INTERACTING PARTICLE SYSTEMS OUT OF EQUILIBRIUM. Davide Gabrielli University of L Aquila. (EURANDOM 2011) joint with

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1 INTERACTING PARTICLE SYSTEMS OUT OF EQUILIBRIUM Davide Gabrielli University of L Aquila (EURANDOM 2011) joint with 1

2 Continuous time Markov chains State space: configurations of particles η = configuration, η M Λ N = X N, N N is a parameter Λ N Z d or Λ N = (Z/NZ) d = T d N = d-dimensional discrete torus M N {0}, ex. M = {0, 1} (exclusion rule) η x = k there are k particles at site x Λ N 2

3 Stochastic evolution: η(t) = configuration of particles at time t. Generator: L N f(η) = η c(η, η ) ( f(η ) f(η) ) Particles are indistinguishable Out of equilibrium not reversible P η ( ) = probability measures induced on D([0, T ], X N ) with initial condition η at time 0. When the initial condition is distributed according to ν we call it P ν ( ) 3

4 µ N = invariant measure ) P µn (η(t) = η = µ N (η) t Main problems: determine µ N determine its asymptotic behavior when N + in a large deviations regime. 4

5 FIRST EXAMPLE 1-d boundary driven ZERO RANGE Λ N = {1, 2,... N}, η x N 5

6 dynamics on the bulk g : N {0} R + such that g(0) = 0 + suitable growth conditions η η δ x + δ x±1 with rate g(η x ) 2 Reversibility ϕ = ϕ + Always product invariant measures µ N (η) = N x=1 ϕ η x x Z(ϕ x )g(η x )! where g(k)! := g(k)g(k 1)... g(2)g(1) and ϕ solves { ϕx = 1 2 ( ϕx 1 + ϕ x+1 ) ϕ 0 = ϕ, ϕ N+1 = ϕ + 6

7 The equation is equivalent to 1 2 ( ϕx ϕ x+1 ) = constant = j When N is large ϕ x Φ ( x N ), where { Φ = 0 Φ(0) = ϕ, Φ(1) = ϕ + Product invariant measure is very general for Z.R.: for example it holds in any dimension and/or when η η δ x + δ x+1 with rate g(η x )r x η η δ x + δ x 1 with rate g(η x )l x 7

8 LARGE DEVIATIONS {ν n } n N sequence of probability measures on X a Polish metric space satisfies a LDP with rate I if lim sup n + lim inf n + 1 n log ν n(c) inf y C I(y), 1 n log ν n(o) inf y O I(y), C closed O open A random element X n satisfies a LDP if the corresponding law satisfies a LDP P(X n a) e ni(a) A typical situation is ν n w δx, and I(x) 0 with I(x) = 0 x = x 8

9 LD asymptotic for Zero Range η π N (η) = Empirical measure π N (η) := 1 N N x=1 η x δ x N M + ([0, 1]) (Coarse graining) 9

10 When η is distributed according to µ N then π N (η) satisfies a LDP on M + ([0, 1]) endowed with the weak topology Gärtner-Ellis rate function S P (f) := lim N + S(ρ) = sup f 1 N log E µ N { in particular is convex [0,1] ( e N 1 0 fdπ N(η) ) fdρ P (f) } 10

11 P (f) = lim N + and then S(ρ) = = lim N + = lim N + = N log E µ N 1 N N log 1 N 1 0 log Z N x=1 log ( x=1 Φ(u)e f(u) ) Z(Φ(u)) ρ(u) log ψ(ρ(u)) Φ(u) ( e N 1 0 fdπ N(η) ) E µn ( e f ( x N ) η x ) Z (ϕ x e f ( x N )) Z(ϕ x ) du Z(ψ(ρ(u))) log Z(Φ(u)) du if ρ = ρ(u)du and S(ρ) = + otherwise. This means S(ρ) = 1 0 s(ρ(u), u)du 11

