, solving the following linear stochastic di7erential equation:
dvt =dwt + "tvt dt; with initial condition: v0 =0.
2.2. Gibbsian integration by parts formula on path space
Our aim in the present section is to present brieRysome of the main results con- tained in (Cattiaux et al., 1996); to this end we ÿrst give general deÿnitions for Gibbs measures based on the lattice Zd.
Deÿnition 2.2. Let X be a Polish space (spin space) and let Td denote the set of all ÿnite subsets ⊂⊂ Zd. G. Ben Arous, M. Sortais / Stochastic Processes andtheir Applications 105 (2003) 211–255 223
Let = ⊗i∈Zd 9i be an inÿnite tensor product of -ÿnite measures on X and : be a probabilitymeasure on X Zd .
(1) A family ; =(; ) of ¿ 0, measurable functionals on X Zd is said to be a ∈Td (:; )-modiÿcation when the following conditions are satisÿed: (i) ∀ ∈ Td; ; (z ∨ x c )d (z)=1;: c -a:s:(x c )
(ii) ∀ ⊂ < ∈ T ;:c -a:s:(x c ); ( ⊗ )-a:s:(y); d < < < x ;<(y)=; (y) · ;<(z ∨ y c )d (z) (here and in the sequel, (z ∨ y c ) denotes the combination of conÿgurations c z ∈ X and y c ∈ X ). (2) An interaction =( ) is a familyof measurable functionals on X Zd such ∈Td that: (i) Zd ∀ ∈ Td; : X → R is F -measurable (ii) Zd ∀ ∈ Td; | (x)| ¡ ∞; ∀x ∈ X ∩ =? Zd d (F being the -algebra generated bythe projections pi : X → X; i ∈ Z ). One maythen consider the Hamiltonian potential H =(H ) ⊂⊂Zd corresponding to the interaction . H is deÿned through the identities: H (x)= (x) ∩ =? Zd holding for all ∈ Td and all x ∈ X . In the situation where the partition function corresponding to the interaction is well deÿned, i.e.: ∀ ∈ Td;: c -a:s:(x c ); Z (x c )= d (z) exp(−H (z ∨ x c )) ¡ + ∞; one maythen easilycheck that exp(−H (x ∨ x c )) ; (x ∨ x c )= Z (x c ) deÿnes a (:; )-modiÿcation on X Zd . (3) Given any( :; )-modiÿcation ;, one says that the probability : is a Gibbs measure on X Zd whenever ∀ ∈ Td;: c -a:s:(x c );:(dx |x c )=; (x ∨ x c )d (x ): 224 G. Ben Arous, M. Sortais / Stochastic Processes andtheir Applications 105 (2003) 211–255 Note. (a) When the modiÿcation ; is constructed via an interaction potential , the corresponding Gibbs measures are called (; )-Gibbs measures. (b) When one knows that the familyof ¿ 0 functionals ; =(; ) on X Zd is a ∈Td (:; )-modiÿcation, to make sure that : isa(;; )-Gibbs measure one simplyhas to check that the identity :(dx|3)=;{i}(x ∨ 3)d9i(x) d holds :{i}c -a.s.(3), for each site i ∈ Z . Several authors (FQollmer and Orey, 1988; Olla, 1988; Comets, 1989) have inves- ( ) tigated the LD asymptotics of the empirical process ˆx in the situation where x is distributed according to some (; )-Gibbs measure G on X Zd . In each of these articles, the Gibbsian interaction is assumed to satisfytranslation invariance, so that (i) (x )= i+ (x) for all ; i; x; as well as the following summabilityproperty = sup | (x)| ¡ + ∞: x O Assuming additionallythat each of the functionals deÿning the interaction is contin- uous and that the reference measure is simplythe inÿnite tensor product of a single site probabilitymeasure 90, one maythen state that the law of the empirical process ( ) Zd ˆx under dG(x) obeys a LDP on Ms(X ), on the scale | | and according to the good rate function I : M (X Zd ) → [0; +∞] given by s Zd ∀P ∈ Ms(X ); I (P)=H(P) − V (x)dP(x) − p ; X Zd where 1 V (x)=− (x) |A| A A O and p = inf H(P) − V (x)dP(x) ; Zd Zd Q∈Ms(X ) X ⊗Zd ⊗Zd H(P)=H(P|90 ) being the speciÿc entropyrelative to = 90 . One should also add that the verysame LD results are valid when considering the ( ) law of the empirical process ˆx under dG(x|Q c ), the probabilityd G(x|Q c ) being Zd a conditional version of dG(x) knowing that x c = Q c , and Q ∈ X being anyÿxed boundarycondition. Before stating the Integration byParts Formula that was developed byCattiaux, Zd Roellyand Zessin in the context of Gibbs measures on path space =WT , we would like to motivate such developments and show how the inÿnite dimensional dynamics Q∞ (deÿned after Corollary 2.1) mayalso be presented as a Gibbs measure on .Itis actuallya rather straightforward task to ÿnd a satisfactoryexpression for the interaction a ⊗Zd corresponding to Q∞ when is equipped with the reference measure RT , RT G. Ben Arous, M. Sortais / Stochastic Processes andtheir Applications 105 (2003) 211–255 225 denoting the Wiener measure on WT having initial condition 0 (cf. Corollary 2.1). Indeed, setting periodic boundaryconditions on the ÿnite cubic box =[−N; N]d ∩Zd, one maynotice that the Radon–Nykod ÃUm derivative of the averaged regime probability ⊗ Q with respect to the reference probabilitymeasure RT writes T dQ i j i j i (x ) = exp −U (x )+ÿ x + "t y − ÿ z dx dR⊗ t t t t t T i∈ 0 j∼i j∼i 2 1 T − −U (xi)+ÿ x j + " yi − ÿ z j dt ; 2 t t t t t i∈ 0 j∼i j∼i j∼i meaning here that sites i; j ∈ are nearest neighbours when is considered with its periodic boundaryconditions, and " still denoting the function given by: 2 2 2 2 "t = ÿ =(1 + ÿ t). 3 Introducing the functionals Fi;Gi;Hi : R+ × → R given by Fi(t; x; y; z)=Gi(t; x; y; z)+Hi(t; x; y; z) and i i i i j j i j Gi(t; x; y; z)=−U(x )+"tx y ;Hi(t; x; y; z)=ÿx (x −"tz )−ÿ"tz x j∼i j∼i we then have dQ (x ) = exp{−K (x )}; ⊗ T dRT ⊗ where (under dRT (x )): T @F 1 T @F 2 −K (x )= i (t; x ; y ; z ) dxi − i (t; x ; y ; z ) dt T @xi t t t t 2 @xi t t t i∈ 0 i∈ 0 T @G T @H = i (t; x ; y ; z ) dxi + i (t; x ; y ; z ) dxi @xi t t t t @xi t t t t i∈ 0 i∈ 0 1 T @G 2 − i (t; x ; y ; z ) dt 2 @xi t t t i∈ 0 T @G @H − i · i (t; x ; y ; z ) dt @xi @xi t t t i∈ 0 1 T @H 2 − i (t; x ; y ; z ) dt: 2 @xi t t t i∈ 0 226 G. Ben Arous, M. Sortais / Stochastic Processes andtheir Applications 105 (2003) 211–255 Using Ito’s formula, we then obtain −KT (x )= [Gi(T; xT ; yT ; zT ) − Gi(0; x0; y0; z0) i∈ 1 T @G 2 − i (t; x ; y ; z ) dt i t t t 2 0 @x T @G T @G − i (t; x ; y ; z )(dxi + U (xi)dt) − i (t; x ; y ; z )dt i t t t t t t t t 0 @y 0 @t 1 T @2G @2G − i +2 i (t; x ; y ; z )dt i 2 i i t t t 2 0 @(x ) @x @y + [Hi(T; xT ; yT ; zT ) − Hi(0; x0; y0; z0) i∈ 1 T @H − i (t; x ; y ; z )xi dt i t t t t 2 0 @z T T j k i l 2 i j j i + ÿ "t xt xt + xt xt dt − ÿ "t zt xt + zt xt dt 0 k∼i l∼j 0 2 1 T − −U (xi)+ÿ x j + " yi − ÿ z j dt: 2 t t t t t i∈ 0 j∼i j∼i In this last expression for the Girsanov exponent −KT (x ), the ÿrst sum obviously corresponds to a sum of self interactions, the second sum corresponds to a sum of nearest neighbours two spins interactions and the