Convergence of path measures arising from a mean field or polaron type interaction

Convergence of path measures arising from a mean field or polaron type interaction Erwin Bolthausen 1, Jean-Dominique Deuschel 2, and Uwe Schmock 1 1 Institut für Angewandte Mathematik der Universität Zürich, Rämistr. 74, CH-81 Zürich, Switzerland 2 Mathematik, ETH Zentrum, CH-892 Zürich, Switzerland Received February 1, 1992; accepted June 24, 1992 Summary. We discuss the iting path measures of Markov processes with either a mean field or a polaron type interaction of the paths. In the polaron type situation the strength is decaying at large distances on the time axis, and so the interaction is of short range in time. In contrast, in the mean field model, the interaction is weak, but of long range in time. Donsker and Varadhan proved that for the partition functions, there is a transition from the polaron type to the mean field interaction when passing to a it by letting the strength tend to zero while increasing the range. The discussion of the path measures is more subtle. We treat the mean field case as an example of a differentiable interaction and discuss the transition from the polaron type to the mean field interaction for two instructive examples. Keywords. maximum entropy principle large deviations weak convergence polaron problem mean field interaction interacting Markov processes 1. Introduction Let {X t } t be a Markov process with Polish state space E. We assume that the sample paths are in (C([, ), E), F) or(d([, ), E), F), where F denotes the corresponding Borel σ-algebra. Let V : E 2 R be a suitable function. A mean field interaction between positions at different points of time, which has often been considered, is given by the Hamiltonian H T = 1 T T T V (X s,x t ) ds dt, T >, (1.1) Research supported by the Swiss National Foundation (21-29833.9)

2 E. Bolthausen, J.-D. Deuschel, and U. Schmock which introduces an interaction which is weak but of long range in time. With this Hamiltonian, we can transform the original path measure P of the process by defining P T (A) =E[1 A exp(h T )]/Z T, A F, (1.2) where Z T = E[exp(H T )]. In contrast to this long-range but weak interaction, there is a model with a strong but short-range (with respect to time) interaction, which we call polaron type interaction, given by the Hamiltonian H α,t = 1 2 T T and the corresponding transformed path measure αe α s t V (X s,x t ) ds dt, α, T >, (1.3) P α,t (A) =E[1 A exp(h α,t )]/Z α,t, A F, (1.4) where Z α,t = E[exp(H α,t )]. The aim of this paper is to describe and compare T P T and α T provided that these its exist. Under some technical conditions, one knows that α T 1 T log Z α,t = T 1 T log Z T = P α,t, (1.5) (Ṽ ) sup (µ, µ) J(µ), (1.6) µ M 1 (E) where M 1 (E) denotes the probability measures on E, the abbreviation Ṽ (µ, ν) stands for the integral of V with respect to µ ν for µ, ν M 1 (E), and J, defined in (2.3), is the Donsker-Varadhan rate function which governs the large deviations of the empirical process. Equation (1.6) has been proved by Donsker and Varadhan [6] for the Fröhlich polaron, where {X t } t is the Brownian motion in R 3 and V (x, y) = x y 1 for x, y R 3 with x y is the Coulomb potential. Recently, (1.6) has been generalized in [14] and [15]. An investigation of the iting mean field path measure with a Hamiltonian, which is more general than (1.1), has been done for discrete-time Markov processes in [2] and [2], and for (possibly non-symmetric) continuous-time Markov processes it is given in Section 2 below. The case of a symmetric continuous-time Markov process with a Hamiltonian given by (1.1) has already been treated in [12], and, with more general Hamiltonians having at least C 2 -regularity, corresponding results have recently been obtained in [13]. A discussion of the iting polaron path measure is given in [22] and [23], partially on a heuristic level. Let us summarize our results about the mean field model contained in Section 2. We consider a uniformly mixing Markov process on a compact state space E, a real-valued continuous function Ψ on M 1 (E), which is differentiable in a suitable sense (Condition 2.12), and define a Hamiltonian by HT Ψ = TΨ(L T ), where L T is the empirical distribution of the process {X t } t up to time T. The Hamiltonian in (1.1) corresponds to Ψ(µ) :=Ṽ (µ, µ) for µ M 1(E), see Example 2.15. In Theorem 2.2 we show that { P T } T> is relatively compact in the weak topology as T and that each accumulation point P is a mixture of homogeneous Markovian path measures {Q µ } µ KΨ, where K Ψ is the set of all µ M 1 (E) which maximize Ψ J. In particular, for the

Path measures arising from a mean field or polaron type interaction 3 Hamiltonian given by (1.1), the set K Ψ consists of all solutions of the variational problem in (1.6) and we obtain a characterization of the first it in (1.5). For µ K Ψ, the measure Q µ is given in terms of Ψ(µ) and the derivative of Ψ at µ. If additional symmetry assumptions are satisfied, then Theorem 2.32 shows that { P T } T> converges weakly to a specified mixture of {Q µ } µ KΨ as T. Given the first equality in (1.6), one could expect that for large T and small α the measure P α,t should be close to P T. There is, however, a somewhat subtle boundary effect for the measures P α,t at the starting point. This boundary effect shows up because the Hamiltonian in (1.3) is not defined in terms of translations of paths which are periodic continuations of the paths in [,T]. Such periodic paths are used in [5] to treat processlevel large deviations. We conjecture that, quite generally, the its in (1.5) are different and that α T P α,t = R α P α,r/α. (1.7) Let us give, on a heuristic level, a description of the right-hand side of (1.7) and explain, why we expect this equality to hold. We are not yet able to prove (1.7) in any generality, especially not for the Fröhlich polaron, but we discuss two instructive examples in Sections 3 and 4. The crucial point is that the right-hand side in (1.7) is a mean field type it which, however, is more delicate to handle than the it of { P T } T> as T. To show this, let us compare the it behaviour of the partition functions {Z T } T> and {Z α,r/α } α,r>. After the time transformations s s/α and t t/α, the latter ones are given by [ ( 1 R R )] Z α,r/α = E exp e s t V (X s/α,x t/α ) ds dt. (1.8) 2Rα One should remark that, for fixed R>, the it of {Z α,r/α } α> as α is a mean field type it. To see this, divide the time interval [,R] into small intervals such that e s t is approximately constant for (s, t) in rectangles formed by these intervals. Then the restriction of the integration in (1.8) to such rectangles is essentially just a mean field type expression. Patching things together, one gets, under appropriate conditions, α log Z α,r/α (1.9) α ( 1 R R = sup e s t Ṽ (µ R,s,µ R,t ) ds dt 1 R ) J(µ R,t ) dt, µ R 2R R where µ R runs over an appropriate space of functions which map [,R] into M 1 (E), compare [3, Exercise 4.2.7]. We expect that the it of { P α,r/α } α> for α is related to the solutions of the variational problem in (1.9) in a similar way as the it of { P T } T> for T is related to the solutions of the variational problem in (1.6). Although the solutions of the variational problem in (1.9) are in general not constant in time, the it of { P α,r/α } α> as α should be a mixture of homogeneous Markov processes, because, due to the transformations leading to (1.8), the inhomogeneity of these solutions is on the time scale 1/α. However, it turns out that, for the description of the Markov processes appearing in this mixture, the inhomogeneity of the solutions of the variational problem is important.