12 LDP for Gibbs measures (Comets, Föllmer, Lanford, Olla,...) Λ N = T N, finite range interactions Pressure Gärtner-Ellis H λ (η) = λ P (λ) := s λ0 (α) = sup λ µ λ N (η) = e H λ(η) Z N (λ) x T N η x + J lim N + { αλ x T N η x η x+1 1 N log Z N(λ) ( )} P (λ + λ 0 ) P (λ 0 ) convex. Then π N (η), when η is distributed according to µ λ 0, satisfies LDP with rate N S λ0 (ρ) = { 10 s λ0 (ρ(u))du ρ = ρ(u)du + otherwise 12

13 Second model 1-d boundary driven SEP In the bulk exchange dynamics η η x,x+1 with rate 1, x = 1,..., N 1 where η x,x+1 z = η z if z x, x + 1 η x if z = x + 1 η x+1 ifz = x 13

14 Reversibility : α + = β + and α = β. In this case product invariant measure µ N (η) = N x=1 p ηx (1 p) 1 η x where p = α + α + +α = β + β + +β. A special case: α + = β = 0, (Kingmann 1969!!) µ N (η x1 = 1,... η xk = 1) = (A k x 1 )(A k + 1 x 2 )... (A 1 x k ) (B k)(b k + 1)... (B 1) where 1 x 1 < x 2 < < x k N and A = N β 1, B = α + 1 β + 14

15 from now on α + = (1 α ) = α and β + = (1 β ) = β, α, β [0, 1] Duality where P k j=0 µ N (η x1 = 1,... η xk = 1) = P ( ) left x 1... x k j (α) j (β) k j ( x 1... x k left j ) denotes the probability that of k particles starting at 1 x 1 < x 2 < < x k N and evolving in exclusion, j will be absorbed at 0 and k j at N + 1. Long range correlations Closed expression for 1-marginals solving µ N (Lη x ) = 0, x = 1,..., N µ N (η x ) = ρ where ρ(u) = α + u(β α) ( x N ) 15

16 Closed expression for correlation functions (Spohn 1983) solving µ N (L(η x η y ) ) = 0. For 1 x < y N C N (x, y) : = µ N (η x η y ) µ N (η x )µ N (η y ) = = (α β)2 N 1 1 ( x N, y N where 1 (u, v) is the Green function of the Laplacian on [0, 1] with Dirichelet boundary conditions 1 (u, v) = u(1 v) 0 u v 1 In particular negatively correlated. ) 16

17 Matrix representation (Derrida, Scütz,...) where w N (η) = l µ N (η) = w N(η) η w N(η ) N x=1 (Dη x + E(1 η x )) r where the matrices E, D and the vectors l, r satisfy 1 (DE ED) = D + E 2 1 l (αe + (1 α)d) = l ((1 β)e + βd) r = r 17

18 LDP for the empirical measure when η is distributed according to µ N then π N (η) satisfy a LDP in M + ([0, 1]) with rate where S(ρ) = [ 1 0 h F (ρ) := ρ log ρ F h F (u) (ρ(u)) + log F (u) ρ (u) and F is monotone and solves ] du + (1 ρ) log (1 ρ) (1 F ) F (1 F ) F (F ) 2 + F = ρ F (0) = α, F (1) = β In particular S is not additive S [0,c] (ρ) + S [c,1] (ρ) S [0,1] (ρ) S(ρ) = sup F G(ρ, F ) 18

19 (Shannon) Entropy (Bahadoran, Derrida-Lebowitz-Speer) Let Then where µ N (η) := N x=1 ρ lim N + = lim = N ( x N 1 N 1 N ) ηx ( ( )) x 1 ηx 1 ρ N η η du h(ρ(u)) µ N (η) log µ N (η) µ N (η) log µ N (η) h(ρ) := ρ log ρ + (1 ρ) log(1 ρ) 19