third sum corresponds to a sum of self interactions, nearest neighbours two spins interactions and three spins {xi;xj;xk } interactions, where i ∼ j ∼ k in and i = k, so that ÿnally per: a KT (x )= :(x); :⊂ a being the interaction given by 1 2 2 a(x)=U(xi ) − U(xi )+ (" (xi − 2xi yi ) − xi ) i T 0 2 T T T T 0 T T 1 i i 2 1 i i i − (U (xt) − U (xt) )dt + (1+2(xt − yt )U (xt))"t dt 2 0 2 0 T 1 i i 2 + ("t(xt − yt )) dt; 2 0 a i j i j j i i j i;j(x)=−ÿ(xT xT − x0x0 )+ÿ"T (zT xT + zT xT ) T 2 T i j j i ÿ i2 j2 − ÿ (U (xt)xt + U (xt )xt)dt + (xt + xt )dt 0 2 0 G. Ben Arous, M. Sortais / Stochastic Processes andtheir Applications 105 (2003) 211–255 227 T i j j i i j j i i j + ÿ (U (xt)zt + U (xt )zt + xtyt + xt yt − 2xtxt )"t dt 0 T T 2 i i j j i j j j i i 2 − ÿ (xtzt + xt zt )"t dt − ÿ (zt (yt − xt )+zt (yt − xt))"t dt 0 0 2 T ÿ i2 j2 2 + (zt + zt )"t dt if : j∼i; 2 0 T T T a 2 i k j k k j j k 2 i;j;k (x)=ÿ xtxt dt − (xt zt + xt zt )"t dt + zt zt "t dt 0 0 0 if : j ∼ i ∼ k; j = k; a A (x) = 0 else: Note. Remembering that: yi = xi − xi + t U (xi )ds, we also have t t 0 0 s T T i i i i i i i i "t(xt − yt )U (xt)dt = − "ty˜ t d˜y t for :y ˜ t = yt − xt; 0 0 so that T T i i i 1 i i 2 "t(xt − yt )U (xt)dt + "t(xt − yt ) dt 0 2 0 1 = − (" (˜yi )2 − " (˜yi )2) 2 T T 0 0 T 2 "T i 1 i 2 = − U (xt)dt + (x0) ; 2 0 2 which yields the following alternative expression for i: 1 2 2 a(x)=U(xi ) − U(xi )+ (" (xi − 2xi yi ) − xi ) i T 0 2 T T T T 0 T 1 i i 2 1 2 2 − (U (xt) − U (xt) )dt + ln(1 + ÿ T) 2 0 2 T 2 "T i 1 i 2 − U (xt)dt + (x0) : 2 0 2 Let us next present the elements of Malliavin calculus enabling us to give a proper statement of an integration byparts formula for Gibbs measures on . Deÿnition 2.3. • W 1;2(C[0;T]) denotes the Sobolev space of all functionals F : C[0;T] → 2 R such that: F is square integrable with respect to RT (F ∈ L (C[0;T])) and there exists a family {(DtF(!))06t6T ; ! ∈ C[0;T]} 228 G. Ben Arous, M. Sortais / Stochastic Processes andtheir Applications 105 (2003) 211–255 in L2([0;T]; C[0;T]) such that : 2 1 ∀g ∈ L ([0;T]);DgF (!) = lim F ! + F gs ds − F(!) F0 F 0 exists as a strong limit in L2(C[0;T]) and equals: T g (D F)(!)dt. 0 t t • W 1;∞(C[0;T]) is the subspace of W 1;2(C[0;T]) consisting of all those F for which ∞ 2 the family {(DtF(!))06t6T ; ! ∈ C[0;T]} takes its values in L (C[0;T]; L ([0;T])). • We also let W(C[0;T]) ⊂ W 1;∞(C[0;T]) denote the set of all regular functionals F : C[0;T] → R such that: F(!)=f(! t0 ;!t1 ;:::;!tn ) ∞ for some ÿnite sequence 0 6 t0 ¡t1 ¡ ···¡tn 6 T and some C , compactly supported f. Let us recall at this stage that the veryÿrst integration byparts formula of Malli- avin calculus (see Nualart, 1998, Chapter 1) states that, relativelyto the standard Wiener measure on C[0;T]: ∀F ∈ W(C[0;T]); ∀g ∈ L2([0;T]); T T E gtDtF dt = E(DgF)=E F gt dwt : 0 0 We next deÿne Sobolev spaces corresponding to the inÿnite product . • For any i ∈ Zd and g = gi ∈ L2([0;T]), for ! ∈ , let ! + F : gi ds denote the T 0 s element ! of such that : i !j = !j for j = i and !i = !i + F gs ds: 0 The Sobolev space W 1;2( ) then consists of all functionals F : → R that are ⊗Zd square integrable with respect to the reference measure RT and such that there i 2 Zd exists a family {(DtF(!))i∈Zd;06t6T ; ! ∈ WT } in (L ([0;T]; WT )) satisfying: d 2 ∀i ∈ Z ; ∀gi ∈ L ([0;T]); 1 : T lim F ! + F gi ds − F(!) = Di F(!)= gi(DiF)(!)dt; s gi t t F0 F 0 0 the preceding limit being taken in the strong sense in L2( ). • W 1;∞( ) is the subspace of W 1;2( ) consisting of all those F for which the family i 2 ∞ Zd {(DtF(!))i∈Zd;06t6T ; ! ∈ C[0;T]} takes its values in (L ( ; L ([0;T]))) . • Wloc( ) is the set of all regular functionals F : → R satisfying: F(!)=f(!:;!:;:::;!:) t0 t1 tn d for some ÿnite subset : ⊂⊂ Z and some ÿnite sequence 0 6 t0 ¡t1 ¡ ···¡tn 6 T, f being C∞ and compactlysupported. G. Ben Arous, M. Sortais / Stochastic Processes andtheir Applications 105 (2003) 211–255 229 1;∞ 1;2 1;∞ • Wloc ( ) (resp. Wloc ( )) consists of all functionals F = F(!) ∈ W ( ) (resp. 1;2 W ( )) depending on ! onlythrough a ÿnite number of coordinates: F(!)=F(!:) for some : ⊂⊂ Zd. 2 • Finally, let E[0;T] denote the set of all elementary L functions g :[0;T] → R: ∀g ∈ L2([0;T]); (g ∈ E[0;T]) ⇔ (∃0=t0 ¡t1 ¡ ···¡tn = T; g is constant in each interval [ti−1;ti[): We maynow state an integration byparts formula holding for all Gibbs mea- sures Q on inÿnite dimensional path space corresponding to some (reasonable) interaction G. Theorem 2.2. Let H G denote the Hamiltonian potential corresponding to some inter- ⊗Zd d action G on ( ; RT ) (c.f. Deÿnition 3.1) andsuppose that for each i ∈ Z andfor d (Z \{i}) G 1;2 i G all Á ∈ WT , Hi (!; Á) is a W functional of ! ∈ C[0;T] and (DtHi (!; Á))06t6T has a measurable version. Let Q be a probability measure on satisfying: T d i i G ∀t ∈ [0;T]; ∀i ∈ Z ; EQ(|!t|) ¡ ∞ and EQ |DtHi | dt ¡ ∞: 0 ⊗Zd If Q is a Gibbs measure with respect to G and RT , then 1;∞ d ∀F ∈ Wloc ( T ); ∀i ∈ Z ; ∀gi ∈ E[0;T]; T E F gi d!i = E (Di F) − E (FDi H G): Q t t Q gi Q gi i 0 Proof. See (Cattiaux et al., 1996, Theorem 2.11, ÿrst part). 2.3. Characterisation of Q∞ as a Gibbs measure Although the inÿnite-dimensional system of interacting di7usions S∞ given by i i i j i j dxt =dwt − U (xt)dt + ÿ xt dt + "t yt − ÿ zt dt j∼i j∼i i i j i j dyt =dwt + ÿ xt dt + "t yt − ÿ zt dt j∼i j∼i dzi = xi dt t t ⊗Zd d Law(x|t=0)=0 ; y|t=0 = z|t=0 = 0 (i ∈ Z ; 0 6 t 6 T) 230 G. Ben Arous, M. Sortais / Stochastic Processes andtheir Applications 105 (2003) 211–255 is not a gradient one, we maystill use the Gibbsian integration byparts formula as an essential tool to prove the following. Proposition 2.2. Let Q be a probability measure on . If Q is Gibbsian with respect a ⊗Zd to the interaction andto the reference measure RT , then Q is the x-marginal of a weak solution corresponding to the system S∞, consequently: Q = Q∞ (since S∞ has a unique strong solution). Proof. We shall simplycheck that the second half of the proof of Theorem 3.7 in (Cattiaux et al., 1996) is also valid in our context. First of all, let us remark that our Gibbs measure Q is such that d ∀i ∈ Z ;Q RT ⊗ QZd\{i}; hence there is a unique adapted process (ÿi) for which {xi − xi − t ÿi ds} t 06t6T t 0 0 s 06t6T is a standard brownian motion under dQ(x). Secondly, Q satisÿes d i ∀t ∈ [0;T]; ∀i ∈ Z ; EQ(|!t|) ¡ ∞ and T d i ∀i ∈ Z ; EQ |DtHi| dt ¡ ∞ 0 (cf. Lemma 3.2 in the next section). Therefore we mayuse the Gibbsian integration byparts formula to show that @F Q-a:s:(x); ∀i ∈ Zd;ÿi(x)=− i (t; x ; y ; z ); t @xi t t t the functional F being deÿned by i i i j i i j Fi(t; x; y; z)=U(x ) − ÿx x − "tx y − ÿ z : j∼i j∼i i d Indeed, {(ÿt)06t6T ; i ∈ Z } is uniquelycharacterised (up to Q indistinguishability) by the fact that t d i i i ∀0 6 s¡t6 T; ∀i ∈ Z ; EQ As xt − xs − ÿr dr =0 s As : T → R being any Wloc, Fs-measurable functional on T . We mayalso write the preceding equation as T t i i EQ As 1]s;t] dxr = EQ As ÿr dr 0 s and after integrating byparts we obtain t E A ÿi dr + Di H a =0 (∗) Q s r 1]s; t] i s since Di A ≡ 0. 