Large deviations of the trajectory of empirical distributions of Feller processes on locally compact spaces 5 1 Richard Kraaij 1 0 2 May 29, 2015 y a M Abstract 8 Westudythelargedeviationbehaviourofthetrajectoriesofempirical 2 distributionsof independentcopiesofFeller processes onlocally compact metricspaces. Undertheconditionthatwecanfindasuitablecoreforthe ] A generatoroftheFellerprocess,weareabletodefineanotionofabsolutely continuoustrajectories of measures in terms of this core. Also, we define F a Hamiltonian in terms of the linear generator and a Lagrangian as its . h Legendretransform. t a The rate function of the large deviation principle can then be de- m composed as a rate function for the initial time and an integral over the Lagrangian,finiteonlyforabsolutelycontinuoustrajectoriesofmeasures. [ The theorem partly extends the Dawson and G¨artner theorem [7], in 3 the sense that it holds for diffusion processes where the drift en diffu- v sion coefficients are sufficiently smooth. On the other hand, the result 2 is sufficiently general to cover both Markov jump processes and discrete 0 interacting particle systems [23]. 8 2 . 1 1 Introduction 0 4 Dawson and G¨artner [7] proved the large deviation principle for the trajectory 1 : of empirical distributions of weakly interacting copies of diffusion processes. v Additionally, they proved that the rate function can be decomposed as an en- i X tropytermforthelargedeviationsattimezeroandanintegraloveraquadratic r Lagrangian, depending on position and speed. Recently, new proofs have been a given in [2] using weak convergence methods and control theory and in [13] based on the comparison principle for an infinite dimensional Hamilton-Jacobi equation. A similar results for Markov jump-processes, with a less explicit rate function, hasbeengiveninS.Feng[14],alsosee[15,21]. Maes,Netoˇcny´andWynants[3] study the largedeviations oftrajectoriesof the empiricaldistributions together with the empirical flow of a finite state space Markov jump process and give a Lagrangianform of the rate function. 1DelftInstituteofAppliedMathematics,DelftUniversityofTechnology,Mekelweg4,2628 CDDelft,TheNetherlands. 1 These two sets of results raise the question whether a general approach is pos- sible to prove large deviations for trajectories of weakly interacting, or even independent copies of processes with a ‘Lagrangian’form of the rate function: ∞ I(γ):=I (γ(0))+ L(γ(s),γ˙(s))ds. (1.1) 0 Z0 forabsolutelycontinuoustrajectoriesγ andinfinity otherwise. Ourmainaimis to allow a large class of state spaces and processes including e.g. independent copies of whole interacting particle systems [23]. Feng and Kurtz [13] propose a generalmethod to prove large deviations on the path-spacebasedontheanalogyofthelargedeviationprincipletoweakconver- gence. The approach involves the construction of a non-linear semigroup V(t) viathecomparisonprincipleforequationsinvolvingthegeneratingHamiltonian H and the Crandall-Liggett theorem [5]. Under some suitable conditions, they show that the semigroupV(t) can be re-expressedusing a Nisio or Lax-Oleinik semigroup ∞ V(t)f(x)= sup f(γ(t))− L(γ(s),γ˙(s))ds γ,γ(0)=x Z0 which in turn gives the Lagrangianform of the rate function. In this paper, we give a proof of the large deviation principle for trajectories of empiricalaveragesof independent copies of Feller processesonsome space E without explicitly specifying the structure of the underlying process. Addition- ally, we express the rate function as in the form of (1.1). Weusethefunctionalanalyticstructureunderlyingthelargedeviationprinciple introduced in [13], but our approach to the problem is different. The indepen- dence assumption implies that the large deviation principle can be proven via Sanov’s theorem and the contraction principle. Also, we can explicitly give the limiting non-linear semigroup V(t) on E as logS(t)ef where S(t) is the semi- group of