4 E. Bolthausen, J.-D. Deuschel, and U. Schmock It is reasonable to expect that, as R, the right-hand side of (1.9) approaches the right-hand side of (1.6) and the solutions (µ R,t ) t [,R] of the variational problem in (1.9) converge to the corresponding solutions of the mean field variational problem in (1.6) for t around the center of the interval [,R]. The above mentioned boundary effect shows up because this convergence does not take place for t near the boundary of the time interval [,R]asR. This behaviour originates from the factor (1/2)e s t in (1.9), which, for t {,R}, gives less than 1/2 when integrated over [,R] instead of converging to 1 as R. If (1.7) is true, then one has a characterization of the iterated it in (1.5). Equality in (1.7) would follow if one can prove that P α,t is close to P α,r/α for large R uniformly in T R/α. To achieve this, one actually needs quite precise information about the solutions of the variational problem in (1.9); to prove the uniformity in T R/α, one has to determine and control the iting behaviour of these solutions in terms of an added condition for µ R,R as R, see Section 3 for details. As already mentioned, we are far away from proving (1.7) and characterizing the iterated it in (1.5) in a general setting, but the two examples in Sections 3 and 4 fully confirm the picture presented above. We actually do not prove (1.7) for these examples, but (1.7) is our guide for identifying the iterated it in (1.5). The first model, treated in Section 3, is a symmetric Markovian jump process on E = {, 1} with exponential waiting times of expectation one and an interaction function V which is given by V (1, ) = V (, 1) = and V (, ) = V (1, 1) = τ, where τ R denotes a strength parameter. We determine the iting path measures in (1.5) explicitly. If τ 1, then they are equal to P. Ifτ > 1, then the iting measures in (1.5) are different but they are both mixtures of asymmetric Markovian jump processes. It turns out that the second iting measure in (1.5) corresponds to the first one with an adjusted strength parameter, namely τ =(τ +1/τ)/2. The second model, treated in Section 4, is the one-dimensional Brownian motion with V (x, y) =τ 2 (x y) 2 /4 for x, y R and a strength parameter τ >. Since Brownian motion is not sufficiently mixing, the results of Section 2 are not applicable. Because V is quadratic, everything is in the realm of Gaussian processes and we can determine the its in (1.5) explicitly by investigating the corresponding covariances. It turns out that the its in (1.5) exist, that they are different, and that they are both mixtures of Ornstein-Uhlenbeck processes with a normally distributed random center. Similar to the result in Section 3, the second iting measure in (1.5) corresponds to the first one with an adjusted strength parameter, namely τ = τ/ 2. 2. Convergence of Path Measures in a Mean Field Model Let E be a compact metric space with Borel σ-algebra E. IfI is a non-empty subset of R, then C(E,I) denotes the set of all I-valued continuous functions on E and C(E) is an abbreviation for C(E,R). We write for the supremum norm. Let M 1 (E) bethe set of all probability measures on (E,E). Note that M 1 (E) with the Prohorov metric, which induces the weak topology [1, Chap. 3, Theorem 3.1], is a compact metric space [1, Chap. 3, Theorem 1.7 and 2.2]. Let B(M 1 (E)) denote the corresponding Borel σ-algebra. Consider the path space Ω = D([, ),E) and, for each t [, ), define the evaluation map X t : Ω E by X t (ω) =ω(t). Let F be the σ-algebra on Ω generated by {X t } t. Note that Ω with the metric given by [1, Chap. 3, (5.2)] is a Polish space

Path measures arising from a mean field or polaron type interaction 5 [1, Chap. 3, Theorem 5.6] and that F coincides with the Borel σ-algebra generated by this metric [1, Chap. 3, Proposition 7.1]. For t let F t denote the sub-σ-algebra generated by {X s } s [,t]. We consider an E-measurable family {P x } x E of time-homogeneous Markovian probability measures on (Ω,F) with P x (X = x) = 1 for all x E. Let {P t } t denote the corresponding semigroup of stochastic transition kernels as well as the semigroup of bounded linear operators on C(E). We assume: Condition 2.1 There exists a {P t } t -invariant probability measure π M 1 (E) with supp(π) = E and, for each t >, there exists a jointly continuous transition density p t C(E 2, (, )) of P t with respect to π. Since supp(π) = E, the continuous transition densities are unique. As an abbreviation define c t = max{p t, 1/p t } = exp( log p t ) for all t>. Note that {P t } t is Feller continuous by Condition 2.1. For t the empirical distribution process of the position process after t is defined by 1 T Ω (t, ) (ω, T) L t,t (ω) = δ Xu (ω) du M 1 (E). (2.2) T t t If t =, then we write L T instead of L,T. The right-continuity of the paths in Ω implies that P t (x, ) converges weakly to δ x as t, for every x E. Therefore, the operator semigroup {P t } t is strongly continuous [25, p. 115]. Let L denote the corresponding (strong) generator with domain D(L) and define the position level rate function J : M 1 (E) [, ] by { Lφ } J(µ) = sup E φ dµ φ D(L) C(E,[1, )). (2.3) Examination of [3, pp. 123 126] shows that this definition coincides with J P in [3]. According to [3, Theorem 4.2.43], the measures { P x L 1 T T>, x E } satisfy a uniform full large deviation principle with the rate function J. For µ M 1 (E), let P µ M 1 (Ω) denote the path measure with starting distribution µ, hence P µ (A) = E P x(a) µ(dx) for all A F. In particular, P δx = P x. We denote the expectation with respect to P µ or P x by E µ and E x, respectively. Given ϕ C(E), define the semigroup of transition kernels {P ϕ t } t by ( t ) ] P ϕ t (x, A) =E x [exp ϕ(x u ) du 1 A (X t ), x E, A E, t. (2.4) The corresponding semigroup of bounded linear operators on C(E) is denoted by {P ϕ t } t, too. The logarithmic spectral radius of P ϕ 1, given by Λ ϕ 1 = t t log P ϕ t op, (2.5) satisfies Λ ϕ ϕ. Ift, then exp(λ ϕ t) is the spectral radius of P ϕ t. Using [3, Exercise 2.1.15, (4.2.21), and Corollary 4.2.27], it follows that Λ ϕ = sup{ ϕ, µ J(µ) µ M 1 (E) }, (2.6) where ϕ, µ denotes the integral of ϕ with respect to µ. Let p denote the usual norm on L p (π).

6 E. Bolthausen, J.-D. Deuschel, and U. Schmock Lemma 2.7 Let ϕ C(E) be given. (a) For each t>, there exists a transition density p ϕ t C(E 2, (, )) of P ϕ t with respect to π satisfying log p ϕ t ϕ t + log c t. (b) There exists a unique function h ϕ C(E,(, )) which satisfies h ϕ 2 =1and P ϕ t h ϕ =e Λϕt h ϕ for all t. (c) There exists an E-measurable set {Q ϕ x} x E of time-homogeneous Markovian probability measures on (Ω,F) such that, for all x E, t, and A F t, [ Q ϕ x(a) = e Λϕ t ( t ) ] h ϕ (x) E x 1 A exp ϕ(x u ) du h ϕ (X t ). (2.8) (d) Transition densities of {Q ϕ ( )Xt 1 } t> with respect to π are given by q ϕ t (x, y) = e Λϕ t h ϕ (x) pϕ t (x, y)h ϕ (y), x,y E, t>. (2.9) (e) There exists a unique {Q ϕ ( )Xt 1 } t -invariant distribution µ ϕ M 1 (E). (f) There exists a unique l ϕ C(E,(, )) such that dµ ϕ /dπ = h ϕ l ϕ. (g) log h ϕ 2 ϕ + log c 1 and log l ϕ 4 ϕ + log c 2 1. (h) For every C>there exist ε, C > such that, for all t 1, sup sup q ϕ t (x, ) l ϕ h ϕ C e εt. ϕ: ϕ C x E Remark 2.1 From Lemma 2.7(b) we see that h ϕ is the 2 -normalized, positive eigenfunction associated with the principle eigenvalue Λ ϕ of the (strong) generator L + ϕ of {P ϕ t } t.ifπ is {P t } t -reversing [3, p. 128], then l ϕ = h ϕ. Remark 2.11 The transition densities {q ϕ t } t> in part (d) of Lemma 2.7 are the Doob h ϕ -transforms of {e Λϕt p ϕ t } t>, and the corresponding measures {Q ϕ x} x E on (Ω,F) describe the h ϕ -path process [8, Section 2.VI.13]. The proof of Lemma 2.7 is given after Theorem 2.2. We consider a real-valued function Ψ on M 1 (E) which satisfies the following condition. Note that this condition determines DΨ only up to a constant but that this constant does not enter into ϕ µ given by (2.18). Condition 2.12 Let the function Ψ : M 1 (E) R be continuous and differentiable in the sense that there exists a continuous map DΨ : M 1 (E) C(E) such that the map R :(, 1] (M 1 (E)) 2 R, defined by R λ (µ, ν) = 1 ( ) Ψ((1 λ)µ + λν) Ψ(µ) λ DΨ(µ),ν µ (2.13) λ for all λ (, 1] and µ, ν M 1 (E), is bounded and satisfies for each ν M 1 (E). λ sup µ M 1 (E) R λ (µ, ν) = (2.14) Example 2.15 Given any V C(E 2 ), define the quadratic functional Ψ by Ψ(µ) = V,µ 2 for all µ M 1 (E). Without loss of generality we may assume that Ψ is