1]s; t] s G. Ben Arous, M. Sortais / Stochastic Processes andtheir Applications 105 (2003) 211–255 231 One then simplyhas to compute Di H a in order to make sure that (∗) holds true 1]s; t] i when setting: @F ÿi(x)=− i (t; x ; y ; z ): t @xi t t t Taking into account the fact that a Hi (x)=Fi(T; xT ; yT ; zT )−Fi(0; x0; y0; z0) T @F T @F T @F T @F − i dyi − i xi dr − i x j dr − i dr @y r @z r @z r @r 0 i 0 i j∼i 0 j 0 1 T @2F @2F − i +2 i dr 2 2 0 @xi @xi@yi 1 T @F 2 T @F @F + i dr + i · j dr 2 @x @x @x 0 i j∼i 0 j j 1 T @F 2 − i dr 2 @x j∼i 0 j T @F T @F 1 T @F 2 = i dxi + i dx j − i dr @x r @x r 2 @x 0 i j∼i 0 j 0 i 1 T @F 2 − i dr; 2 @x j∼i 0 j we maythen proceed to the (rather lengthy)Malliavin di7erentiation of each term in the preceding sum. To this end, we deÿne r r ∀r ∈ [0;T];’r = 1]s;t](v)dv =((r ∧ t) ∨ s − s);Ir = ’v dv 0 0 and i;F i ∀F¿0;xr = xr + F’r r i;F i;F i;F i;F yr = xr − x0 + U (xv )dv 0 r i;F i;F zr = xv dv; 0 we then observe that 1 i;F i (xr − xr)−→ ’r F F0 r 1 i;F i i (yr − yr)−→ ’r + U (xv)’v dv F F0 0 232 G. Ben Arous, M. Sortais / Stochastic Processes andtheir Applications 105 (2003) 211–255 1 i;F i (zr − zr)−→ Ir; F F0 these convergences taking place in L2 . RT One thus obtains for example T @F t @F T @2F @2F Di i dxi = i dr + i · ’ + i 1]s; t] r 2 r 0 @xi s @xi s @xi @xi@yi r 2 i @ Fi i i × ’r+ U (xv)’v dv + · Ir (dMr +ÿr dr); 0 @xi@zi i where (Mr ) is a standard brownian motion under dQ, and 1 T @F 2 Di − i dr 1]s; t] 2 0 @xi T @F @2F @F @2F = − i i · ’ + i i 2 r s @xi @xi @xi @xi@yi r 2 i @Fi @ Fi × ’r + U (xv)’v dv + · Ir dr; 0 @xi @xi@zi i d so that all in all the process {(ÿt)06t6T ; i ∈ Z } satisÿes Eq. (∗) if and onlyif t i @Fi EQ As ÿr + (r; xr; yr; zr) dr s @xi T @2F @2F r = E A − i · ’ + i · ’ + U (xi )’ dv Q s 2 r r v v s @xi @xi@yi 0 @2F T @2F @2F + i · I ÿi dr − i · ’ + i @x @z r r @x @x r @x @y i i j∼i s j i j i r 2 i @ Fi j × ’r + U (xv)’v dv + · Ir ÿr dr 0 @xj@zi T @F @2F @F @2F r + i i · ’ + i i · ’ + U (xi )’ dv 2 r r v v s @xi @xi @xi @xi@yi 0 @F @2F T @F @2F + i i · I dr + i i · ’ @x @x @z r @x @x @x r i i i j∼i s j j i 2 r @Fi @ Fi i + · ’r + U (xv)’v dv @xj @xj@yi 0 2 @Fi @ Fi + · Ir dr : @xj @xj@zi G. Ben Arous, M. Sortais / Stochastic Processes andtheir Applications 105 (2003) 211–255 233 This last equalityis certainlysatisÿed for anyof our test functionals As if @F ÿi (x)=− i (r; x ; y ; z ); r @xi r r r and the proposition is proved. Since we are in a situation where the spin space WT is non-compact and where the translation invariant interaction a is not a summable one, it is not obvious at ÿrst sight that there exists indeed a Gibbs measure Q ∈ M( ) corresponding to a and to ⊗Zd the reference measure RT ; however, this fact maybe seen to follow from two results contained in Chapter 4 of (Georgii, 1988), namelyCorollary(4.13, p. 63) together a ⊗Zd with Theorem (4.17, p. 67). So the set consisting of all ( ; RT )-Gibbs measures is non-empty, and according to the preceding proposition this set is actually reduced to {Q∞}. 3. Averaged large deviations of the empirical process a Although Q∞ is the unique Gibbs measure associated with the interaction and ⊗Zd reference measure RT , some veriÿcations are needed in order to state a LDP when ( ) considering the law of the empirical process ˆx under dQ (x), since the interac- tion we are dealing with is not a summable one. Nevertheless, proceeding just as in Comets (1989) we mayobserve, still using periodic boundaryconditions for , that the following identityholds true for each Borel set A ⊂ Ms( ): ⊗ a − ⊂ (x) ! (A)= dRT (x)1 ( ) e ⊗ { x ∈A} WT a (x) − ⊗ i∈ i | | = dRT (x)1 ( ) e ⊗ { x ∈A} WT 0 | | V ad = d! ( )e ; A 0 ( ) ⊗ a ! denoting the law of the empirical process x under dRT (x), and V : → R being naturallydeÿned by a(x) ∀x ∈ ;Va(x)=− < : |<| < O Letting L be some positive real number for which 0([−L; L])=1 and Ms;L( ) denote the subset of Ms( ) consisting of all Q such that i d −L 6 x0 6 L; Q-a:s:(x)(∀i ∈ Z ); we ÿrst establish the large deviation upper bound for {! } bymaking use of Varadhan’s method (see Varadhan, 1966, Section 3 or Dembo and Zeitouni, 1998, Section 4.3): this amounts essentiallyto check that the functional V : → V a d 234 G. Ben Arous, M. Sortais / Stochastic Processes andtheir Applications 105 (2003) 211–255 maybe deÿned on the whole space Ms;L as an upper semicontinuous functional taking values in the interval [ −∞; M] for some M¡+ ∞ depending onlyon the choices of ÿ, , T and L. ⊗Zd H still denoting the speciÿc entropyrelative to the reference measure RT , we then check that the good rate functional Ia = H − V vanishes onlyat Q∞. To be more precise, we establish that our nonsummable interac- tion a also satisÿes the variational principle of standard Gibbsian theory(see Georgii, 1988, Theorem 15.39, p. 325), and this is enough to make sure that Ia vanishes ex- a ⊗Zd clusivelyat Q∞ since we know that the set of all ( ; RT )-Gibbs measures reduces to {Q∞}. We ÿnallyprove the large deviations lower bound for {! } following the method of FQollmer and Orey(see FQollmer and Orey, 1988, Section 3) and using the fact that for anyopen set O⊂Ms;L( ): inf {H( ) − V( )} = inf {H( ) − V( )} ; ∈O ∈O∩M V a MV denoting the set of all points ∈ Ms;L( ) at which the functional V : → V d is lower semicontinuous. 