conditional expectations of the Markovprocess. This approachavoids the difficult problem of constructing a semigroup. To obtain a Lagrangianform of the rate function, the main technical challenge is toshowthatV(t)equalsaNisiosemigroupV(t). Thedefinitionofthe Nisio- semigroup poses us with two problems. First, we need a context-independent way to define absolutely continuous trajectories of measures, and secondly, we need a way to define a Lagrangian. To this end, we assume the existence of a suitable topology on a core of the generator (A,D(A)) of the Feller process. The equality of V(t) and V(t) is then proven using resolvent approximation arguments and Doob-transform techniques. Therestofthepaperisorganisedasfollows. We startoutinSection2withthe preliminaries and state the two main theorems. Theorem 2.1 gives, under the condition that the processes solves the martingale problem, the large deviation principle. Undertheconditionthatthereexistsasuitablecoreforthegenerator of the process, Theorem 2.8 gives the decomposition of the rate function. In Section 3, we prove Theorem 2.1 using Sanov’s theorem for large deviations on the Skorokhod space and the contraction principle. We show that the rate function is given by a rate for the initial law, and a second term that is given as the supremum over sums of conditional large deviation rate functions. The 2 Legendre transforms of such conditional rate functions is given in terms of the non-linear semigroup V(t). Additionally, we give a short introduction to the Doob transform, which we will use to study the non-linear semigoup. In Sections 4 and 5, we prove Theorem 2.8. In the first section, we study the Hamiltonian, Lagrangian and a family of ‘controlled’ generators. Finally, in Section 5, we introduce the Nisio semigroup V(t) in terms of absolutely continuous trajectories and the Lagrangian, and show that it equals the non- linear semigroup V(t). InSection6,wegivethreeexampleswhereTheorem2.8applies. Westartwitha Markovjumpprocess. Afterthat,wechecktheconditionsforspatiallyextended interacting particle systems of the type that are found in Liggett [23]. Lastly, we check the conditions for a class of diffusion processes and show that, at least if the process is time-homogeneous and the diffusion and drift coefficients are sufficiently smooth, we recover the result for averages of independent and time-homogeneous processes by Dawson and G¨artner [7]. 2 Preliminaries and main results We start with some notation. Let (E,d) be a complete separable metric space with Borel σ-algebra E. M(E) is the set of Borel measures of bounded total variation on E be equipped with the weak topology and P(E) is the subset of probability measures. We denote with D (R+) the Skorokhod space of E E valued ca`dla`g paths [12, Section 3.5], R+ = [0,∞). We write hf,µi for the integral of f ∈C (E) with respect to µ∈M(E). b We define the relative entropy H(µ|ν) of µ with respect to ν by logdµdµ if µ<<ν H(µ|ν)= dν (R∞ otherwise. OnE, we havea time-homogeneousMarkovprocess {X(t)} givenby a path t≥0 space measure P on D (R+). Let X1,X2,... be independent copies of X and E let P the measure that governs these processes. We look at behaviour of the sequence L := LX(t) , n n t≥0 n o n 1 LX(t) := δ , n n {Xi(t)} i=1 X under the law P. L takes values in D (R+), the Skorokhod space of paths n P(E) taking values in P(E). We also consider C (R+) the space of continuous P(E) paths on P(E) with the topology inherited from D (R+). P(E) WesaythatL satisfiesthelargedeviationprinciple(LDP)onD (E)(R+)with n P lower semi-continuous rate function I : D (R+) → R+ if for every open set P(E) A 1 liminf logP[L ∈A]≥− inf I(µ) n n→∞ n µ∈A and for every closed set B 1 limsup logP[L ∈B]≤− inf I(µ). n n→∞ n µ∈B 3 I is called good if its level sets {I ≤c} are compact. Suppose that A : D(A) ⊆ C (E) → C (E), is a linear operator with a domain b b that separates points: for every x,y ∈ E, there exists a f in this set such that f(x)6=f(y). We say that X solves the martingale problem for (A,D(A)) with starting measure P , if P is the law of X(0) and if for every f ∈D(A), 0 0 t f(X(t))−f(X(0))− Af(X(s))ds Z0 is a martingale for the natural filtration {F } given by F =σ(X(s)|s≤t). t t≥0 t In Section 3, we obtain the following preliminary result. Theorem 2.1. Let X, represented by the measure P on D (R+) solve the E martingale problem for (A,D(A)) with starting measure P . Then, the sequence 0 L satisfies the large deviation principle with good rate function I, which is n given for ν ={ν(t)} ∈D (R+) by t≥0 P(E) k H(ν(0)|P )+sup I (ν(t )|ν(t )) if ν ∈C (R+) I(ν)= 0 {ti}i=1 ti−ti−1 i i−1 P(E) ∞ X otherwise, where {t }is a finite sequence of times: 0 = t < t < ··· < t . For s ≤ t, we i 0 1 k have I (ν |ν )= sup {hf,ν i−hV(t)f,ν i}, (2.1) t 2 1 2 1 f∈Cb(E) where V(t)f(x)=logE ef(X(t)) X(0)=x . Forfurtherresults,wein(cid:2)troduce(cid:12)someaddit(cid:3)ionalnotation. Foralocallyconvex (cid:12) space (X,τ), we write X′ for its continuous dual space and L(X,τ) for the space of linear and continuous maps from (X,τ) to itself. Also, for x ∈ X and x′ ∈ X′, we write hx,x′i := x′(x) ∈ R for the natural pairing between x and x′. For two locally convex spaces X,Y and a continuous linear operator T : X → Y, we write T′ : Y′ → X′ for the adjoint of T, which is uniquely defined by hx,T′(y′)i=hTx,y′i, see for example Treves[27, Chapter 19]. For a neighbourhood N of 0 in X, we define the polar of N◦ ⊂X′ by N◦ :={u∈X′||hx,ui|≤1 for every x∈N}. (2.2) We saythat a locallyconvexspace X is barrelledif everybarrelis a neighbour- hood of 0. A set S is a barrelif it is convex,balanced, absorbing and closed. S is balanced if we have the following: if x ∈S and α∈ R, |α| ≤1 then αx ∈ S. S is absorbing if for every x ∈ X there exists a r ≥ 0 such that if |α| ≥ r then x ∈ αS. We give some background and basic results on barrelled spaces in Appendix 8. To rewrite the rate function obtained in Theorem 2.1, we restrict to locally compact metric spaces (E,d) and we consider the situation where S(t)f(x) = E[f(X(t))|X(0)=x]isastronglycontinuoussemigrouponthespace(C (E),||·||): 0 for every t ≥ 0, the map S(t) : (C (E),||·||) → (C (E),||·||) is continuous, and 0 0 for every f ∈C (E), the trajectory t7→S(t)f is continuous in (C (E),||·||). 0 0 4 Let (A,D(A)) be the generatorof the semigroupS(t). It is a well known result that X solves the martingale problem for (A,D(A)) [12, Proposition 4.1.7], so the above result holds for the process {X(t)} . t≥0 Our goal is to rewrite I as t I(ν)=H(ν(0)|P )+ L(ν(s),ν˙(s))ds 0 Z0 for a trajectoryν of probability measures that is absolutely continuous in some sense. Thus our first problem is to define differentiation in a context for which no suitable structure on E or P(E) is known. Therefore, we will have to tailor the definition of differentiation to the process itself. Suppose that µ(t) is the law of X(t) under P. Then we know that t 7→ µ(t) = S(t)′µ(0) is a weakly continuous trajectory in P(E), so can ask whether for f ∈D(A) the trajectory t7→hf,µ(t)i is differentiable as a function from R+ →R: ∂ ∂ hf,µ(t)i= hS(t)f,µ(0)i=hS(t)Af,µ(0)i=hAf,µ(t)i. (2.3) ∂t ∂t So our candidate for µ˙(t) would be A′µ(t), which is a problematic because (A,D(A)) could be unbounded. To overcome this, and other problems, we introduce two sets of conditions on (A,D(A)). Recall that D is a core for (A,D(A)) if for every f ∈ D(A), we can find a sequence f ∈D such that f →f and Af →Af. n n n Condition2.2. ThereexistsacoreD ⊆D(A)densein(C(E),||·||)thatsatisfies (a) D is an algebra, i.e. if f,g ∈D then fg ∈D, (b) if f ∈ D and φ : R → R a smooth function on the closure of range of f, then φ◦f −φ(0)∈D, In the case that E is compact, C (E)=C(E), then (b) can be replaced by 0 (b’) iff ∈Dandφ:R→Rasmoothfunctionontherangeoff,thenφ◦f ∈D. Note if a dense subspace D ⊆ D(A) satisfies S(t)D ⊆ D, then D is a core for (A,D(A)) [12, Proposition 1.3.3]. Under Condition 2.2, we define the operator H : D → C (E) and for every 0 g ∈D the operator Ag :D →C (E) by 0 Hf =e−fAef, Agf =e−gA(feg)−(e−gAeg)f. If E is non-compact,these definitions needs some care as ef ∈/ C (E). This can 0 be solvedby looking at the one-pointcompactificationof E, see Section 4.1. In Section 4, we will show that {V(t)} turns out to be a non-linear semigroup t≥0 on C (E) which has H as its generator. The operators Ag are generators of 0 Markov processes with law Qg on D ([0,t]) that are obtained from P by E dQg t t(X)=exp g(X(t))−g(X(0))− Hg(X(s))ds , (2.4) dP t (cid:26) Z0 (cid:27) where P and Qg are the measures P and Qg restricted to times up to t, see t t Theorem 4.2 in [25]. 5 Condition 2.3 (Conditions on the core). D satisfies Condition 2.2 and there exists a topology τ on D such that D (a) (D,τ ) is a separable barrelled locally convex Hausdorff space. D (b) The topology τ is finer than the sup norm topology restricted to D. D (c) Themapsexp−1:(D,τ )→(D,τ )and×:(D,τ )×(D,τ )→(D,τ ), D D D D D defined by f 7→ef −1, respectively (f,g)7→fg are continuous. (d) We have S(t)D ⊆ D, V(t)D ⊆ D and the semigroups {S(t)} and t≥0 {V(t)} restricted to D are strongly continuous for (D,τ ). t≥0 D (e) The map A:(D,τ )→(C (E),||·||) is continuous. D 0 (f) There exists a barrel N ⊆D such that sup||Hf||≤1, f∈N and for every c>0 sup ||Hf||<∞. f∈cN Conditions (a) to (e) are related to the differentiation of the trajectories of measuresthatwillturnupinourlargedeviationproblem. Condition(a)implies that (D,τ ) is wellbehavedas a locallyconvexspace and,among otherthings, D makes sure that we are able to define the Gelfand integral, see Appendix 8. Condition (b) implies that (M(E),wk) is continuously embedded in (D′,wk∗), so that every weakly continuous trajectory of measures can in fact be seen as a weak* continuous trajectory in D′. Important is this light is that the conditions in (d) on {S(t)} imply that t 7→ S(t)′µ is weak* continuous in t≥0 D′ for all measures µ ∈ P(E). (e) implies that we take the adjoint of A : (D,τ ) → (C (E),||·||), so that we obtain a weak to weak* continuous map D 0 A′ : M(E) →D′. Returning to Equation (2.3), we now have a good definition for µ˙(t) := A′µ(t) ∈ D′. Furthermore, we can also differentiate trajectories of measures that are obtained from X via a tilting procedure, e.g. Equation (2.4) by Lemma 2.5 below. Condition (f) is the main technical condition on H which implies for example the compactness of the level sets of L. Using these compactness arguments, we are able to rewrite I. Remark 2.4. The conditions on {V(t)} in (d) seem to be to strong. They t≥0 are used to prove that V(t) equals V(t), see Proposition 5.10. If, for example, the range condition for H, i.e. Ran( −λH)(D) = C (E) for λ > 0, can be 0 1 checked directly, or if the comparison principle for H can be proven, the result of Proposition 5.10 would follow without V(t)D ⊆ D or the strong continuity of V(t). ThefollowinglemmaisaconsequenceofCondition2.3(c)and(e)andtheproof is elementary. Lemma 2.5. Let (D,τ ) satisfy Condition 2.3, then the maps A : (D,τ )× D D (D,τ )→(C (E),||·||) given by Φ(g,f)=Agf and the operator H :(D,τ )→ D 0 D (C (E),||·||) are continuous. 0 6 Remark 2.6. TheresultsofthispaperalsoholdinthecasethatCondition2.3 (c) fails as long as the conclusions of Lemma 2.5 hold. In all examples that we consider in Section 6 (c) is satisfied. For the next definition we will need the Gelfand or weak* integral, which is introduced in Appendix 8, but the rigorousconstruction of this integralcan be skipped on the first reading. Definition 2.7. Define D−AC, or if there is no chance of confusion, AC, the space of absolutely continuous paths in C (R+). A path ν ∈ C (R+) is P(E) P(E) called absolutely continuous if there exists a (D′,wk∗) measurable curve s 7→ u(s) in D′ with the following properties: t (i) for every f ∈D and t≥0 |hf,u(s)i|ds<∞, 0 (ii) for every t≥0, ν(t)−ν(0)R= tu(s)ds as a D′ Gelfand integral. 