Path measures arising from a mean field or polaron type interaction 7 given by a symmetric V. To prove that Ψ is continuous, note that V is bounded and uniformly continuous on the compact set E 2. Hence, given any ε>, there exist n N and x 1,...,x n E such that the n balls in C(E) with centers V (x 1, ),...,V(x n, ) and radius ε cover { V (x, ) x E }. Hence, if V (x j, ),µ ν ε for all j {1,...,n}, then V (x, ),µ ν 3ε for all x E and Ψ(µ) Ψ(ν) 6ε. Using the same argument, the continuity of DΨ, given by DΨ(µ)(x) =2 V (x, ),µ, follows. Since R λ (µ, ν) = λ V,(µ ν) 2 for all λ (, 1] and µ, ν M 1 (E), Condition 2.12 holds for Ψ. We fix a starting distribution m M 1 (E). For every function ϕ C(E) define m ϕ M 1 (E) and Q ϕ M 1 (Ω), for all A Fand B E,by Let m ϕ (B) = 1 Bh ϕ,m h ϕ,m K Ψ = { µ M 1 (E) and Q ϕ (A) = Ψ(µ) J(µ) = For each µ M 1 (E) define ϕ µ C(E)by E Q ϕ x(a) m ϕ (dx). (2.16) } sup (Ψ(ν) J(ν)). (2.17) ν M 1 (E) ϕ µ = DΨ(µ)+Ψ(µ) DΨ(µ),µ. (2.18) To simplify the notation, we replace the superscript ϕ µ by µ. Let P M 1 (Ω) be the path measure with starting distribution m. Similar to (1.2) we define the transformed path measures { P T } T> M 1 (Ω)by P T (A) =E[1 A exp(tψ(l T ))]/Z T, A F, T>, (2.19) where Z T = E[exp(TΨ(L T ))]. The main result of this section is the following theorem; for further results about the mixture Σ see Theorem 2.32. Theorem 2.2 The set K Ψ is non-empty and compact, { P T } T> is relatively compact in the weak topology on M 1 (Ω) as T, and for each accumulation point P there exists Σ M 1 (K Ψ ) such that P(A) = Q µ (A) Σ(dµ), A F. K Ψ Proof of Lemma 2.7. (a) Since P ϕ t (x, ) π by Condition 2.1, we can define p ϕ t (x, ) = dp ϕ t (x, )/dπ for each x E and t>. For s, t > with 2s <tdefine p ϕ s,t(x, y) = p s (x, x) p ϕ t 2s ( x, ỹ)p s(ỹ, y) π(dỹ) π(d x), x,y E. E Then p ϕ s,t C(E 2, (, )) by Condition 2.1. Furthermore, E P ϕ t (x, A) (P s P ϕ t 2s P s)(x, A) c t e (t 2s) ϕ (e 2s ϕ 1)π(A), for each A Eand x E, hence sup p ϕ t (x, ) p ϕ s,t(x, ) c t e t ϕ (e 2s ϕ 1) x E

8 E. Bolthausen, J.-D. Deuschel, and U. Schmock whenever < 2s <t. Since supp(π) =E by Condition 2.1, we may and will assume that p ϕ t C(E 2, (, )) for all t>. The estimate for p ϕ t follows from Condition 2.1 and the definition of P ϕ t. (b) Since p ϕ t C(E 2, (, )), it follows that P ϕ t is a compact (compare [17, VI.5]), strictly positive [19, Def. II.2.4], and irreducible [19, p. 186] operator on C(E). The logarithmic spectral radius of e Λϕt P ϕ t is, hence 1 is in the spectrum of e Λϕt P ϕ t [19, Proposition V.4.1] and therefore a pole of the resolvent [9, Theorem V.5.2]. By [19, Theorem V.5.2 and its corollary], there exists a unique h ϕ t C(E,(, )) which satisfies h ϕ t 2 = 1 and h ϕ t = e Λϕt P ϕ t h ϕ t. Define h ϕ = h ϕ 1. Since {P ϕ t } t is a semigroup, P ϕ t h ϕ =e Λϕt h ϕ for all rational t in [, ). Using the right-continuity of the paths in Ω and the dominated convergence theorem, it follows that (P ϕ t f)(x) = E x [exp(t ϕ, L t )f(x t )] and converges to f(x)ast for every x E and f C(E). Applying [25, proof on p. 115] to {e ϕ t P ϕ t } t, it follows that {P ϕ t } t is strongly continuous. Hence, P ϕ t h ϕ =e Λϕt h ϕ for all t. (c) Given x E, it follows from (b) that (2.8) defines a probability measure on (Ω,F t ) for each t and that these measures are consistent. Hence, they can be uniquely extended to a measure Q ϕ x on (Ω,F) [16, Chap. V, Theorem 4.2]. The other properties of {Q ϕ x} x E follow from those of {P x } x E. (d) follows from (a), (c), and the definition of P ϕ t. (e) Let {Q ϕ t } t denote the semigroup of stochastic transition kernels corresponding to {Q ϕ x} x E in (c). According to [3, Exercise 4.1.48], there exists a unique Q ϕ t -invariant µ ϕ t M 1 (E) for each t>. Define µ ϕ = µ ϕ 1. The semigroup property of {Qϕ t } t implies that µ ϕ Q ϕ t = µ ϕ for all rational t in [, ). Since {P ϕ t } t is a strongly continuous semigroup by the proof of (b), the semigroup {Q ϕ t } t of linear operators on C(E) is strongly continuous, too. Choose ε, t > and f C(E). Using [3, (4.1.5)], there exist n N and a rational s in (,nt) such that µ ϕ Q ϕ nt µ ϕ t var ε and Q ϕ nt sf f ε. Since µ ϕ Q ϕ nt = µ ϕ Q ϕ nt s, it follows that µ ϕ t,f µ ϕ,f 2ε. Using [1, Theorem 1.3], it follows that µ ϕ t = µ ϕ for all t>, hence µ ϕ Q ϕ t = µ ϕ for all t. (f) Using µ ϕ Q ϕ 1 = µϕ, (2.9), the continuity of p ϕ 1 and hϕ, and supp(π) =E, it follows that there exists a unique f ϕ C(E,(, )) with f ϕ = dµ ϕ /dπ. Define l ϕ = f ϕ /h ϕ. (g) Note that h ϕ =e Λϕ P ϕ 1 hϕ by (b). Using the estimate for p ϕ 1 in (a) as well as Λ ϕ ϕ and h ϕ 1 h ϕ 2 = 1, the estimate for h ϕ follows. Note that l ϕ h ϕ = d(µ ϕ Q ϕ 1 )/dπ by (e) and (f). Rewriting with (2.9), dividing by hϕ, using Λ ϕ ϕ and the estimates for p ϕ 1 and hϕ, the estimate for l ϕ follows. (h) Choose ϕ C(E) with ϕ C.Fors, t > define f(t) = sup q ϕ t (x, ) µ(dx) l ϕ h ϕ qs ϕ (x, y) and α s = inf x,y E l ϕ (y)h ϕ (y). µ M 1 (E) E Note that α s (, 1]. If α s < 1, then, for each µ M 1 (E), qs+t(x, ϕ ) µ(dx) l ϕ h ϕ =(1 α s ) q ϕ t (y, ) µ s (dy) l ϕ h ϕ, E E where µ s M 1 (E) isgivenby ( ) 1 µ s (A) = qs ϕ (x, y) µ(dx) α s l ϕ (y)h ϕ (y) π(dy), 1 α s A E A E.