3.1. Large deviations upper bound Let us begin with the following a Proposition 3.1. V : → V d may be deÿned on Ms;L( ) as an upper semicon- tinuous functional taking values in the interval [ −∞; M], for some M¿0 depending only on the choices of ÿ, , T and L. Proof. A quick glance at the expression for the interaction a given earlier enables one to make sure that Zd ∀x ∈ WT ; T a i i i i i i i i i V (x)=P1("T ; {x0;xT ;y0;yT ;z0;zT }i∈<)+ P2("t; {xt;yt ;zt }i∈<)dt; 0 where < = {i ∈ Zd | (i = O)or(i ∼ O)or(i ∼ j for some j ∼ O)}; P1 and P2 being polynomial functions. To make sure that T i i i d (x) P2("t; {xt;yt ;zt }i∈<)dt 0 is bounded from above uniformlywhen varies in M ( ), we shall ÿrst deal with s the terms in (1) = T P (" ; {xi;yi;zi} )dt coming from the “single site interaction 0 2 t t t t i∈< G. Ben Arous, M. Sortais / Stochastic Processes andtheir Applications 105 (2003) 211–255 235 a potential” 0 ; theysum to T T ÿ O O 2 ÿ O O O (1) = (U (xt ) − U (xt ) )dt − (1+2(xt − yt )U (xt ))"t dt 2 0 2 0 T ÿ O O 2 − ("t(xt − yt )) dt; 2 0 which is also equal to T ÿ O O 2 ÿ 2 2 (U (xt ) − U (xt ) )dt − ln (1 + ÿ T) 2 0 2 T 2 ÿ O ÿ O 2 + "T U (xt )dt − (x0 ) 2 0 2 according to the remark preceding Deÿnition 2.3. Next, according to Jensen’s inequality: 2 1 T 1 T U (xO)dt 6 U (xO)2 dt; 2 t t T 0 T 0 so that ÿnally T T ÿ O ÿ O 2 (1) 6 K + U (xt )dt − T U (xt ) dt 2 0 2 0 2 2 for T =1− T"T =1=(1 + ÿ T) (here and in the sequel of the proof, K; K1;K2;::: are some positive constants depending merelyon the choices of ÿ, , T and ). 0 The terms remaining in the integral: d (x)( T P dt) (coming from the “two sites 0 2 and three sites interaction potentials”) maybe dealt with byusing Young inequalities and the translation invariance of . For example: the integral term 2 T ÿ i O d (x) U (xt)xt dt 2 0 a (coming from the interaction O;i for some i ∼ O) is not greater than 2 T T ÿ 3 O 5=3 2 O 5=2 d (x) |U (xt )| dt + |xt | dt 2 5 0 5 0 and the integral term 5 T ÿ i j d (x) "txtzt dt 3 0 a (coming from the interaction O;i;j for some j ∼ i ∼ O) is not greater than 5 T T ÿ 1 2 O 2 1 O 2 d (x) "t (xt ) dt + (zt ) dt ; 3 2 0 2 0 236 G. Ben Arous, M. Sortais / Stochastic Processes andtheir Applications 105 (2003) 211–255 which maybe bounded from above by 5 T 2 T ÿ 1 2 O 2 T O 2 d (x) "t (xt ) dt + (xt ) dt ; 3 2 0 2 0 remembering that: zO = t xO ds and using Jensen’s inequality. t 0 s All in all, one ÿnds out that for anytranslation invariant probabilitymeasure ∈ Ms( ), the integral: T i i i d (x) P2 "t; {xt;yt ;zt }i∈< dt 0 is bounded from above by T ÿ O 2 O 5 d (x) − T U (xt ) +(K1|xt | + K2) dt 0 2 for some positive constants K1 and K2. We are now in a position to complete the proof of the proposition bytaking into account the term P1 arising in the expression for V given above. We have ÿ3 P = −ÿU(xO)+ÿU(xO) − " (xO)2 − 2" xOyO − xO 1 T 0 2 T T T T T 0 2 i O i O 2 i O O i + ÿ (xT xT − x0x0 ) − ÿ "T (zT xT + zT xT ); i∼O i∼O and we mayhere again use Jensen’s inequality,Young inequalities and the translation invariance of . For example, the term 2 i O d (x){ÿ xT xT } (coming from a nearest neighbour interaction with the origin) is bounded from above by 2 O 2 ÿ d (x)(xT ) and the term 2 i O d (x) −ÿ zT xT is not greater than 2 2 T ÿ O 2 ÿ T O 2 d (x)(xT ) + d (x) (xt ) dt : 2 2 0 G. Ben Arous, M. Sortais / Stochastic Processes andtheir Applications 105 (2003) 211–255 237 Handling similarlythe terms remaining in d (x)P1, one maythen assert that T a d (x)V (x)= d (x) P1 + P2 dt 0 T ÿ O 2 O 5 6 d (x) − T U (xt ) +(K3|xt | + K4) dt 0 2 O O 3 + d (x){−ÿU(xT )+K5|xT | + K6}: Hence, letting ÿ 2 5 3 M = T · sup − T U (x) +(K3|x| + K4) + sup{−ÿU(x)+K5|x| + K6}; x∈R 2 x∈R we ÿnallyhave V( )= V (x)d (x) 6 M; ∀ ∈ Ms;L( ): Let us ÿnallyestablish the upper semicontinuityof V : → V a(x)d (x)on Ms;L( ). According to the two preceding points, the continuous functional V a : → R may be decomposed as V a = V + − V −; + a − a + with V = V · I{V ¿0}, V = V · I{V¡0} and V is taking its values in [0; M] while V −( ) = [0; +∞[. Fixing ∈ Ms;L( ) and a sequence ( n)n¿1 on Ms;L( ) converging weaklyto , we certainlyhave + + V d n −→ V d n→∞ so that we onlyneed to check that − − lim inf V d n ¿ V d n→∞ in order to establish that V is upper semicontinuous. Consider ÿrst the case where V − ∈ L1( ), so that V − d =+∞: Using Fatou’s lemma and Portmanteau’s theorem we obtain +∞ − − lim inf V d n = lim inf n{V ¿y} dy n→∞ n→∞ 0 +∞ − ¿ lim inf n{V ¿y} dy 0 n→∞ +∞ = ( {V − ¿y})dy =+∞ 0 238 G. Ben Arous, M. Sortais / Stochastic Processes andtheir Applications 105 (2003) 211–255 so that − − lim inf V d n ¿ V d : n→∞ In the case where V − d =a¡+∞, one mayconsider for each A¿0 a bounded − − − continuous functional VA : → R+ such that VA takes its values in [0; A] and VA (x) coincides with V −(x) whenever V −(x) 6 A. One maythen ÿx F¿0 and choose A large enough so that − − (V − VA )d ¡F: − −1 Setting: A =(V ) ([0; A]), we then have − − − − V d n = VA d n + (V − VA )d n; c A the ÿrst term in the preceding sum converges to V − d , which is greater than (a−F), A while the second term is ¿ 0 for each n. Hence − ∀F¿0; lim inf V d n ¿a− F n→∞ so that − − lim inf V d n ¿ V d ; n→∞ and V is indeed upper semicontinuous. As a consequence of Proposition 3.1, we maynow state a large deviations upper bound for the family {! } ⊂⊂Zd . Corollary 3.1. {! } ⊂⊂Zd satisÿes a large deviation upper bound on Ms;L( ) according to the good rate function Ia = H − V, i.e.: 1 lim sup log! (C) 6 − inf {I( )} Zd | | ∈C for each closedsubset C ⊂ Ms;L( ). Proof. Lemma 4.3.6 in (Dembo and Zeitouni, 1998, Section 4.3) maybe applied here when considering the functional I deÿned on Ms;L( ) through the identities −∞ if ∈ C; I( )= V( )if ∈ C: G. Ben Arous, M. Sortais / Stochastic Processes andtheir Applications 105 (2003) 211–255 239 Before turning to the proof of the Gibbsian variational principle in the case of our nonsummable interaction a, we ÿnd it appropriate to make here the following (elementary) remarks (1) An easyconsequence of Proposition 3.1 is that M| | ⊗ Q (A) 6 e RT (A) for each ÿnite cubic box and each Borel set A ⊂ WT . (2) According to the remarks made at the end of Section 2.1, we mayalso state that Q∞ is such that T O 6 O 4 dQ∞(x) |xt | dt ¡ ∞ and dQ∞(x)|xT | ¡ ∞: 0 (3) Still using Proposition 3.1 and the estimations established during its proof, one mayalso check that any Á ∈ Ms;L( ) satisfying V(Á)= V a(x)dÁ(x) ¿ −∞ cannot have “thick tails” in the sense that the variables T O 6 O 4 |xt | dt and |xT | 0 are both integrable under dÁ(x). 