0 We denote ν˙(s) := u(s). FurthermRore, we will denote AC for the space of µ absolutely continuous trajectories starting at µ , and ACT for trajectories that 0 are only considered up to time T. Similarly, we define ACT. µ A directconsequence of property(ii) is that for almosteverytime t≥0 and all f ∈D the limit hf,ν(t+h)i−hf,ν(t)i lim h→0 h ˙ exists and is equal to hf,ν(t)i. This justifies the notation u(s)=ν˙(s). Using these definitions, we are able to sharpen Theorem 2.1. In Section 4, we study the semigroup V(t) and its generator H. Also, we give a number of properties of the level sets of the Lagrangian L, defined in the theorem below. The proof of the theorem is given in Section 5. Theorem 2.8. Let (E,d) be locally compact. Let (A,D(A)) have a core D equipped with a topology τ such that (D,τ ) satisfies Condition 2.3. Then, the D D rate function in Theorem 2.1 can be rewritten as H(ν(0)|P )+ ∞L(ν(s),ν˙(s))ds if ν ∈AC I(ν)= 0 0 ν(0) (∞ R otherwise, where L:P(E)×D′ →R+ is given by L(µ,u):= sup{hf,ui−hHf,µi}. f∈D Remark 2.9. If we restrict ourselves to [0,T] instead of R+, then we obtain IT({ν(s)} ) 0≤s≤T H(ν(0)|P )+ T L(ν(s),ν˙(s))ds if ν ∈ν ∈ACT = 0 0 ν(0) (∞ R otherwise, by applying the contraction principle. 7 3 The large deviation principle via Sanov’s the- orem and optimal trajectories Let (E,d) is a complete separable metric space. We start by proving the large deviation principle for a generalclass of processes via Sanov’s theorem and the contraction principle. This will lead to the proof of Theorem 2.1. Define for every t the map π :D (R+)→E by π (x):=x(t). By Proposition t E t III.7.1 in Ethier and Kurtz, π is a measurable map. Complementary to π , t t we introduce the map π . For every path x ∈ D (R+), the value x(t−) := t− E lim x(s) is welldefined, which makesit possible to define π :D (R+)→E s↑t t− E by π (x):=x(t−). As π is the point-wise limit ofthe measurable maps π , t− t− tn for t <t, t ↑t, also π is measurable. n n t− Let P be a probability measure on D (R+), and let X = (X(t)) be the E t≥0 process with law P. Define µ(t) = P◦π−1 and µ(t−) = P◦π−1 the laws of t t− X(t) and X(t−). Also define the map φ : P(D (R+)) → P(E)R+ by setting E φ(P) = (µ(t)) and finally define the maps φ : P(D (R+)) → P(E) by t≥0 t E setting φ (P)=µ(t). t Lemma 3.1. φ is a map from P(D (R+)) to D (R+). E P(E) Proof. First, we prove that if s ↓ t then µ(s) → µ(t) weakly. Because the paths of X are right-continuous, we have X(s) → X(t). Hence, we have a.s. convergence,which in turn implies that µ(s)→µ(t) weakly. If s ↑ t, then we need to show that lim µ(t) exists, but as above X(s) → s↑t X(t−), hence, the weak limit lim µ(s) is equal to µ(t−). s↑t We wouldlike to provethatφ and {φ } arecontinuous maps,but this is not t t≥0 always true as can be seen from the following example. Example 3.2. Pick two distinct points e ,e in E. Define 1 2 e for t<1+1/n x+(t)= 1 n (e2 for t≥1+1/n e for t<1−1/n x−(t)= 1 n (e2 for t≥1−1/n andletPn ∈P(D (R+))bedefinedbyPn = 1δ +1δ . Clearly,thesequence E 2 x+n 2 x−n Pn converges weakly to P˜ = δ where x˜(t) is equal to e for t < 1 and e for x˜ 1 2 t≥1. If we look at the images φ(Pn) = (µn(t)) and φ(P˜) = (µ˜(t)) , then we t≥0 t≥0 obtain δ for t<1−1/n e1 µn(t)= 1δ + 1δ for 1−1/n≤t<1+1/n 2 e1 2 e2 δe2 for 1+1/n≤t, µ˜(t)=δe1 for t<1 (δe2 for t≥1. Clearly, µn(1) → 1δ + 1δ , which is not equal to µ˜(1) or µ˜(1−). We obtain 2 e1 2 e2 that both φ and φ are not continuous. Obviously, it follows that φ for t ≥ 0 1 t are not continuous either. 