Path measures arising from a mean field or polaron type interaction 9 Therefore, f(s + t) (1 α s )f(t) in both cases. Using (2.9), (a), (g), and the estimate Λ ϕ ϕ, it follows that α 1 exp( 8C)/c 4 1 and f(1) 2c3 1 exp(6c). Hence, part (h) follows. Lemma 2.21 Let M be the set of all functions f : Ω R, which are bounded, continuous, and F s -measurable for some s. Then M is convergence determining [1, p. 112] for the weak topology on M 1 (Ω). Proof. By [1, Chap. 3, Theorem 3.1], the set of all bounded f : Ω R, which are uniformly continuous with respect to the metric d given by [1, Chap. 3, (5.2)], is convergence determining. We show that M is dense in this set. Given f as above, choose ε>. There exists s 1 such that f(ω) f( ω) ε for all ω, ω Ω with d(ω, ω) e 1 s. Define π t : Ω Ω by π t (ω)(u) =ω(s u) for all t, u and ω Ω. Note that d(ω, π t (ω)) e t by the definition of d.ifg : Ω R is defined by g(ω) = [s 1,s] f(π t(ω)) dt, then g is bounded and measurable with respect to F s. Furthermore, g f ε.givent and ω Ω, it follows from [1, Chap. 3, (5.2), (5.3), and Proposition 5.2] that π t is continuous at ω if t is a continuity point of ω. Since ω has at most countably many points of discontinuity [1, Chap. 3, Lemma 5.1], it follows with the dominated convergence theorem that g is continuous at ω. Lemma 2.22 (a) The map C(E) ϕ Λ ϕ is continuous. (b) The map C(E) ϕ h ϕ C(E,(, )) is continuous. (c) The map C(E) ϕ Q ϕ M 1 (Ω) is continuous. (d) The map C(E) ϕ l ϕ C(E,(, )) is continuous. Proof. (a) If ϕ, ψ C(E), then Λ ϕ Λ ψ ϕ ψ by (2.6). (b) If ϕ, ψ C(E), then P ϕ t (x, A) P ψ t (x, A) c t e t ϕ (e t ϕ ψ 1)π(A) for all x E and A E, hence p ϕ t p ψ t c t e t ϕ (e t ϕ ψ 1) (2.23) for all t>. If t> and ϕ C(E), then (2.9) shows that g ϕ t (x, y, z) := qϕ 1 (x, y)qϕ t (y, z) q ϕ 1+t (x, z) = pϕ 1 (x, y)pϕ t (y, z) p ϕ 1+t (x, z), x,y,z E. (2.24) Lemma 2.7(a) and (2.23) imply the continuity of ϕ g ϕ t. Define g ϕ in C(E 3 )by g ϕ (x, y, z) =q ϕ 1 (x, y). If C>, then Lemma 2.7(h) implies that sup g ϕ t g ϕ sup q ϕ 1 C e εt + C e ε(1+t) ϕ : ϕ C ϕ : ϕ C l ϕ h ϕ C e ε(1+t) (2.25) for all sufficiently large t 1. Since this estimate is uniform on bounded subsets of C(E), it follows by using Lemma 2.7(a), (g), (2.9), and Λ ϕ ϕ and letting t, that the map ϕ q ϕ 1 is continuous. Since 1 h ϕ (x) = hϕ 2 h ϕ (x) = q ϕ 1 (x, ) ϕ (x, )eλ, x E, (2.26) 2 p ϕ 1 by (2.9), it follows with (a) that ϕ 1/h ϕ and ϕ h ϕ are continuous. (c) By part (b), the map ϕ h ϕ,m is continuous. Using (a) and (b), it follows that the map ϕ exp( Λ ϕ t)e ( ) [f exp(t ϕ, L t )h ϕ (X t )],m is continuous for each t>

1 E. Bolthausen, J.-D. Deuschel, and U. Schmock and each bounded, continuous, and F t -measurable f : Ω R. Using (2.8), (2.16), and Lemma 2.21, part (c) follows. (d) If C>, then (2.9) and Lemma 2.7(g), (h) show that sup sup e Λϕ t t h ϕ (x) pϕ t (x, ) l ϕ =. ϕ: ϕ C x E Since this it is uniform on bounded subsets of C(E), part (d) follows from (a), (b), and (2.23). Lemma 2.27 There exists a constant M 1 such that E x [exp(tψ(l T ))] ME π [exp(tψ(l T ))] M 2 E y [exp(tψ(l T ))] for all x, y E and T>1. Proof. Let C denote the common bound of the maps DΨ and R given by Condition 2.12. Define M = c 1 exp(4c). Note that T Ψ(L T ) Ψ(L 1,T +1 ) = DΨ(L 1,T ),L 1 L T,T+1 + R 1/T (L 1,T,L 1 ) R 1/T (L 1,T,L T,T+1 ) 4C. Using the Markov property and Condition 2.1, it follows that E x [exp(tψ(l T ))] e 4C E x [exp(tψ(l 1,T +1 ))] =e 4C E x [E X1 [exp(tψ(l T ))]] ME π [exp(tψ(l T ))] and E y [exp(tψ(l T ))] M 1 E π [exp(tψ(l T ))] for all x, y E. By Condition 2.12 and Lemma 2.22(d), the map M 1 (E) E (µ, y) l µ (y) is continuous in each argument and therefore jointly measurable. For t, T > define Γ t,t M 1 (M 1 (E)) by Γ t,t (A) = 1 l µ (y) exp(tλ µ + TΨ(µ)) (P y L 1 T )(dµ) π(dy) (2.28) c t,t E A for all A B(M 1 (E)), where c t,t denotes the appropriate normalizing constant. Proof of Theorem 2.2. Due to Condition 2.12 and [3, Theorem 4.2.43], the map Ψ J is upper semi-continuous. Therefore, Ψ J attains its supremum on a closed subset of the compact set M 1 (E). Hence, K Ψ is non-empty and compact. Condition 2.12 and (2.18) imply that the map µ ϕ µ is bounded. Take any t>. Using Lemma 2.7(g) and Λ ϕ ϕ, it follows that there exists a constant C> such that tλ µ + log l µ C for all µ M 1 (E). Since Ψ is bounded and continuous by Condition 2.12, it follows from the full large deviation principle for { PL 1 T T>, x E } and Varadhan s theorem (compare [3, Exercise 2.1.24] and [2, Lemma 4.4]) that Γ t,t (U c ) ast for every t>and every neighbourhood U M 1 (E) of K Ψ. Since M 1 (E) is compact, the set {Γ t,t } T 1 is relatively compact for each t>. By the above paragraph, each accumulation point of {Γ t,t } T 1 as T is concentrated on K Ψ. Let {T n } n N be a strictly increasing sequence tending to infinity. By the preceeding paragraph and a diagonal argument we may assume that, for each k N, the sequence {Γ Tk,T n T k } n>k converges weakly to a measure Γ Tk in M 1 (M 1 (E)) as n. Note that Γ Tk (K Ψ ) = 1. By choosing a further subsequence if necessary, we may assume

Path measures arising from a mean field or polaron type interaction 11 that {Γ Tk } k N converges weakly to a measure Γ with Γ (K Ψ )=1.Letf: Ω [1, 2] be a continuous function, which is F s -measurable for some s. By Lemma 2.21, it suffices to show that Ê Tn [f] = n h µ,m K Ψ Ω / fdq µ Γ (dµ) h µ,m Γ (dµ). K Ψ (2.29) If s<t<t, then L T =(1 t/t )L t,t +(t/t )L t and, by (2.13) and (2.18), TΨ(L T )=t ϕ L t,t,l t +(T t)ψ(l t,t )+tr t/t (L t,t,l t ). (2.3) If A F t and B B(M 1 (E)), then the Markov property for P implies that E[1 A B (,L t,t ) X t = y] =P(A X t = y)p y (L T t B) = E[1 A B (,µ) X t = y](p y L 1 T t )(dµ) M 1 (E) for PXt 1 -almost all y E; therefore, by (2.3), E[f exp(tψ(l T ))] = E[f exp(t ϕ µ,l t + tr t/t (µ, L t )) X t = y] p t (y) E M 1 (E) exp((t t)ψ(µ)) P y L 1 T t (dµ) π(dy), where p t = d(pxt 1 )/dπ. Using (2.14) and the dominated convergence theorem, it follows that [ E sup exp(tr t/t (µ, L t )) 1 ] = T µ M 1 (E) for every t>. Since µ ϕ µ is bounded, it follows with Lemma 2.27 that, for each t>s, there exist {ε t,t } T>t (, ) with ε t,t 1asT such that E[f exp(tψ(l T ))] = ε t,t E[f exp(t ϕ µ,l t ) X t = y] p t (y) E M 1 (E) By Lemma 2.7(c), (d), and (2.16), E[f exp(t ϕ µ,l t ) X t = y] p t (y) =e Λµt h µ,m l µ (y) exp((t t)ψ(µ)) P y L 1 T t (dµ) π(dy). Ω f qµ t s(x s,y) h µ (y)l µ (y) dqµ for π-almost all y E. Since µ ϕ µ is bounded, it follows with Lemma 2.7(g) and (h) that there exists { ε t } t>s (, ) with ε t 1ast such that ( E[f exp(tψ(l T ))] = ε t,t ε t fdq ) h µ µ,m l µ (y) E M 1 (E) exp(tλ µ +(T t)ψ(µ)) P y L 1 T t (dµ) π(dy). Using (2.19) and (2.28), it follows that, for s<t<t, / Ê T [f] =ε t,t ε t h µ,m fdq µ Γ t,t t (dµ) h µ,m Γ t,t t (dµ), M 1 (E) Ω M 1 (E) Ω