3.2. Gibbsian variational principle for a Recall that the speciÿc entropy H(Á) deÿned with respect to the inÿnite dimensional ⊗Zd Wiener measure RT is such that for each translation invariant Á ∈ Ms( ): 1 ⊗ 1 ⊗ H(Á) = lim H(Á |RT ) = sup H Á |RT ; Zd | | ⊂⊂Zd | | ⊗ ⊗ H(:|RT ) denoting the usual relative entropywith respect to RT . We now prove the variational principle for a as a consequence of the following lemmata: Lemma 3.1. One has for each Á ∈ Ms;L( ): H(Á) − V(Á) ¿ 0: a 1 Proof. We mayassume that V ∈ LÁ, since there is nothing to prove otherwise. In this case, the L1 version of the multidimensional ergodic theorem (stated as Corol- lary14.A5 in Georgii (1988, p. 304)) enables us to view V(Á) as a limit in the following way: 1 V(Á)= V a dÁ = lim V a(x(i))dÁ(x): | | i∈ 240 G. Ben Arous, M. Sortais / Stochastic Processes andtheir Applications 105 (2003) 211–255 On the other hand, Á has “thin tails” in the sense of remark (3) above, and this enables one to see that the preceding limit coincides with 1 a ( );(i) lim V (x )dÁ (x ): | | i∈ Indeed, the di7erence between the two preceding limits maybe viewed as 1 ( ) −ÿ lim dÁ(x c ) dÁ(x )( (x ) − (x ∨ x c )) ; | | ∩ =∅; ∩ c =∅ the number n of nonzero terms in the preceding sum (over such that: ∩ = ∅; ∩ c = ∅) satisÿes n → 0; | | Zd since a has ÿnite range, and these nonzero terms have absolute values that are uniformlybounded from above bysome positive constant K (depending onlyon Á, cf. remark (3)). Hence 1 a ( );(i) dÁ V(Á) − H(Á) = lim dÁ (x ) V (x ) − ln (x ) | | dR⊗ i∈ T and Jensen’s inequalityapplied to ln yieldsfor each : a ( );(i) dÁ dÁ V (x ) − ln (x ) dR⊗ i∈ T exp( V (x( );(i))) 6 ln dÁ i∈ =0 ⊗ (dÁ =dRT (x )) (because the computation of the pressure corresponding to a becomes trivial when is equipped with its periodic boundaryconditions). Lemma 3.2. Q∞ is such that a H(Q∞)= V dQ∞: Proof. To prove this equalitywe shall merelyuse the fact that Q∞ is a Gibbs state corresponding to a, so that for each ⊂⊂ Zd: a exp −ÿ :∩ =∅ : (x ∨ x c ) dQ (x )= dQ (x c ) ; ∞ ∞ a Z (x c ) a a ⊗ −ÿ :∩ =∅ : (x ∨x c ) where: Z (x c )= dRT (x )e . G. Ben Arous, M. Sortais / Stochastic Processes andtheir Applications 105 (2003) 211–255 241 Using again the L1 multidimensional ergodic theorem and taking into account remark (2), we also have a 1 a (i) V dQ∞ = lim dQ∞(x)V (x ) | | i∈ a −ÿ :∩ =∅ : (x ∨x c ) 1 ⊗ e = lim dQ (x c ) dR (x ) ∞ T a | | Z (x c ) i∈ a (x ∨ x c ) × −ÿ : |:| : i a −ÿ :∩ =∅ : (x ∨x c ) 1 ⊗ e = lim dQ∞(x c ) dRT (x ) a | | Z (x c ) a × −ÿ (x ∨ x c ) : : :∩ =∅ On the other hand: a −ÿ :∩ =∅ : (x ∨x c ) 1 ⊗ e H(Q∞) = lim dQ∞(x c ) dRT (x ) a | | Z (x c ) a −ÿ (x ∨x c ) e :∩ =∅ : ×ln a Z (x c ) hence a 1 a H(Q∞) − V dQ∞ = lim dQ∞(x c ) − ln(Z (x c )): | | −ÿ a (x( )) Jensen’s inequalityapplied to −ln and to the probabilitymeasure e :∩ =∅ : ⊗ dRT (x ) now yields: 1 a dQ (x c ) − ln(Z (x c )) | | ∞ 1 a ⊗ −ÿ (x ∨x c ) = dQ (x c ) − ln dR (x )e :∩ =∅ : | | ∞ T 1 −ÿ a(x( )) ⊗ = dQ (x c ) − ln (e :∩ =∅ : dR (x )) | | ∞ T a a ( ) −ÿ( (x ∨x c )− (x )) ×e :∩ =∅ : :∩ =∅ : 242 G. Ben Arous, M. Sortais / Stochastic Processes andtheir Applications 105 (2003) 211–255 1 −ÿ a(x( )) ⊗ 6 dQ (x c ) (e :∩ =∅ : dR (x )) | | ∞ T a a ( ) × ÿ ( (x ∨ x c ) − (x )) ; : : :∩ =∅;:∩ c =∅ and the preceding term maybe easilyseen to converge to 0, using again the fact that the number n of nonzero terms in the sum ranging over : such that: :∩ =∅; :∩ c=∅ satisÿes n → 0 | | Zd and Proposition 3.1. ByLemma 3.1, we also know that a H(Q∞) ¿ V dQ∞; and the proof is now complete. Lemma 3.3. Any Á ∈ Ms;L( ) satisfying H(Á) − V(Á)=0 is also such that 1 dÁ lim ln (x ) dÁ (x )=0 | | dQ∞; (here andin the sequel , Q∞; simply denotes the -marginal of Q∞). Proof. Consider Á ∈ M ( ) satisfying: H(Á) ¡ ∞ and V a dÁ = H(Á). s;L One has for each ⊂⊂ Zd: 1 dÁ ln (x ) dÁ (x ) | | dQ∞; 1 dÁ 1 dQ = ln (x ) dÁ (x ) − ln ∞; (x ) dÁ (x ) | | ⊗ | | ⊗ dRT dRT 1 dÁ 1 = ln (x ) dÁ (x ) − dÁ (x ) | | ⊗ | | dRT a −ÿ (x ∨x c ) e :∩ =∅ : × ln dQ c (x c ) ∞; a Z (x c ) G. Ben Arous, M. Sortais / Stochastic Processes andtheir Applications 105 (2003) 211–255 243 1 dÁ 1 6 ln (x ) dÁ (x ) − dÁ (x ) dQ c (x c ) | | ⊗ | | ∞; dRT a a × −ÿ (x ∨ x c ) − ln(Z (x c )) ; : :∩ =∅ in the right hand side of the preceding inequality, the ÿrst term converges to H(Á) while the second term has a limit that mayeasilybe seen to coincide with V(Á), using once a 1 again the multidimensional ergodic theorem (since V ∈ LÁ) and then remarks (2), (3). Proposition 3.2 (Variational principle for ). For Á ∈ Ms;L( ), the following are equivalent: a (a) Á is a Gibbs measure corresponding to (i.e.: Á = Q∞) (b) H(Á) ¡ ∞ and V a dÁ = H(Á) (c) lim 1=| | ln(dÁ =dQ∞; (X )) dÁ (X )=0. Proof. (a) ⇒ (b) is the content of Lemma 2.4.2, (b) ⇒ (c) is the content of Lemma 2.4.3, and (c) ⇒ (a) is a fact holding in great generalityfor translation invariant Gibbs measures (see Georgii, 1988, Theorem 15.37, p. 323). 3.3. Large deviations lower bound Following FQollmer and Orey’s approach to level 3 large deviations in a Gibbsian setup (see FQollmer and Orey, 1988), we decompose the proof of the large deviations lower bound for {! } ⊂⊂Zd into two steps: a Step 1: we ÿrst prove that any ∈ Ms;L( ) such that: I ( ) ¡ + ∞ maybe ap- proximated bymeans of a sequence { n}n¿1 of ergodic probabilitymeasures on satisfying (a) n ∈ Ms;L( ); n =⇒ , n→∞ (b) H( n) −→ H( ), and n→∞ (c) V( n) −→ V( ). n→∞ a Step 2: ∈ Ms;L( ) being anyergodic probabilitymeasure such that I ( ) ¡+∞, the inequality 1 a lim inf log ! (G) ¿ − I ( ) Zd | | holds for all open neighbourhoods G of . Proof of Step 1. Just as in (FQollmer and Orey, 1988), we mayconsider the sequence d d of volumes n =]− n; n] ∩ Z , let ˜n be the probabilitymeasure coinciding with 244 G. Ben Arous, M. Sortais / Stochastic Processes andtheir Applications 105 (2003) 211–255 on each of the -ÿelds i d Jn;k = {xu;06 u 6 T; i ∈ ( n +2n:k)}; k ∈ Z and making these -ÿelds independent, and then set 1 −1 n = ˜noÂi ; | n| i∈ n Zd where Âi simplydenotes a shift on = WT . Each n ∈ Ms;L( ) deÿnes an ergodic probabilitymeasure on , and ( n)n¿1 converges weaklyto . Moreover lim inf H( n) ¿ H( ); n→∞ since H is lower semicontinuous. In order to prove the complementaryinequality H( ) ¿ lim sup H( n); n→∞ we maysimplyobserve that for each n ¿ 1: 1 H( )= H( | ; R⊗ n ) n n Jn; O T | n| since n deÿnes a n-periodical probabilityon . We then have 1 H( ) 6 H(( ˜ oÂ−1)| ; R⊗ n ) n 2 n i Jn; O T | n| i∈ n since H(·; R⊗ n ) is convex. Moreover, for each i ∈ , the measure ( ˜ oÂ−1)| may T n n i Jn; O be viewed as a tensor product based on (2d (see ) block marginals taken from |Jn; O