8 So problems arise when the time marginals of the limiting measure P are dis- continuous in time. However, this is the only thing that can happen. Proposition 3.3. φ : P(D (R+)) → D (R+) is continuous at measures P E P(E) for which it holds that for every t>0: P[X(t)=X(t−)]=1. A similar statement for the finite dimensional projections φ , can be found in t Ethier and Kurtz [12, Theorem 3.7.8]. Proof. Let Pn,P ∈ P(D (R+)) such that Pn → P weakly and P such that E for every t P[X(t) = X(t−)] = 1. By the Skorokhod representation Theorem [12, Theorem 3.1.9], we can find a probability space (Ω,F,P) and D (R+) E valued random variables Yn,Y distributed as Xn and X under Pn,P such that Yn →Y P a.s. Let {t } be a sequence converging to t>0. Define the sets n n≥0 A:={Y(t)=Y(t−)}, B :={d(Yn(t ),Y(t))∧d(Yn(t ),Y(t−))→0}. n n By the assumption that P[X(t) = X(t−)] = 1, it follows that P[A] = 1. By Proposition 3.6.5 in [12], and the fact that Yn → Y P a.s. it follows that P[B]=1. Combining these statements yields P[Yn(t )→Y(t)]≥P[A∩B]=1, n whichimplies that µn(t )→µ(t). As µ(t)=µ(t−)by assumption,Proposition n 3.6.5 in Ethier and Kurtz yields the final result. 3.1 Large deviations for measures on the Skorokhod space Suppose that we have a process X on D (R+) and a corresponding measure E P ∈ P(D (R+)). Then Sanov’s theorem, Theorem 6.2.10 in [8], gives us the E largedeviationbehaviouroftheempiricaldistributionLX ofindependentcopies n of the process X: X1,X2,...: n 1 LX := δ ∈P(D (R+)). n n {Xi} E i=1 X Theorem 3.4 (Sanov). The empirical measures LX satisfy the large deviation n principle on P(D (R+)) with respect to the weak topology with the good and E convex rate function dQ dQ I∗(Q)=H(Q |P):= log dP. dP dP Z We are interested in obtaining a large deviation principle on D (R+). In P(E) Proposition 3.3, we saw that we have a map φ that is continuous on a part of its domain. Hence, we we are in the position to use the contraction principle. Theorem 3.5. Suppose that P satisfies P[X(t) = X(t−)] = 1 for every t ≥ 0, then the large deviation principle holds for n 1 LX(t) = δ n t≥0 n Xi(t)! (cid:16) (cid:17) Xi=1 t≥0 9 on D (R+) with rate function P(E) I((ν ) )=inf{H(Q|P)|Q∈P(D (R+)),φ(Q)=(ν(t)) } t t≥0 E t≥0 and I is finite only on C (R+). P(E) Proof. The measures Q for which it holds that I(Q) < ∞ satisfy Q << P hence it follows that for every t: Q[X(t) = X(t−)] = 1. This yields that φ is continuous at Q by Proposition 3.3. Bythecontractionprinciple,Theorem4.2.1andremark(c)afterTheorem4.2.1 inDemboandZeitouni[8],weobtainthelargedeviationprincipleonD (R+) P(E) with I as given in the theorem. 3.2 The large deviation principle for Markov processes AlthoughTheorem3.5canbe appliedto awide rangeof(time-inhomogeneous) processes,weexploreits consequencesfortime-homogeneousMarkovprocesses. RecallthedefinitionofasolutiontothemartingaleproblemprecedingTheorem 2.1. Lemma 3.6. Suppose that the process X with corresponding measure P on D (R+) solves the martingale problem for (A,D(A)) with starting measure P . E 0 Then, it holds that for every t ≥ 0 P[X(t) = X(t−)] = 1. Hence, the large deviation principle holds for {LX(t)} on D (R+) with rate function n t≥0 P(E) I((ν ) )=inf{H(Q|P)|Q∈P(D (R+)),φ(Q)=(ν(t)) } t t≥0 E t≥0 and I is finite only on C (R+). P(E) Proof. To apply Theorem 3.5, we need to check that P[X(t) = X(t−)] = 1 for every t≥0, but this follows by Theorem 4.3.12 in [12]. Using this result, Theorem 2.1 follows without much effort. Proof of Theorem 2.1. ThelargedeviationprinciplefollowsbyLemma3.6. This lemma alsogives that the rate function is ∞ onthe complement ofC (R+). P(E) To obtain the rate function as a supremumoverrate functions for finite dimen- sional problems sup I[0,t ,...,t ](ν(0),ν(t )...,ν(t )) if ν ∈C (R+) I(ν)= 0,t1,...,tk 1 k 1 k P(E) (∞ otherwise, we use Theorem 4.13 and Theorem 4.30 in Feng and Kurtz [13]. Proposition 7.3 gives us the final decomposition of the rate function. 3.3 Approximating V(t) Before we turn to the proof of Theorem 2.8, we start with some general results on approximating hV(t)f,µi = sup hf,νi−I (ν|µ) by using the Doob ν∈P(E) t transform. For related results on Schr¨odingerbridges, see F¨ollmer and Gantert [17] or the survey paper by L´eonard [22]. 10