20 Freidlin-Wentzell theory dx ɛ t = b(x ɛ t )dt + ɛdb t X ɛ t Rn, B t = n-dimensional Brownian motion, b = globally attractive vector field (ɛ N 1 ) 20

21 Sample path LDP (Dynamic LDP) where P (X ɛ t x t ; t [0, T ]) e ɛ 1 I [0,T ] (x) I [T1,T 2 ] (x) = 1 2 T2 T 1 ẋ(s) b(x(s)) 2 ds µ ɛ = invariant measure Main Issue µ ɛ satisfies a LDP (static LDP) on R n with rate functional W = QUASIPOTENTIAL 21

22 W (x) = inf T inf I [ T,0] (y) {y(s) : y( T )=x, y(0)=x} 22

23 Hamilton-Jacobi equation I [T1,T 2 ] (x) = T2 T 1 L(ẋ(s), x(s))ds L = Lagrangian H(p, x) = sup y {p y L(y, x)} H = Hamiltonian Then W solves (weakly) the stationary Hamilton- Jacobi equation H( W (x), x) = 0 If b = U (reversible) then W = 2U. In general W is not differentiable (Lagrangian phase transitions) 23

24 Example: 1-d torus (Faggionato-G.) dx ɛ t = b(x ɛ t )dt + ɛdb t W = quasipotential + combinatorial optimization 1 b(u)du = 0 reversible 0 If 1 0 b(u) du 0 U(u) = u 0 b(z)dz W (u) = 2U(u) + c W (u) = S[2U](u) + c S = sunshine transformation 24

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26 Hydrodynamic scaling limit (Exclusion) Initial condition η such that π N (η) ρ 0 (u)du fdπ N(η) N + [0,1] [0,1] f(u)ρ 0(u)du L N N 2 L N Diffusive rescaling π N (η(t)) ρ(u, t)du where ρ t = 1 2 ρ uu ρ(u, 0) = ρ 0 (u) ρ(0, t) = α, ρ(1, t) = β Law of π N (η) converges weakly to δ ρ on D([0, T ], M + ) 26

27 Dynamic LDP (Kipnis-Olla-Varadhan) L N L H N weakly asymmetric perturbation exchange η η x,y with rate ( ) c H {x,y} (η) := c x η y ) H( y {x,y} (η)e(η N,t ) H( N x,t ) H : [0, 1] [0, T ] R, H(0, t) = H(1, t) = 0 c H {x,y} (η) := c {x,y} (η) + O ( 1 N ) dp H η,n dp η,n Graphical construction Upper bound: Chebyshev inequality arguments Lower bound: a relative entropy computation lim N + 1 N Ent ( P H η,n P ) η,n Key step: superexponential replacement Lemma 27

28 The rate functional Variational representation If ρ(u, t) is smooth and far from 0 and 1 I [T1,T 2 ] (ρ) = 1 2 where T2 T 1 dt 1 0 du ρ(u, t)(1 ρ(u, t))h2 u(u, t) { ρt = 1 2 ρ uu (ρ(1 ρ)h u ) u H(0, t) = H(1, t) = 0 if ρ(0, t) = α, ρ(1, t) = β and + otherwise The quasipotential (Bodineau-Giacomin, Farfan): Exclusion processes: the quasipotential associated to the dynamic rate function from hydrodynamic rescaling coincides with LD rate function for the invariant measure 28

29 Generalized Onsager-Machlup symmetry Particle systems satisfying a dynamic and static LDP ) P µn (π N (η(t)) ρ(u, t), t [ T, 0] = P µ N (π N (η(t)) ρ(u, t), t [0, T ] where P µ N ( )= law of the stationary time reversed (adjoint) process. At LD level using Markov property we get e NS(ρ( T )) e NI [ T,0] (ρ) = e NS(ρ(0)) e NI [0,T ] (θρ) where I is the dynamic rate functional of the adjoint process and θ is the time-reversal θρ(u, t) := ρ(u, t) ) 29