Fig. 1); recalling that relative entropyis sub-additive in the sense that m ⊗Al ⊗ n H( |J ; R ) 6 H( |J ; R ); Al T n; O T l=1 ;A ;:::;A is a partition of (with corresponding sub--ÿelds J ;J ;:::; whenever A1 2 m n A1 A2 J ⊂ J ), one obtains Am n;O H(( ˜ oÂ−1)| ; R⊗ n ) 6 H( | ; R⊗ n ); ∀i ∈ : n i Jn; O T Jn; O T n All in all, one ÿnds out that 1 H( ) 6 H( | ; R⊗ n ) 6 H( ): n Jn; O T | n| A fortiori lim sup H( n) 6 H( ); n→∞ so that (H( n))n¿1 converges to H( )asn→∞. G. Ben Arous, M. Sortais / Stochastic Processes andtheir Applications 105 (2003) 211–255 245 Fig. 1. In d = 2, each measure ( ˜ oÂ−1)| maybe viewed as a tensor product where four distinct block n i Jn;O marginals of | are being used. Jn;O Finally, the convergence V( n) −→ V( ) n→∞ also takes place simplybecause V( ) is ÿnite, so that integrates the variables { T |xO|6 dt} and {|xO|4}, and because most of the terms (Vo )d˜ appearing in 0 t T i n the decomposition a 1 a V( n)= V d n = (V oÂi)d˜ n | n| i∈ n coincide with (V ao )d . To be more precise, there are at most 3 ×|@ | sites i n i ∈ n for which a a (V oÂi)d˜ n = (V oÂi)d ; these di7erences are averaged out in the large volume limit. This ÿnishes the proof of Step 1. Proof of Step 2. As a consequence of the L1 version of the multidimensional ergodic theorem (cf. Georgii, 1988) we know that the convergence ( ) lim {x ∈ | ˆx ∈ O} =1 Zd 246 G. Ben Arous, M. Sortais / Stochastic Processes andtheir Applications 105 (2003) 211–255 holds true for anyopen neighbourhood O of , and that the following L1 convergences 1 1 1 a ( );(i) L a 1 d L ⊗Zd V (x ) ←− V d ; log (x) ←− H( ; RT ) | | Zd | | dR⊗ Zd i∈ T also take place. Let us now consider an elementaryopen neighbourhood O ⊂ G of , of the form n O = Á ∈ Ms;L( ) fk dÁ − fk d ¡F k=1 for some F¿0 and some bounded continuous functions f1;f2;:::;fn : → R, each fk being measurable with respect to a ÿxed -algebra J = {xi ;06 u 6 T; i ∈ }; 0 u 0 d for some ÿnite 0 ⊂⊂ Z . Observe that for each ⊂⊂ Zd, the event ( ) E = {x ∈ | ˆx ∈ O} decomposes into n 1 E = x ∈ (f o ) − f d ¡F : | | k i k k=1 i∈ Setting o = {i ∈ | (i + 0) ⊂ }; we maynext observe that for suOcientlylarge the event n 1 F E = x ∈ (f o ) − f d ¡ o k i k 2 k=1 | | o i∈ lies in the -algebra J and satisÿes: E ⊂ E . Taking into account the fact that Q , with d d dR⊗ d -a:s:(x); (x)= (x) · T (x)= (x) exp V a(x( );(i)) ; ⊗ ⊗ dQ dR dQ dR T T i∈ we then have ( ) ! (G)=Q {x | ˆx ∈ G} ( ) ¿ Q {x| ˆx ∈ O} ¿ Q (E ) G. Ben Arous, M. Sortais / Stochastic Processes andtheir Applications 105 (2003) 211–255 247 d 1 d ¿ Q E ∩ (x) ¿ 0; log (x) ⊗ | | ⊗ dRT dRT 1 + V a(x( );(i)) 6 Ia( )+ ; | | i∈ being some positive number. Hence d 1 d ! (G) ¿ exp(−| |(Ia( )+)) E ∩ (x) ¿ 0; log (x) ⊗ | | ⊗ dRT dRT 1 + V a(x( );(i)) 6 Ia( )+ | | i∈ and taking into account the convergences obtained earlier via the L1 multidimensional ergodic theorem ÿnishes the proof of Step 2. We maynow state and prove a large deviations lower bound for the empirical ( ) process ˆx considered in the averaged regime. Proposition 3.3. The family (! ) ⊂⊂Zd of probability measures corresponding to the averagedregime of the empirical process also satisÿes a large deviationslower bound on Ms;L( ), in the sense that for any Borel set B ⊂ Ms;L( ): 1 a lim inf log ! (B) ¿ − inf I ( ); Zd | | o ∈B o B denoting the interior of B (with respect to the topology of weak convergence). Proof. The proof is a straightforward consequence of Steps 1 and 2: assuming that inf Ia( )=A¡+ ∞; o ∈B o one maychoose an ergodic ∈B satisfying: Ia( ) ¡A+ F, for some ÿxed positive o F, and then consider an open neighbourhood O of such that O ⊂ B. According to Step 2: 1 lim inf log ! (O) ¿ − (A + F); | | and letting F 0 ÿnishes the proof of the proposition. The large deviation principle we have just proved maynaturallybe viewed as an a LDP taking place in Ms( ), once we extend I bysetting Ia(Q)=+∞; 248 G. Ben Arous, M. Sortais / Stochastic Processes andtheir Applications 105 (2003) 211–255 whenever Q ∈ (Ms( )\Ms;L( )). Moreover, as maybe seen from the proofs of Propo- sition 3.1 and Lemmata 3.1–3.3, one mayrelax the compact support condition on 0 byrequiring simplythat 6 x d0(x) ¡ + ∞ R and still derive such LD bounds. One maygeneralise further the hypothesesmade on the initial conditions, e.g. byrequiring these initial conditions to be Gibbsian with respect to some (reasonable) interaction potential deÿned on R(Zd); such generalisations are carried out brieRyin Section 5. But let us ÿrst come to a studyof the quenched ( ) LD asymptotics of the empirical process ˆx . 4. Quenched large deviation estimates We now consider a ÿxed (typical) realisation k0 of the external ÿeld variables, and let k0 ⊗ R denote the joint law of x and k when x is distributed according to RT while k is ÿxed at the value k0 (or rather, at the projection of k0 onto R ). According to Theorem III.1 in Comets (1989), we know that: a.s. in the realisation of the disorder variables 2 ( ) k0 k0, the law of the “joint empirical process” ˆx;k considered under dR (x; k) obeys (Zd) a LDP on Ms((WT × R) ), on the scale | | and according to the deterministic rate functional Hq given by H d ( )if has a second marginal coinciding R⊗Z x k T q d H ()= with N(0; 2)⊗Z ; +∞ else: k0 k0 Denoting by Q the joint law of x and k when x is distributed according to dQ and k is ÿxed at the value k0, we also know that k0 T dQ i j i i = exp −U (xt)+ÿ xt + ÿk dxt dRk0 i∈ 0 j∼i 2 1 T − −U (xi)+ÿ x j + ÿki dt : 2 t t i∈ 0 j∼i Considering (x; k) as a spin conÿguration where each spin variable (xi; ki) lies in (WT × R), one maythen express the preceding Radon–Nykod ÃUm derivative as dQk0 (x; k) = exp Fi(x; k) ; dRk0 i∈ 2 ( ) The joint empirical process ˆx;k is deÿned as the empirical process corresponding to the conÿguration i i i =(N )i∈ =((x ; k ))i∈ . G. Ben Arous, M. Sortais / Stochastic Processes andtheir Applications 105 (2003) 211–255 249 i (Zd) the functional F :(WT × R) → R being of the form: q Fi = − : |:| : i for some ÿnite range, translation invariant Gibbsian interaction q q =( : ):⊂⊂Zd on (W × R)(Zd). The “quenched” interaction q is preciselygiven by T T q i i i T 1 i i i 2 2 i 2 {i}(x; k)=[U(xt)−ÿk xt]0 + U (xt)+(ÿk −U (xt)) +2ÿ d(xt) dt; 2 0 T q i j T i i j j j i {i;j}(x; k)=−ÿ[xtxt ]0 +ÿ (−U (xt)+ÿk )xt +(−U (xt )+ÿk )xt dt 0 if |j − i| =1; T √ q i j {i;j}(x; k)=2ÿ xtxt dt if |j − i| = 2; 0 T q i j {i;j}(x; k)=ÿ xtxt dt if |j − i| =2; 0 q : ≡ 0 for anyother :: Just as in Comets (1989), we maynow use the fact that dQk0 (x; k) = exp | | FO dˆ ( ) ; k0 x;k dR and applying the Laplace–Varadhan method in this context leads us to conjecture that ( ) i i the joint empirical process x;k should also satisfyan LDP when (( x ; k ))i∈ is dis- k0 tributed according to dQ (x; k). The proposition is more preciselythe following: Proposition 4.1. Almost surely in the realisation of k0, the law of the “joint empirical ( ) k0 process”ˆ x;k under dQ (x; k) satisÿes a LDP on the scale | | andaccordingto the goodrate function Jq : M ((W ×R)(Zd)) → [0; +∞] given by s T Hq() − FO d if Hq() ¡ + ∞; Jq()= +∞ else: Here again, some veriÿcations are needed since the functional FO lacks a property of uniform boundedness; one should preciselycheck that the quantity 1 ( ) ·| | FO;ˆ k0 A = lim sup ln e dR d | | Z (WT ×R) is ÿnite for some ¿1 in order to prove the LD upper bound. In the present case one mayactuallyprove the following stronger statement 250 G. Ben Arous, M. Sortais / Stochastic Processes andtheir Applications 105 (2003) 211–255 Lemma 4.1. a.s. (k0), the quantity A is ÿnite for all ¿1. ( ) Proof. Let us ÿx ¿1 and notice ÿrst that the exponential of ·| | FO;ˆ may be decomposed as T i j i i exp −U (xt)+ÿ xt + ÿk dxt i∈ 0 j∼i 2 T − 2 −U (xi)+ÿ x j + ÿki dt t t i∈ 0 j∼i 2 22 − T ×exp −U (xi)+ÿ x j + ÿki dt : 2 t t i∈ 0 j∼i Using the Cauchy–Swartzinequalitytogether with the martingale propertyof the square of the ÿrst term thus leads to the following inequality: 1 (22−) T (−U (xi)+ÿ x j +ÿki)2 dt k i∈ 0 t j∼i t 0 A 6 lim sup ln e dR : Zd 2| | k0 Taking into account the translation invariance of the x-marginal of R , we then have (42−2) T (2U (xi)2+4 dÿ2(xi)2)dt 1 0 t t i∈ k0 A 6 lim sup ln e dR Zd 2| | 1 2 2 i 2 + lim sup · ÿ (4 − 2) (k0) : d 2| | Z i∈ Now the ÿrst of these two terms is clearlyÿnite, while the second coincides a.s. with 2ÿ2(22 − ), due to the strong law of large numbers. The LD lower bound maythen be proved similarlyas in Section 3.3, i.e. by (Zd) considering an arbitraryopen neighbourhood N of ∈ Ms((WT ×R) ), where may be chosen as an ergodic p.m. such that Jq() ¡ + ∞, and establishing the inequality 1 k0 ( ) q lim inf ln Q { ˆ: ∈ N} ¿ − J () Zd | | via the L1 ergodic theorem. ( ) One thus obtains a quenched LDP for the joint empirical process ˆ(x;k) (Proposition 4.1), which maynaturallybe contracted to a quenched LDP for the empirical process ( ) ˆx , yielding part (ii) of Theorem 1.1. Proposition 4.2. Almost surely in the realisation of k0, the law of the empirical pro- ( ) k0 cess ˆx under dP obeys a large deviations principle on Ms( ), on the scale | | G. Ben Arous, M. Sortais / Stochastic Processes andtheir Applications 105 (2003) 211–255 251 q andaccordingto the (deterministic) goodrate functional I : Ms( ) → [0; +∞] given by: Iq(Á) = inf (Jq()): 1st marg:()=Á q The preceding expression for I does not enable one to see immediatelythat Q∞ is also the unique minimiser associated with Iq; this fact follows however from the general inequality Iq ¿ Ia ¿ 0; a proof of which maybe found in ( Zeitouni, 2003, Lemma 2.2.8). 5. General initial and boundary conditions Our aim in the present section is to show that one mayalso consider the quenched dynamics i i i j i dxt =dwt − U (xt)dt + ÿ xt dt + ÿk dt k j∼i (P ) law(x|t=0)= (i ∈ ; 0 6 t 6 T) obtained when using a (nonproduct) probabilitymeasure ∈ M(R ) as initial condi- j tion and some ÿxed boundarycondition ( 3t )06t6T;j∈ c , and still derive averaged and quenched LDPs for the empirical process. Our general strategyconsists in viewing the ÿnite dimensional, averaged dynamics (P )asthe -dimensional projection corresponding to an inÿnite volume Gibbs mea- sure Q∞; as we shall see this strategymaystill be carried out when the initial condition is simplythe -dimensional projection of a (reasonable) translation invariant Gibbs Zd measure corresponding to some deterministic interaction P =(pA)A⊂⊂Zd on R . As a ÿrst step towards this goal, let us remark that we mayreplace the compactly ⊗ supported probabilitymeasure u0 ∈ M(R ) (corresponding to the initial condition u0 used in the preceding section) bya probability m0 supported on the whole real line and having thin enough tails; one should preciselyrequire that: 6 x dm0(x) ¡ + ∞ R since, according to the remarks made after Corollary1.2.1 (cf. proof of Theorem 4.6 in FQollmer and Wakolbinger, 1986), this fact guarantees the ÿniteness of the integrals T i 6 i 6 I1 = dQ∞(x)|xT | and I2 = dQ∞(x) |xt| dt ; 0 Q being the x-marginal corresponding to the system ∞ i i i j i j dxt =dwt − U (xt)dt + ÿ xt dt + "t yt − ÿ zt dt j∼i j∼i 252 G. Ben Arous, M. Sortais / Stochastic Processes andtheir Applications 105 (2003) 211–255 i i j i j dyt =dwt + ÿ xt dt + "t yt − ÿ zt dt j∼i j∼i i i dzt = xtdt ⊗Zd d Law(x|t=0)=m0 ; y|t=0 = z|t=0 = 0 (i ∈ Z ; 0 6 t 6 T) ⊗Zd (such system certainly has a unique strong solution since m0 is supported by S(Zd)). In view of the computations carried out in the preceding section, the ÿniteness of I1 and I2 suOces to establish that Q∞ is the unique translation invariant Gibbs measure a ⊗Zd corresponding to the interaction (on inÿnite dimensional path space =WT ) and ⊗Zd to the reference measure R = RT , RT now denoting the law of a one dimensional Brownian motion having initial condition m0. We maythus choose e.g. e−2U(x) dm (x)= dx 0 e−2U(y) dy as the reference probabilitycorresponding to a “deep quench” initial condition. At this stage we should make an important remark stated as Proposition 2.5, (ii), in (Cattiaux et al., 1996). d y y Lemma 5.1. For each y ∈ S (Z ), denote by R (resp. Q∞) the probability R (resp. Q∞) conditioned to start at y: Ry(! ∈ A)=R(! ∈ A | !(0) = y): y For Q∞-almost all y, Q∞ deÿnes a Gibbs measure on with respect to the interaction a andto the reference measure Ry. Let be a (reasonable) Gibbs measure associated to an interaction P on RZd and to the reference measure R; according to the preceding lemma, the inÿnite volume dynamics i i i j i j dxt =dwt − U (xt)dt + ÿ xt dt + "t yt − ÿ zt dt j∼i j∼i (S ) dyi =dwi + ÿ x j dt + " yi − ÿ z j dt ∞ t t t t t t j∼i j∼i i i dz = x dt t t d Law(x|t=0)=; y|t=0 = z|t=0 = 0 (i ∈ Z ; 0 6 t 6 T) maynow be viewed as a Gibbsian average of Gibbs measures, and we next give a necessaryand suOcient condition devised byCattiaux, Roellyand Zessin for such an average to deÿne a Gibbs measure on path space : G. Ben Arous, M. Sortais / Stochastic Processes andtheir Applications 105 (2003) 211–255 253 Lemma 5.2. Assume that the Gibbs measure on RZd is supportedby the space S(Zd) of all temperedsequences on Zd; let (Qy; y ∈ S(Zd)) be a measurable family of (; Ry) Gibbs measures, and: Q = Qy d(y): H d Assume further that the expectation EQ[e ] is ÿnite for each ⊂⊂ Z , andset H H N (y) = log EQ[e | !