12 E. Bolthausen, J.-D. Deuschel, and U. Schmock where { ε t} t>s and {ε t,t } T>t have the same properties as { ε t } t>s and {ε t,t } T>t. Condition 2.12 and Lemma 2.22(c) imply that the map µ Q µ is continuous. Using the assumptions about {T n } n N, equation (2.29) follows. Usually, it is difficult to determine whether { P T } T> converges as T and, if it converges, to which it law. There is, however, a special case, where this it can be determined. Let G be a compact and metrizable topological group. Then there exists a unique normalized left-invariant Haar measure σ [4, 14.1.5, 14.2.3], which is also right-invariant [4, 14.3.3]. Let {T a } a G be a collection of bijective transformations of E onto itself such that the map G E (a, x) T a x is continuous and T a T b = T ab for all a, b G. If e denotes the neutral element of G, then T e =id E, hence Ta 1 = T a 1. For each a G define T a : M 1 (E) M 1 (E) by(t a µ)(a) =µ(ta 1 (A)) for all A Eand µ M 1 (E). Note that T a is continuous, T a T b = T ab, and Ta 1 = T a 1 for all a, b G. We assume: Condition 2.31 (a) The function Ψ and the transition kernels {P t } t are G-invariant, which means that Ψ = Ψ T a and P t (T a x, A) =P t (x, Ta 1 (A)) for all t, x E, A E, and a G. (b) There exists ν K Ψ such that the map G a Φ(a) :=T a ν is injective and K Ψ = {T a ν a G }. Typical examples satisfying (a) are Brownian motion and suitable Markovian jump processes on groups. Condition 2.31(b) is more restrictive and usually delicate to investigate. Of course, if K Ψ contains just one measure, then Condition 2.31 is satisfied with G = {e} and T e =id E. The next section contains an example with G = Z 2. Theorem 2.32 If the Conditions 2.1, 2.12, and 2.31 hold, then / P T = Q Taν h Taν,m σ(da) h Taν,m σ(da). (2.33) T G G Proof. Choose a G arbitrarily. Define T a : Ω Ω by T a (ω)(t) =T a X t (ω) for all t and ω Ω. Then X t T a = T a X t and, since {P t } t is T a -invariant, P Ta x = P x T 1 a for each x E. By Condition 2.1, the measure T a π is {P t } t -invariant and the invariant measure is unique [3, Exercise 4.1.48], hence T a π = π. Since T a is continuous and bijective, J Ta 1 is the rate function for { P x L 1 T T a 1 x E, T > } [3, Lemma 2.1.4]. Since L T T a = T a L T and P Ta xl 1 T = P x L 1 T T a 1 for each T>, the uniqueness of the rate function [3, Lemma 2.1.1] implies that J = J Ta 1, hence J = J T a. Choose ϕ C(E). Then Λ ϕ = Λ ϕ T a by (2.6). Since h ϕ T a = exp( Λ ϕ T a t)p ϕ T a t (h ϕ T a ) for all t and h ϕ T a 2 = 1, Lemma 2.7(b) shows that h ϕ T a = h ϕ T a. Since Q ϕ T a x = Qϕ T a x T 1 a for all x E, the measure T a µ ϕ T a is {Q ϕ ( )Xt 1 } t> -invariant and Lemma 2.7(e) implies that T a µ ϕ T a = µ ϕ. Since Ta 1 = T a 1 and dµ ϕ T a /dπ =(h ϕ l ϕ ) T a, Lemma 2.7(f) implies that l ϕ T a = l ϕ T a. Since Ψ = Ψ T a by assumption, T a µ K Ψ for each µ K Ψ. Since DΨ is (up to a constant) uniquely determined by (2.13) and (2.14), it follows that DΨ(T a µ)(t a x) DΨ(µ)(x) is constant and, by (2.18), ϕ Taµ T a = ϕ µ for all µ M 1 (E). Therefore, Λ Taµ = Λ µ and l Taµ T a = l µ. Finally, Γ t,t Ta 1 = Γ t,t for all T >t>. Since T a is continuous, Γ Ta 1 = Γ for the it Γ in the proof of Theorem 2.2.

Path measures arising from a mean field or polaron type interaction 13 Since G E (a, x) T a x is continuous, Φ is continuous, too. By Condition 2.31(b), the map Φ is injective. Therefore, if G denotes the Borel σ-algebra of G, then Φ(A) B(M 1 (E)) for every A Gby Kuratowski s theorem [16, Chap. I, Corollary 3.3]. Since Γ (K Ψ ) = 1 by the proof of Theorem 2.2 and K Ψ = Φ(G) by Condition 2.31(b), the measure σ M 1 (G) with σ(a) =Γ (Φ(A)) for all A Gis well-defined. Since Φ(a 1 b)=t a 1Φ(b), T a 1 = T 1 a, and Γ T 1 a = Γ for all a, b G, it follows that σ(a 1 A)=ΓΦ(a 1 A)=Γ (T a 1Φ(A)) = Γ Ta 1 (Φ(A)) = Γ (Φ(A)) = σ(a) for all A G. Therefore, σ = σ and the theorem follows from (2.29). 3. Convergence of Path Measures Arising from a Jump Process Let {X t } t denote a symmetric jump process on E = {, 1} with exponential holding times of expectation one. We denote by P x the law of the process on the path space Ω = D([, ), {, 1}) which starts in x {, 1}. Fort>define l t =(1/t) t X s ds. Since the measures in M 1 ({, 1}) can be parametrized by µ p = pδ 1 +(1 p)δ for p [, 1], the empirical distribution L t, defined after (2.2), is given by µ lt for t>. The random variables {l t } t> satisfy a uniform large deviation principle [3, Theorem 4.2.43]. The corresponding rate function J : R [, ] can either be calculated via (2.3) or, since the generator of the jump process is symmetric, via [3, Theorem 4.2.58 and (4.2.49)]. Both ways show that { 1 2 p(1 p) ifp [, 1], J(p) = if p R \ [, 1]. Fix a constant τ R and define an interaction function V : {, 1} 2 R by V (x, y) = τ(xy +(1 x)(1 y)) for x, y {, 1}. As in Example 2.15 define Ψ(µ) = V,µ 2. Then Ψ(µ p )=τ(p 2 +(1 p) 2 ) for p [, 1] and H T = TΨ(µ lt ). We choose m = δ as starting distribution, hence P = P. The transformed probability measures P T and P α,t corresponding to P are defined by (1.2) and (1.4), respectively. Then the right-hand side of (1.6) is given by sup{ Ψ(µ p ) J(p) p [, 1] }. Ifτ 1, then the supremum is attained only for p = 1/2. If τ > 1, then there are exactly two maxima at p ± =(1± 1 1/τ 2 )/2. Compared to the situation considered in [6], it would be easy to prove (1.6) in the present setting, but we do not need (1.6) explicitly. For γ>let Q γ be the path measure of a jump process on {, 1} starting in with generator ( L γ L γ, L γ ) ( ),1 γ γ = L γ 1, L γ =, 1/γ 1/γ 1,1 and define the mixture Theorem 3.1 Define P γ = 1 1+γ Qγ + γ 1+γ Q1/γ. { 1 if τ 1, γ(τ) = τ + τ 2 1 if τ>1, and τ = { τ if τ 1, (τ +1/τ)/2 if τ > 1.