30 S(ρ( T )) + I [ T,0] (ρ) = S(ρ(0)) + I [0,T ] (θρ) if ρ( T ) = ρ, since S(ρ) = 0 it becomes I [ T,0] (ρ) = S(ρ(0)) + I [0,T ] (θρ) consequently since I 0 inf T inf I [ T,0] (ρ) S(ρ 0) {ρ:ρ( T )=ρ ρ(0)=ρ 0 } Let ρ such that I (ρ ) = 0. The minimizer for the quasipotential ρ m is such that ρ m = θρ 30

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32 How to compute the quasipotential (Exclusion) Reversible case α = β = c. Rate function for the invariant measure 1 S(ρ) = h c(ρ(u)) d u 0 Hydrodynamic limit ρ associated to P solves the heat equation. We expect ρ m = θρ ρ m t = 1 2 ρm uu Let it holds Γ[ρ] := log S(ρ(T 2 )) S(ρ(T 1 )) = A computation gives ρ 1 ρ log c 1 c T2 T 1 d t 1 0 d uγ[ρ(t)](u)ρ t(u, t) I [ T,0] (ρ) = S(ρ(0)) S(ρ( T )) T d t 1 0 d u(h Γ)2 uρ(1 ρ) 32

33 If ρ( T ) = ρ S(ρ( T )) = 0 I [ T,0] (ρ) S(ρ(0)). Minimizing sequence for any n N ρ m H m = Γ[ρ m ] ρ n (t) = { ρ m (t) t [ n, 0] interpolation t [ n 1, n] 33

34 I [ n 1,0] (ρ n ) = I [ n,0] (ρ m ) + I [ n 1, n] (ρ n ) = S(ρ(0)) S(ρ m ( n)) + ɛ(n) n + S is the quasipotential not reversible α β S(ρ(0)) Γ[ρ] := log ρ 1 ρ log F [ρ] 1 F [ρ] where F [ρ] is the unique monotone solution of F (1 F ) F (F ) 2 + F = ρ F (0) = α, F (1) = β The rate function of µ N is S(ρ) = G(ρ, F [ρ]) A computation S(ρ(T 2 )) S(ρ(T 1 )) = T2 T 1 d t 1 0 d uγ[ρ(t)](u)ρ t(u, t) 34

35 and ds(ρ(t)) dt A computation = δg 1 0 d u [ δg δρ ρ t + δg ] δf F t δρ (ρ, F [ρ]) = Γ[ρ] (ρ, F [ρ]) = 0 δg δf I [ T,0] (ρ) = S(ρ(0)) S(ρ( T )) d t d u(h T 0 Γ)2 uρ(1 ρ) This suggests ρ m H m = Γ[ρ m ] 35

36 The minimizer ρ m solves ) = 2 1 ρm uu (ρ m (1 ρ m )(Γ[ρ m ]) u ρ m t ρ m (u, 0) = ρ 0 (u) and b.c. It is the following coupled differential problem F (1 F ) F ) u u ρ m t = 1 2 ρm uu ( ρ m (1 ρ m ) ρ m (u, 0) = ρ 0 (u) F (1 F ) F uu (F u ) 2 + F = ρm and b.c. A computation shows it is equivalent to F t = 1 2 F uu F (1 F ) F uu (F u ) 2 + F = ρm ρ m (u, 0) = ρ 0 (u) and b.c. u 36

37 KMP model (Kipnis-Marchioro-Presutti) Λ N = {1, 2,... N} and η x R + η x = Energy of the harmonic oscillator located at x Stochastic dynamics in the bulk at rate 1 (η x, η x+1 ) (U(η x +η x+1 ), (1 U)(η x +η x+1 )) U= uniform on [0, 1] 37

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39 at the boundary η 1 U(η 1 + X τl ) L(X τl ) = e τ L u τ L du η N U(η N + X τr ) L(X τr ) = e Reversibility τ L = τ R = τ, product invariant measure µ N (η)dη = N x=1 e η x τ τ dη x If τ L τ R not reversible, no matrix representation τ R u τ R du 39

40 Hydrodynamic diffusive rescaling when L N N 2 L N then π N (η) converges to where I [T1,T 2 ] (ρ) = 1 2 ρ t = 2 1 ρ uu ρ(u, 0) = ρ 0 (u) ρ(0, t) = τ L ρ(1, t) = τ R Dynamic LDP T2 T ρ2 (u, t)h 2 u(u, t) du { ρt = 1 2 ρ uu (ρ 2 H u ) u H(0, t) = H(1, t) = 0 if ρ(0, t) = τ L, ρ(1, t) = τ R and I = + otherwise 40

41 Static LDP When η is distributed according to µ N (η)dη then π N (η) satisfies a LDP with rate S(ρ) = G(ρ, F [ρ]) where G(ρ, F ) := 1 0 K F (u) (ρ(u)) log F (u) ρ (u) du K F (ρ) := ρ F 1 log ρ F and F [ρ] is the unique monotone solution to It holds F 2 F (F ) 2 F = ρ F (0) = τ L F (1) = τ R not convex! S(ρ) = inf F G(ρ, F ) 41

42 A physicist proof Infinite dimensional Hamilton-Jacobi equation for S 1 2 δs δρ, ρ2 δs δρ + δs δρ, ρ = 0 Search for a solution of the form δs δρ = 1 F 1 ρ A few smart integrations by parts give (ρ F ) F 4, ( F 2 F + (ρ F )( F ) 2 ) = 0 if F = F [ρ] H-J is satisfied. If S(ρ) = G(ρ, F [ρ]) then a solution δs δg(ρ, F [ρ]) = δρ δρ = 1 F [ρ] 1 ρ + δg(ρ, F [ρ]) δf [ρ] δf δρ 42

43 contraction principle? (yes) F = hidden temperature profile A toy model for the SNS (U 1,..., U N ) i.i.d. uniform random variables on [τ L, τ R ] (U [1],..., U [N] ) the order statistics and γ N (u)du the corresponding law. Define ν N (η, u)dηdu := γ N (u)du N x=1 e η x u x u x dη x probability measure on [τ L, τ R ] N (R + ) N. The toy measure is ) µ T N (η)dη := ( [τ L,τ R ] N ν N(η, u)du dη a mixture of product of exponentials 43

44 π N (U) := 1 N N x=1 U [x] δ x N When ((U, η) is distributed ) according to ν N (u, η)dudη then π N (U), π N (η) satisfies a LDP with rate ( ) P (π N (U), π N (η)) (F, ρ) It follows by P = P ( ( (π N (U), π N (η)) (F, ρ) π N (U) F ) ( P e NG(ρ,F ) ) π N (η) ρ π N(U) F ) 44

45 ( ( ) P π N (U) F e N ) 1 0 log F (u) ρ (u) (A) ( P π N (η) ρ π N(U) F ) e N 1 0 K F (u) (ρ(u))du (B) (A): i.i.d exponentials conditioned to have fixed sum order statistics of uniforms (B): by LDP for product measures 45

46 An exact result 1 Oscillator N = 1 µ 1 (η 1 )dη 1 = 1 π τ R τ L u e η 1u du dη 1 (τ R u)(u τ L ) It means γ 1 (u)du = arcsine distribution and not uniform. N > 1? 46

47 External fields Stochastic lattice gas η x {0, 1} on T d N c {x,y} (η) = rate of exchange (only n.n.) Detailed balance e H(η) c {x,y} (η) = e H(ηx,y) c {x,y} (η x,y ) H = Hamiltonian: translation invariant finite range. High temperature regime. µ N (η) = e H(η) Z N reversible, also the canonical. Switch on a vector field E, function on ordered edges discrete vector field E xy = E yx 47