(0) = y] = log EQy [e ]: In such conditions, Q is a Gibbs measure corresponding to the reference measure R andto the Boltzmann weights given by: −1 P ; =(Z (! c )) exp{−(H (!) − N (!(0)) + H (!(0)))} if andonly if the variables ! (0) and ! c are independent under the probability measure S given by: P −1 P dS (!)=(Z (! c (0))) exp −(H (!) − N (!(0)) + H (!(0))) dQ(!): Proof. See (Cattiaux et al., 1996, Proposition 2.6). In the case of interest to us, where Q is the x-marginal corresponding to the inÿnite volume dynamics (S∞), we have N (y) ≡ 0 and Q is a Gibbs measure corresponding to the interaction ( A + pA ◦ pr:|t=0)A⊂⊂Zd ⊗Zd and with the reference measure RT . Of course, one should also require that the Gibbs measures corresponding to the interaction P=(pA)A⊂⊂Zd enjoythe level 3 large deviation property;this will certainly Zd be the case if P deÿnes a translation invariant, summable interaction on R (so that = sup | (x)| is ÿnite), and it will also be the case in the natural O x∈RZd situation where P is the nearest neighbour interaction corresponding to a standard Ising model: i j p{i;j}(x)=ÿx x whenever i ∼ j; pA ≡ 0 else: The preceding observations maynow be gathered into Theorem 5.1. Let P be a translation invariant, summable interaction on RZd , and be a translation invariant Gibbs measure on RZd corresponding to P andto a ⊗Zd reference probability m0 . Assume that 6 O 6 x dm0(x) ¡ + ∞; (x ) d(x) ¡ + ∞; 254 G. Ben Arous, M. Sortais / Stochastic Processes andtheir Applications 105 (2003) 211–255 anddenoteby Q∞ the x-marginal corresponding to the inÿnite dimensional system (S ). For each ÿnite volume , consider the law Pk of the quencheddynamics ∞ i i i j i dxt =dwt − U (xt)dt + ÿ xt dt + ÿk dt j∼i law(x|t=0)= (:|3 c (0)) (i ∈ ; 0 6 t 6 T) j d 3 with boundary condition Q =(3t ; j ∈ Z ; 0 6 t 6 T), anddenoteby P the corre- sponding averaged dynamics: k P = d"(k)P : ( ) Then: (i) Q∞-a:s:(Q), the law of the empirical process ˆx under dP (x) obeys a large deviation principle on the scale | | andaccordingto the goodrate function Ia : M ( ) → [0; +∞] given by s a ⊗Zd ∀P ∈ Ms( ); I (P)=H(P; RT )+H(pr0(P); ) − V dP; where { + p ◦ pr:| }(x) V ()=− A A t=0 ; |A| A O andwhere pr0(P) denotes the projection of P at time t =0. ⊗Zd Moreover, if is the only Gibbs measure corresponding to P and m0 then Q∞ is the unique minimiser corresponding to the good rate function Ia. (ii) Furthermore, for a ÿxed, typical realisation k of the disordered external ÿeld, ( ) k the law of the empirical process ˆx under dP (x) also obeys a large deviation q principle on Ms( ), on the scale | | andaccordingto a goodrate function I satisfying: q a I (P) ¿ I (P); ∀P ∈ Ms( ): Acknowledgements The second author gratefullyacknowledges ÿnancial support from the Swiss National Science Foundation. References Ben Arous, G., Guionnet, A., 1997. Symmetric Langevin spin glass dynamics. Ann. Probab. 25, pp. 1367–1422. Ben Arous, G., Guionnet, A., 1998. Langevin dynamics for Sherrington–Kirkpatrick spin glasses. In: Bovier, A., Picco, P. (Eds.), Mathematical Aspects of Spin Glasses and Neural Networks. BirkhQauser, Basel, London, pp. 355–382. 3 i For i ∈ @ , Q appears in the drift term associated with dxt . G. Ben Arous, M. Sortais / Stochastic Processes andtheir Applications 105 (2003) 211–255 255 Ben Arous, G., Sortais, M., 2002. Large Deviations in the Langevin dynamics of a short range spin glass, Preprint. BrÃezin, E., De Dominicis, C., 1998a. Dynamics versus replicas in the Random Field Ising Model, preprint available on the xxx server under: cond-mat/9807113 8 July1998. BrÃezin, E., De Dominicis, C., 1998b. New phenomena in the Random Field Ising Model, preprint available on the xxx server under: cond-mat/9804266 v2 21 August 1998. Cattiaux, P., Roelly, S., Zessin, H., 1996. Une approche gibbsienne des di7usions browniennes inÿni-dimensionnelles. Probab. Theor. Relat. Fields 104, 147–179. Comets, F., 1989. Large deviation estimates for a conditional probabilitydistribution. Applications to random interacting Gibbs measures. Probab. Theor. Relat. Fields 80, 407–432. Dembo, A., Zeitouni, O., 1998. Large Deviations Techniques and Applications, 2nd Edition. Springer Verlag, Berlin. FQollmer, H., Orey, S., 1988. Large deviations for the empirical ÿeld of a Gibbs measure. Ann. Probab. 16, 961–977. FQollmer, H., Wakolbinger, A., 1986. Time reversal of inÿnite dimensional di7usion processes. Stochast. Proc. Appl. 22, 59–78. Georgii, H.-O., 1988. Gibbs Measures and Phase Transitions. de Gruyter, Berlin. Grunwald, M., 1996. Sanov results for Glauber spin glass dynamics. Probab. Theor. Relat. Fields 106, 187–232. Grunwald, M., 1998. Sherrington–Kirkpatrick spin glass dynamics: the discrete setting. In: Bovier, A., Picco, P. (Eds.), Mathematical Aspects of Spin Glasses and Neural Networks. BirkhQauser, Basel, pp. 355–382. Krylov, N.V., 1990. A simple proof of the existence of a solution of Itˆo’s equation with monotone coeOcients. Theor. Probab. Appl. 35 (3), 583–587. Nualart, D., 1998. Analysis on Wiener space and anticipating stochastic calculus. In: Bernard, P. (Ed.), Lectures on Probabilityand Statistics, St-Flour 1995. Springer Verlag, Berlin, pp. 123–227. Olla, S., 1988. Large deviations for Gibbs random ÿelds. Probab. Theor. Relat. Fields 77, 343–357. Shiga, T., Shimizu, A., 1980. Inÿnite dimensional stochastic di7erential equations and their applications. J. Math. Kyoto Univ. 20 (3), 395–416. Sompolinsky, H., Zippelius, A., 1983. Relaxational dynamics of the Edwards-Anderson model and the mean-ÿeld theoryof spin glasses. Phys.Rev. B 25 (11), 6860–6875. Sortais, M., 2001. Large deviations in the Langevin dynamics of lattice disordered systems, Ph.D. Thesis. Varadhan, S.R.S., 1966. Asymptotic probabilities and di7erential equations. Comm. Pure Appl. Math. 19, 261–286. Zeitouni, O., 2003. Lecture Notes on RWRE, Saint-Flour 2001 Lecture Notes, Springer Verlag, Berlin, to appear.