14 E. Bolthausen, J.-D. Deuschel, and U. Schmock Then and α T P T = P γ(τ) (3.2) T P α,t = P γ( τ). (3.3) Remark 3.4 Given α>, we do not prove explicitly that { P α,t } T> converges to a it as T. Instead of (3.3) we prove that R sup α sup T R/α r Ω ( P α,t, P γ( τ) )=, (3.5) where r Ω denotes the Prohorov metric on M 1 (Ω). Proof of (3.2). Ifτ 1, then K Ψ = {µ 1/2 } by (2.17). Example 2.15 and (2.18) give DΨ(µ 1/2 )(x) =τ and ϕ µ 1/2 (x) =τ/2 for x {, 1}, hence ϕ µ 1/2 Λ µ 1/2 = by (2.6). Therefore, h µ 1/2 from Lemma 2.7(b) is given by h µ 1/2 (x) = 1, hence m µ 1/2 = δ and Q µ 1/2 = P by (2.16), and (3.2) follows from Theorem 2.2. If τ > 1, then we apply Theorem 2.32 as follows: Abbreviating µ p± by µ ±,we obtain K Ψ = {µ +,µ }, and DΨ(µ ± )(x) =2τ(p ± x +(1 p ± )(1 x)) according to Example 2.15. Using (2.18) and (2.6) and writing ϕ ± for ϕ µ ± and Λ ± for Λ µ ±,it follows that ϕ ± (x) =τp ± (2x p ± )+τ(1 p ± )(2(1 x) (1 p ± )) and Λ ± = τp 2 ± + τ(1 p ± ) 2 1+1/τ. By the Feynman-Kac formula [18, Example IV.22.11], the semigroups {exp( Λ ± t)p ϕ ± t } t with P ϕ ± t given by (2.4) have the generators ( ) ( ) ϕ± () Λ ± 1 1 2τp± 1 = 1 ϕ ± (1) Λ ± 1 1 2τp with eigenvalues and 2τ. The non-negative 2 -normalized eigenfunction corresponding to is given by h ϕ ±() = 2(1 p ± ) and h ϕ ±(1) = 2p ±. Since h ϕ ±(1)/h ϕ ±()=2τp ± = γ(τ) ±1, it follows from (2.8) that L γ(τ) and L 1/γ(τ) are the generators of {Q ϕ ± x } x {,1}. We prove (3.5) only in the case τ>1, the other one is simpler. Notice that γ( τ) =τ for all τ 1. We prove (3.5) essentially by the technique used in Section 2. There are, however, additional difficulties. We proceed as outlined in the introduction starting with (1.9), see Lemma 3.6 below. The crucial estimate is given in Lemma 3.11. It would not be difficult to prove the existence of the it of P α,r/α as α, which appears in (1.7). We do not need the it explicitely, therefore we only study the behaviour for small α and large R. The uniformity stated in Lemma 3.11 is crucial for the interchange of the its in (1.7). Lemma 3.11 depends on analytic considerations in Lemma 3.13. Once we have Lemma 3.11, the rest of the proof follows along the lines of Section 2. For R>define C R = { f C([,R], R) f()=}, which is a Banach space with respect to the supremum norm. Let B R denote the set of all Borel measurable functions from [,R]to[, 1]. For α, R > define the map L α,r : Ω C R by

Path measures arising from a mean field or polaron type interaction 15 L α,r (t) =tl t/α for all t [,R]. Note that the random variable L α,r takes values in the set G R = { f C R f(t) f(s) t s for all s, t [,R] with s t }, which is compact by Ascoli s theorem. If f G R, then we denote by f B R the density with respect to Lebesgue measure. For α, R > and x, y {, 1} define the probability measure Q x,y α,r on G R by Q x,y α,r = P x( L 1 α,r ( ) X R/α = y ). Lemma 3.6 If x, y {, 1} and R>, then the measures {Q x,y α,r } α> satisfy a large deviation principle on G R with the good rate function J R (f) = R J(f (t)) dt, f G R, in the sense of [3], i.e. the rate function J R is convex, { f G R J R (f) r } is compact for each r R, and inf J R(f) inf f A α α log Q x,y α,r (A) sup α log Q x,y α,r (A) inf J R (f) α f A (3.7) for each Borel subset A of G R. Here the interior A and the closure A of A are taken with respect to the relative topology on G R. Proof. For the measures P x L 1 α,r, α>, the function space large deviation principle on C R follows from [3, Exercise 4.2.7] with the good rate function J R : C R [, ] given by R J R (f) = n 2 n 2 n k=1 ( f(2 n kr) f(2 n ) (k 1)R) J 2 n. (3.8) R If f C R \G R, then there exists k n {1,...,2 n } for all sufficiently large n N such that the corresponding term in (3.8) is infinite, hence J R (f) =. Iff G R, then [7, VII.8] and the dominated convergence theorem show that J R (f) =J R (f). To prove (3.7), first note that α P x (X R/α = y) =1/2. The upper estimate in (3.7) then follows from P x (L α,r A, X R/α = y) P x (L α,r A) and the corresponding estimate for the measures {P x L 1 α,r } α>. To show the lower estimate in (3.7), let A be an open subset of G R and ε (,R). Let A ε denote the set of all g G R ε such that there exists f A satisfying f [,R ε] = g and dist(f,g R \ A) >ε. Since { f G R f [,R ε] A ε } Aand since L α,r takes values in G R, it follows by using the Markov property that P x (L α,r A, X R/α = y) P x (L α,r ε A ε,x R/α = y) P x L 1 α,r ε (A ε) min{ P (X ε/α = y), P 1 (X ε/α = y) }. Since A ε is an open subset of G R ε, there exists an open subset B ε of C R ε such that A ε = B ε G R ε. Using the lower estimate for P x (L α,r ε B ε )asα, it only remains to show that inf ε (,R) inf J R ε (g) inf J R(f). (3.9) g A ε f A

16 E. Bolthausen, J.-D. Deuschel, and U. Schmock Take any f A and define ε = min{r, dist(f,g R \ A)}/2. Since A is open, ε>. If g := f [,R ε], then g A ε and J R ε (g) J R (f), hence (3.9) holds. To prepare the application of Varadhan s theorem, define, for each R >, the mappings g R : B R R and g R : G R R by g R (φ) = R e t φ(t) dt and g R (f) =e R f(r)+ R e t f(t) dt for all φ B R and f G R. Extend the interaction function V to [, ) 2 by defining V (x, y) =τ(xy+(1 x)(1 y)) for all x, y [, ). Furthermore, define F R :[ 1, 1] B R R and F R :[ 1, 1] G R R by F R (ϱ, φ) = 1 R R R e s t V (φ(s),φ(t)) ds dt + τϱ e (R t) φ(t) dt 2 and F R (ϱ, f) = τ 2 f 2 (R)+ τ 2 (R f(r))2 + τϱf(r) F R (ϱ, f) + τ 2 τ R R { f 2 (t)+(t f(t)) 2} dt e (R t){ f(r)f(t)+(r f(r))(t f(t)) } dt for all ϱ [ 1, 1], φ B R, and f G R. Integration by parts shows that g R (f) =g R (f ) and F R (ϱ, f) =F R (ϱ, f ) (3.1) for all ϱ [ 1, 1] and f G R. Note that g R and F R are continuous. For x {, 1} let P x α,t be the transformed path measure arising from P x via (1.4). For δ> define U δ =(ξ δ, ξ +δ) (ξ 1 δ, ξ 1 +δ), where ξ =1/(2τ 2 ) and ξ 1 =1 ξ. Lemma 3.11 If δ> and x {, 1}, then sup α T R/α P x α,t (Y α,r / U δ )= (3.12) for all sufficiently large R, where Y α,r = R e t X t/α dt. Proof. Givenα, R >, define X α,r : Ω B R by X α,r (t) =X t/α for all t [,R]. If T R/α, then H α,t = H α,[r/α,t ] + τ α Z (1 e R )+ 1 α F R(Z 1 Z,X α,r ), where H α,[r/α,t ] is given by (1.3) but with integration over [R/α, T ] and Z 1 = αt R e (t R) X t/α dt and Z = αt R e (t R) (1 X t/α ) dt.