48 c c E c E {x,y} (η) := c {x,y} (η)ee xy(η x η y ) E xy (η x η y ) = work done by the field Microscopic analysis 1) E is a gradient vector field E xy = 1 2 (V y V x ) µ E N (η) = e HE (η) H E (η) = H(η) ZN E V x η x reversible! x T d N 48

49 2) E constant vector field E x,x+ei = E i, i = 1,... d 2A) c determine a gradient model: exists a cylindric function h(η) such that average current through the edge (x, y) c {x,y} (η)(η x η y ) = τ y h(η) τ x h(η) Exclusion, zero range, KMP, all gradient µ E N (η) = µ N(η) not reversible (ex: asymmetric exclusion has Bernoulli product as invariant measure) 49

50 2B) c is not gradient µ E N =? Conjectures (Garrido-Lebowitz-Maes-Spohn) i) For any density ρ there exists a unique translation invariant thermodynamic limit µ Ē ρ ii) d = 1 then µ Ē ρ exponentially has pair correlations decaying iii) d > 1 pair correlations decay as a power law 3) Divergence free fields G xy := e E xy e E yx divergence free discrete vector field + gradient model µ E N (η) = µ N(η) not reversible 50

51 Macroscopic analysis (Varhadan-Yau) hydrodynamic limit non gradient models, diffusive scaling; (Quastel) dynamic LDP (Bertini-Faggionato-G) hydrodynamic limit and dynamic LDP for weakly asymmetric models E xy E(u) N Hydrodynamic equation ρ t = (D(ρ) ρ) (σ(ρ)e) D(ρ) = diffusion matrix, variational representation σ(ρ) = mobility D(ρ) = σ(ρ)s (ρ) Einstein relation 51

52 1) If E(u) = 1 2 U(u) is gradient, when η is distributed according to µ E N π N(η) satisfies a LDP with rate S E (ρ) = T d s V (u) (ρ(u)) du minimizer for the quasipotential ρ m = θρ E reversible, Onsager-Machlup symmetry 2) and 3) When η is distributed according to µ E N then π N(η) satisfy a LDP with rate S E (ρ) = S(ρ) minimizer for the quasipotential ρ m = θρ E not reversible, generalized Onsager-Machlup symmetry For the case 3) D and σ must be multiple of the identity 52

53 Lagrangian phase transitions (diffusions) Critical trajectories for computing W the quasipotential ( ) d L = L dt ẋ x Euler-Lagrange equations ( x(t), p(t) := L ) (x(t), ẋ(t)) Hamilton Equations ẋ 53

54 Phase space (x, p) 54

55 M u = unstable manifold M s = stable manifold M u, M s {H = 0} and are Lagrangian manifolds pdq = 0 γ M u, M u γ (x, p) M u W (x, p) = Prepotential (Day) W (x, p) := γ pdq 55

56 Quasipotential from prepotential (Day) W (x) = inf W (x, p) p:(x,p) M u 56

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59 Superdifferential is not empty no convexity, instability configuration space 59

60 Lagrangian phase transition on WASEP 1-d boundary driven exclusion weakly perturbed c {x,x+1} c E {x,x+1} := e E N (η x η x+1 ) constant vector field Diffusive rescaling, Hydrodynamic equation { ρt = ρ uu E(ρ(1 ρ)) u ρ(0, t) = α, ρ(1, t) = β α < β E > 0 push push E > h (β) h (α) recall h(ρ) = ρ log ρ + (1 ρ) log(1 ρ) 60

61 Dynamic LDP Lagrangian structure Hamiltonian structure prepotential Hamilton equations { ρt = ρ uu (ρ(1 ρ)(e + 2π u ) u ) π t = π uu + (2ρ 1)(π 2 u Eπ u ) after a symplectic transformation M u = { (φ, ρ) : where 0 < φ u < E. W E = 1 0 h(ρ)+h φ uu φ u (E φ u ) e φ = ρ ( φu E } ) +(1 ρ)φ log(1+e φ ) du 61

62 S E (ρ) = W E (ρ) = inf W E (φ, ρ) φ:(φ,ρ) M u Limiting cases E h (β) h (α) := E 0 then! φ affine { 0 < φu < h (β) h (α) φ(0) = h (α), φ(1) = h (β) M u = { ( φ, ρ) } is a graph W E0 (ρ) = W E0 ( φ, ρ) product of Bernouilli E E 0 no Lagrange phase transition 62

63 where E + W E W + 1 W + (φ, ρ) := 0 h(ρ)+(1 ρ)φ log(1+eφ ) du φ CADLAG increasing + b.c. W + is concave in φ infimum can be restricted to extremal elements (step functions) 63

64 inf φ inf z [0,1] 1 + W + (φ, ρ) = { z inf z [0,1] W + (φ z, ρ) = 0 h(ρ) + h (α)(1 ρ) log(1 + e h (α) ) z h(ρ) + h (β)(1 ρ) log(1 + e h (β) ) LDP for TASEP (Derrida-Lebowitz-Speer) } α < β, E > 0 not weak 64

65 For a density profile ρ such that exists A inf z W + (φ z, ρ) = W + (φ z, ρ) = W + (φ z +, ρ) Lagrangian phase transition, by a perturbative argument it happens also for E large but finite 65

66 2-class TASEP Λ N = T N, η x {0, 1, 2} η x = two conserved quantities 0 empty site 1 first class particle 2 second class particle Invariant measure algorithmic combinatorial representation (O. Angel, Ferrari-Martin) 66

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68 N 1 first class particles, N 2 second class particles ξ i x {0, 1} configurations η (ξ 1, ξ 2 ) ξ 1 ξ 2 ξ 1 x = 1 if there is a first class particle at x ξ 2 x = 1 if there is a particle at x x T N ξ 1 x = N 1 x T N ξ 2 x = N 1 + N 2 The invariant measure Generate ξ 1 with N 1 particles uniformly Independently generate ξ 2 with N 1 + N 2 particles uniformly ( ξ 1, ξ 2 ) collapsing procedure (ξ 1, ξ 2 ) distributed according to µ N 68

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70 flux across (x, x + 1) J(x) = sup y T N z [y,x] ( ξ z 1 ξ z 2 ) Collapsed configuration (ξ 1, ξ 2 ) = C( ξ 1, ξ 2 ). + We have ξ 2 = C( ξ 2 ) = ξ 2. ξ 1 = C( ξ 1 ) is defined by z [a,b] ξ 1 z = z [a,b] ξ 1 z + J(a 1) J(b) continuity equation 70

71 Collapsing procedure for positive measures on T ρ 1, ρ 2 M + (T) such that: where [0,1] d ρ 1 [0,1] d ρ 2 ( ρ 1, ρ 2 ) C( ρ 1, ρ 2 ) = (ρ 1, ρ 2 ) (a,b] dρ 1 = J(u) := sup v Note that ρ 1 ρ 2 Definition [ (a,b] d ρ 1 + J(a) J(b) (v,u] d ρ 1 ρ 2 = ρ 2 (v,u] d ρ 2 ] + 71

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73 (G) when N + and N 1 N α, N 2 N β ( ) π N ( ξ 1 ), π N ( ξ 2 ) S( ρ 1, ρ 2 ) = where [0,1] ρ 1(u)du = α, satisfies LDP with rate [0,1] h α( ρ 1 (u))du+ [0,1] h α+β( ρ 2 (u))du [0,1] ρ 2(u)du = α + β Generalized contraction principle ( ) π N (ξ 1 ), π N (ξ 2 ) satisfies LDP with rate S(ρ 1, ρ 2 ) = Not convex! inf { ρ 1 : C( ρ 1 )=ρ 1 } S( ρ 1, ρ 2 ) = S(ρ 1, ρ 2) 73

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