Path measures arising from a mean field or polaron type interaction 17 Using (3.1), it follows that E x [ exp(h α,t ); Y α,r / U δ ] = E x [ exp ( H α,[r/α,t ] + α 1 τz (1 e R ) ) E x [ exp ( α 1 F R (Z 1 Z, L α,r ) ) ; g R (L α,r ) / U δ F R/α ]], where F R/α is the σ-algebra generated by {X t } t R/α. Since a similar formula holds for E x [exp(h α,t )], we obtain the estimate ( ) Yα,R / U δ P x α,t sup ϱ [ 1,1] y {,1} E x [ exp ( α 1 F R (ϱ, L α,r ) ) ; g R (L α,r ) / U δ X R/α = y ] E x [ exp ( α 1 F R (ϱ, L α,r ) ) X R/α = y ]. Since F R (,f): [ 1, 1] R is increasing for every f G R, it follows that P x α,t ( Yα,R / U δ ) max k { n,...,n 1} y {,1} G δ,r exp ( α 1 F R ((k +1)/n, ) ) dq x,y α,r G R exp ( α 1 F R (k/n, ) ) dq x,y α,r for every n N, where G δ,r = { f G R g R (f) / U δ }. For each ϱ [ 1, 1] the function F R (ϱ, ): G R R is continuous. Hence, it is bounded on the compact set G R and an analogue of [3, (2.1.9)] holds. Since g R is continuous, G δ,r is closed. Therefore, Lemma 3.6 and a version of Varadhan s theorem [3, Exercise 2.1.24(i)] imply that, for every ϱ [ 1, 1], sup α log α exp ( α 1 F R (ϱ, ) ) dq x,y α,r sup ( FR (ϱ, f) J R (f) ). G δ,r f G δ,r Analogously, by [3, Lemma 2.1.7], it follows that inf α log exp ( α 1 F R (ϱ, ) ) dq x,y ( α,r κ(ϱ) := sup FR (ϱ, f) J R (f) ). α G R f G R The last three estimates together imply that sup α 1 α log sup T R/α n P α,t x ( ) Yα,R / U δ ( sup F (min{1,ϱ+1/n},f) JR (f) κ(ϱ) ) (ϱ,f) [ 1,1] G δ,r = sup (ϱ,f) [ 1,1] G δ,r ( F (ϱ, f) JR (f) κ(ϱ) ), where the last equality follows from the uniform continuity of F R on the compact set [ 1, 1] G R. Furthermore, the uniform continuity of F R implies that the map [ 1, 1] ϱ κ(ϱ) is continuous, too. Hence, Λ R :[ 1, 1] G R (, ], defined by Λ R (ϱ, f) =F R (ϱ, f) J R (f) κ(ϱ), is upper semi-continuous and, therefore, attains,

18 E. Bolthausen, J.-D. Deuschel, and U. Schmock its supremum on the compact set [ 1, 1] G δ,r. To prove the lemma, it suffices to show that { (ϱ, f) [ 1, 1] G R Λ R (ϱ, f) =andg R (f) / U δ } = for all sufficiently large R. Using (3.1), this follows from the next lemma. Define Λ R :[ 1, 1] B R (, ] by R ( R ) Λ R (ϱ, φ) =F R (ϱ, φ) J(φ(t)) dt sup F R (ϱ, φ) J( φ(t)) dt. φ B R Lemma 3.13 For each δ>there exists R δ > such that, for all R>R δ, { (ϱ, φ) [ 1, 1] B R Λ R (ϱ, φ) =and g R (φ) / U δ } =. Proof. Define x = τ 1/τ. Since τ > 1, it follows that x >. Since U δ U δ for <δ <δ, it suffices to prove the lemma for δ (,x ]. Define c : R R by c(x) = 4(τ 2 τ 4+x 2 + 1). Note that c is even and strictly increasing on [, ) with c() = 4(τ 1) 2 and c(x ) =. Define R δ > by { R δ = max log 2 ( ) 1 δ, τ τ 1, 4τ c(x + δ), 2 2τ +1} 2τ c(x δ) 1. Fix R (R δ, ). Suppose that ϱ [ 1, 1] and φ B R satisfy Λ R (ϱ, φ) =, which is equivalent to saying that Λ R attains its maximal value at (ϱ, φ). To prove the lemma, it suffices to show that g R (φ) U δ. Define p ± =1/2 ± τ 2 /(1+4τ 2 ) and φ (t) =p (p + φ(t)) for all t [,R]. Since J 2p 1 (p) =2, p (, 1), 1 (2p 1) 2 it follows that J(p) J(p ) 4τ p p for all p [,p ] and J(p) J(p + ) 4τ p p + for all p [p +, 1]. Since F R (ϱ, φ 1 ) F R (ϱ, φ 2 ) 3τ φ 1 φ 2 L 1 for all φ 1,φ 2 B R, it follows that Λ R (ϱ, φ ) Λ R (ϱ, φ) τ φ φ L 1. Since (ϱ, φ) maximizes Λ R, we therefore may assume that φ(t) [p,p + ] for all t [,R]. For each φ B R the function ( p,p ) ε Λ R (ϱ, φ + ε φ) attains its maximum at ε =. Considering the derivatives, we see that φ satisfies τ R (2φ(s) 1)e s t ds + τϱe (R t) J (φ(t))= (3.14) for almost all t [,R]. Define ψ = J φ. Then φ(t) =(g(ψ(t))+1)/2 for all t in [,R], where the bounded function g C (R) isgivenbyg(x) =x(4 + x 2 ) 1/2. Equation (3.14) implies that ψ(t) =τϱe (R t) + τ R g(ψ(s))e s t ds (3.15) for almost all t [,R]. Since the right-hand side of (3.15) is continuous, we may assume that ψ is continuous and that (3.15) holds for all t [,R]. It then follows

Path measures arising from a mean field or polaron type interaction 19 that ψ and φ are in C ([,R], R). Hence, (3.14) holds for t = and shows that g R (φ) =(ψ() + τ)/(2τ) (1 + ϱ)e R /2. If we can prove that ψ() x <δ or ψ() + x <δ, (3.16) then min{ g R (φ) ξ 1, g R (φ) ξ } δ/(2τ)+e R <δ, because R>R δ log(2/δ). This means that g R (φ) U δ. Therefore, it remains to prove (3.16). If a, b [, 1/2) or a, b (1/2, 1], then a(1 b)+(1 a)b <ab+(1 a)(1 b). Furthermore, note that J(a) =J(1 a) for all a [, 1]. If there were s, t [,R] with φ(s) < 1/2 and φ(t) > 1/2, then F R (,φ) <F R (,φ + )=F R (,φ ), where φ + := max{φ, 1 φ} and φ := min{φ, 1 φ}; hence Λ R (ϱ, φ) < Λ R (ϱ, φ + )if ϱ [, 1], and Λ R (ϱ, φ) <Λ R (ϱ, φ )ifϱ [ 1, ], which contradicts the choice of (ϱ, φ). This proves that φ 1/2 ifϱ (, 1], that φ 1/2 ifϱ [ 1, ), and that φ 1/2 does not change its sign if ϱ =. Since p φ p +,asweprovedabove,it follows that ψ 4τ if ϱ (, 1], that 4τ ψ ifϱ [ 1, ), and that either ψ 4τ or 4τ ψ in the case ϱ =. Computing the first and the second derivative of (3.15), we see that ψ is a solution of the boundary value problem y (t) =y(t) 2τg(y(t)), t [,R], (3.17) y () = y(), (3.18) y (R) =2τϱ y(r). (3.19) Since ψ solves (3.17) and (3.18), it follows that, for all t [,R], ( ψ 2 (t) = 2τ 2 4+ψ (t)) 2 + c(ψ()). (3.2) If ψ() is given, then ψ is uniquely determined by (3.17) and (3.18). If ψ() =, then ψ is identically zero and (3.19) shows that ϱ =. Note that ψ = corresponds to φ =1/2. According to the special choice of φ, the function ( 1, 1) ε Λ R (,φ+ ε/2) attains its maximum at ε =, hence, since φ =1/2, d2 dε 2 Λ R(,φ+ ε/2) =(τ 1)R τ(1 e R ), ε= which is impossible because R>R δ τ/(τ 1). If ψ() x + δ, then ψ (t) c 1/2 (x + δ) >c 1/2 (x ) = for all t [,R]by (3.2). Using (3.18) and the continuity of ψ, it follows that ψ(r) c 1/2 (x + δ)r. This is impossible because R>R δ 4τc 1/2 (x + δ) and ψ(r) 4τ. If ψ() (,x δ], then >c(x δ) c(ψ()) c() = 4(τ 1) 2. Define y = (2τ c(ψ()) ) 2 4, (2τ y 1 = c(x δ) ) 2 4, ( ) 1 2τ t 1 = 4+ y 1, 2 1 and note that t 1 (R δ 1)/2 and ψ() = (2τ c(ψ()) + ψ2 () ) 2 4 <y y 1 < 2 τ 2 1.

2 E. Bolthausen, J.-D. Deuschel, and U. Schmock If ψ(t) [ψ(),y ], then (3.17) implies that ( ) ψ 2τ (t) = ψ(t) 4+ψ2 (t) 1 ψ() t 1 <. Hence, since ψ() = ψ () > by (3.18), there exists a smallest t [,t 1 ] such that ψ (t ) = or ψ(t ) = y. By (3.2), each of these two equations implies the other one. According to (3.17), the functions ψ [t,2t ] and ψ :[t, 2t ] R, given by ψ(t) =ψ(2t t), solve the initial value problem y (t) =y(t) 2τg(y(t)), y (t )=, y(t )=y. Due to the uniqueness of the solution, it follows that ψ(2t )=ψ() and ψ (2t )= ψ (). Since ψ() = ψ () > and ψ (t) ifψ(t) [, 2 τ 2 1], there exists t 2 [2t, 2t + 1] where ψ changes its sign. This is impossible because we proved above that ψ does not change its sign in [,R]. A similar argument shows that ψ() [ x + δ, ) is impossible. The preceding three paragraphs prove (3.16). Proof of (3.5) for τ>1. Define π =(1/2, 1/2). For x {, 1} and ϱ [, 1] define ϕ ϱ (x) =τϱx+ τ(1 ϱ)(1 x). Since p ϕ ϱ,µ p J(p) attains its supremum at p ϱ = 1 ( ) 2ϱ 1 1+τ, 2 4+τ2 (2ϱ 1) 2 (2.6) yields Λ ϱ := Λ ϕ ϱ =(τ 2+ 4+τ 2 (2ϱ 1) 2 )/2. Next we determine h ϕ ϱ, µ ϕ ϱ, and l ϕ ϱ, which appear in Lemma 2.7. We use the abbreviations h ϱ, µ ϱ, and l ϱ, respectively. According to the Feynman-Kac formula [18, Example IV.22.11], the semigroup {exp( Λ ϱ t)p ϕ ϱ t } t, where P ϕ ϱ t given by (2.4), has the generator ( ϕϱ () Λ ϱ ) ( ) 1 1 γϱ 1 1 ϕ ϱ (1) Λ ϱ = 1 1 1/γ ϱ with γ ϱ := ( 4+τ 2 (2ϱ 1) 2 + τ(2ϱ 1))/2. The non-negative 2 -normalized eigenfunction corresponding to the eigenvalue is given by h ϱ () = 2/(1 + γ ϱ2 ) and h ϱ (1) = γ ϱ h ϱ (). Since h ϱ (1)/h ϱ () = γ ϱ, it follows from (2.8) that L γ ϱ is the generator of {Q ϕ ϱ x } x {,1}. Note that p ϱ =(h ϱ (1)) 2 π(1) and 1 p ϱ =(h ϱ ()) 2 π(). Since (1,γϱ)L 2 γ ϱ =, it follows that µ ϱ := (1 p ϱ,p ϱ )isthe{q ϕ ϱ ( ) Xt 1 } t -invariant distribution and that l ϱ = h ϱ. Note that γ ξ1 = τ and γ ξ =1/τ. We now follow the proof of Theorem 2.2. For ϱ [, 1] and α, t, T > with t<t define T Θ α,[t,t ] = αe α(s t) X s ds + 1 t) e α(t 2 and t t R α,t,t (ϱ) =H α,t τ (1 e α(t s) )(ϱx s +(1 ϱ)(1 X s )) ds τ 2α (eαt 1)e αt. We use the abbreviation Θ α,t = Θ α,[,t ]. Then, corresponding to (2.3), H α,t = t ϕ Θ α,[t,t ],µ lt + H α,[t,t ] + R α,t,t (Θ α,[t,t ] ), (3.21)

Path measures arising from a mean field or polaron type interaction 21 where H α,[t,t ] is given by (1.3) but with integration over [t, T ] instead of [,T]. Define Γ α,t,t M 1 ([, 1]), for all Borel sets A of [, 1], by Γ α,t,t (A) = y {,1} π(y) c α,t,t E y [1 A (Θ α,t )l Θ α,t (y) exp(tλ Θ α,t + H α,t )], (3.22) where c α,t,t denotes the appropriate normalizing constant. Since Λ ϱ = Λ 1 ϱ and V (x, y) = V (1 x, 1 y) as well as P x (X s = y) = P 1 x (X s = 1 y) and h ϱ (x) = h 1 ϱ (1 x) for all s, ϱ [, 1], and x, y {, 1}, it follows that Γ α,t,t is symmetric, which means that Γ α,t,t (A) =Γ α,t,t ({ 1 a a A }) for all Borel sets A of [, 1]. Since sup{ Θ α,t Y α,r : T R/α } asα and R, it follows from Lemma 3.11 that, for each t>, the measures {Γ α,t,t } α,t > converge weakly to the symmetric measure (δ ξ1 + δ ξ )/2, uniformly for T R/α as α and then R. Let f : Ω [1, 2] be a continuous function which is F s -measurable for some s. By Lemma 2.21 it suffices to show that R sup α sup T R/α Êα,T [f] 1 1+τ Ω fdq τ τ 1+τ Ω fdq 1/τ =, (3.23) which would prove (1.7) for this model, too. For α, T > and y {, 1} define the measure Γ y α,t by Γ y α,t (A) =E y[1 A (Θ α,t ) exp(h α,t )] for all Borel sets A of [, 1]. Using (3.21) and the Markov property, it follows that, for every t (s, T ), E[f exp(h α,t )] = y {,1} 1 E[ f exp(t ϕ ϱ,µ lt + R α,t,t (ϱ)); X t = y ] Γ y α,t t (dϱ). Since R α,t,t (ϱ) ατt 2 +2τ(e αt 1+αt)/α + τe R (e αt 1)/(2α) for all ϱ [, 1] and α, R, t, T > with T R/α, there exists, for each t s, a suitable subset { ε α,t,t α>,t > t} of (, ) with sup{ ε α,t,t 1 : T R/α } asα and R such that E[f exp(h α,t )] = ε α,t,t y {,1} 1 By Lemma 2.7(c) and (d), E[ f exp(t ϕ ϱ,µ lt ); X t = y ]=e Λϱt h ϱ ()l ϱ (y)π(y) E[ f exp( ϕ ϱ,µ lt ); X t = y ] Γ y α,t t (dϱ). Ω f qϱ t s(x s,y) h ϱ (y)l ϱ (y) dqϱ for y {, 1}. Since [, 1] ϱ ϕ ϱ is bounded, it follows from Lemma 2.7(g) and (h) that there exists { ε t } t>s (, ) with ε t 1ast such that 1 ( E[f exp(h α,t )] = ε α,t,t ε t fdq ϱ) h ϱ ()π(y)l ϱ (y)e Λϱt Γ y α,t t (dϱ). y {,1} Using (1.4) and (3.22), it follows that, for s<t<tand α>, 1 / 1 Ê α,t [f] =ε α,t,t ε t h ϱ () fdq ϱ Γ α,t,t t (dϱ) h ϱ () Γ α,t,t t (dϱ), Ω Ω