Hydrostatics, Statical and Dynamical Large Deviations of Boundary Driven Gradient Symmetric Exclusion Processes

Transcrição

1 Instituto acional de Matemática Pura e Aplicada Hydrostatics, Statical and Dynamical Large Deviations of Boundary Driven Gradient Symmetric Exclusion Processes Jonathan Farfan Vargas Tese apresentada para obtenção do título de Doutor em Ciências Orientador: Claudio Landim Rio de Janeiro Dezembro, 28

2 Resumo os últimos anos princípios de grandes desvios estáticos e dinâmicos de sistemas de partículas interagentes com bordos estocásticos têm sido muito estudados como um primeiro passo para o entendimento de estados estacionários fora do equilíbrio. este trabalho consideramos Processos de exclusão simêtrico gradiente com bordos estocásticos em cualquer dimensão e estudamos neste contexto os seguintes problemas. Primeiro apresentamos uma prova da hidrostática baseada no ite hidrodinâmico e o fato que o perfil estacionario é um atrator global da equação hidrodinâmica. Também sâo provados o ite hidrodinâmico e a lei de Fick. Depois apresentamos uma prova do principio dos grandes desvios dinâmico para a medida empirica. A prova apresentada aqui é mais simples do que a usual ja que ao invez de aproximarmos trajetorias com funçã custo finita por trajetorias suaves, aproximamos o campo externo associado a ele com campos externos suaves e provamos que as soluçoes fracas da equação hidrodinâmica com estes campos externos aproximam a trajetoria original. Isto simplifica consideravelmente a prova dos grandes desvios dinâmicos. Por último, apresentamos uma prova do principio de grandes desvios para a medida estacionaria. Mais precisamente, seguindo a estratégia de Freidlin e Wentzell provamos que a medida estacionária de nosso sistema satisfaz um princípio de grandes desvios com função custo dada pelo quase potencial da função custo dinâmica.

3 A mi esposa Vannesa y mis hijas Ximena y Araceli. i

4 Agradecimentos Gostaria de aproveitar esta oportunidade para agradecer às pessoas e entidades que, de uma ou outra forma, ajudaram no desenvolvimento deste trabalho. Primeiramente gostaria de agradecer ao IMPA como instituição e grupo humano, e em especial ao professor Karl Otto Stohr cujos ensinamentos contribuiram profundamente na minha formação acadêmica. Agradeço ao CPq e à CAPES pelo apóio económico recebido. Gostaria também de agradecer ao meu orientador, Prof. Claudio Landim, pelo apoio que me deu durante todos estes anos, pela disposição que teve para ouvir minhas perguntas e idéias, pela escolha dos problemas apresentados neste trabalho, e por ter me permitido trabalhar com liberdade e tranquilidade. Agradeço aos professores Glauco Valle, Roberto Imbuzeiro, Pablo Ferrari e Felipe Linares por terem aceito participar da banca, e pelas sugestões feitas para melhorar a redação desta tese. Ao professor Cesar Camacho pela confiança depositada em mim e apóio que me deu durante todos estes anos tanto no setor professional quanto no pessoal. Ao meu amigo e professor Milton Jara pelas valiosas discussões matemáticas. Aos meu amigos Jimmy Santamaria, Acir Carlos, Miguel Orrillo, Arturo Fernandez, Maycol Falla, a sua mãe Dona Sandra e seu irmão caçula Edson os quais me fizeram sentir em familia nos anos em que estive afastado da minha. Aos meus amigos de toda a vida Johel Beltran, Jesus Zapata, Rudy Rosas, Fidel Jimenez e Liz Vivas, principais artífices do meu retorno á matemática. Aos professores do IMCA, em especial ao meu velho amigo Wilfredo Sosa e a Percy Fernández sem o apoio dos quais não teria sido possível meu retorno. Quiero agradecer de manera especial a mis padres por haberme todo el amor y la disciplina que me brindaron. A mis hermanos que siempre me alentaron para seguir adelante. Finalmente, agradezco a los tres amores de mi vida, mi mujer Vannesa, por la infinita paciencia que tuvo conmigo y el invalorable apoio que me brindo a lo largo de estos años, y mis hijas Ximena y Araceli por ser mi mayor fuente de inspiración. ii

5 Contents Introduction otations and Results 5. Boundary Driven Exclusion Process Hydrostatics Dynamical Large Deviations Statical Large Deviations Hydrodynamics and Hydrostatics 2. Hydrodynamics, Hydrostatics and Fick s Law Proofs of Propositions 2..2, 2..3 and Dynamical Large Deviations The Dynamical Rate Function I T ( γ)-density Large Deviations Superexponential estimates Energy estimates Upper bound Lower bound Statical Large Deviations The Functional I T The Statical Rate Function Large Deviations Lower bound Upper bound Weak Solutions Existence and Uniqueness Energy Estimates iii

6 Introduction Statical and dynamical large deviations principles of boundary driven interacting particles systems has attracted attention recently as a first step in the understanding of nonequilibrium thermodynamics (cf. [5, 7, 9] and references therein). One of the main dificulties is that in general the stationary measure is not explicitly known and moreover it presents long range correlations (cf. [25]). This work has three purposes. First, inspired by the dynamical approach to stationary large deviations, introduced by Bertini et al. in the context of boundary driven interacting particles systems [3], we present a proof of the hydrostatics based on the hydrodynamic behaviour of the system and on the fact that the stationary profile is a global attractor of the hydrodynamic equation. More precisely, if ρ represents the stationary density profile and π the empirical measure, to prove that π converges to ρ under the stationary state µ ss, we first prove the hydrodynamic it stated as follows. If we start from an initial configuration which has a density profile γ, on the diffusive scale the empirical measure converges to an absolutely continuous measure, π(t, du) = ρ(t, u)du, whose density ρ is the solution of the parabolic equation t ρ = (/2) D(ρ) ρ, ρ(, ) = γ( ), ρ(t, ) = b( ) on Γ, where D is the diffusivity of the system, the gradient, b is the boundary condition imposed by the stochastic dynamics and Γ is the boundary of the domain in which the particles evolve. Since for all initial profile γ, the solution ρ t is bounded above, resp. below, by the solution with initial condition equal to, resp., and since these solutions converge, as t, to the stationary profile ρ, hydrostatics follows from the hydrodynamics and the weak compactness of the space of measures. The second contribution of this work is a simplification of the proof of the dynamical large deviations 2 of the empirical measure. The original proof [8,, 6] relies on the convexity of the rate functional, a very special property only fulfilled by very few interacting particle systems as the symmetric simple exclusion process. The extension to general processes [22, 23, 6] is relatively technical. The main difficulty appears in the proof of the lower bound where one needs to show that any trajectory λ t, t T, with finite rate function, I T (λ) <, can be approximated by a sequence of smooth trajectories λ n : n } such that λ n λ and I T (λ n ) I T (λ). (..) This is part of one work in partnership with Claudio Landim and Mustapha Mourragui. 2 This result is also part of the same work mentioned above

7 This property is proved by approximating in several steps a general trajectory λ by a sequence of profiles, smoother at each step, the main ingredient being the regularizing effect of the hydrodynamic equation. This part of the proof is quite elaborate and relies on properties of the Green kernel associated to the second order differential operator. We propose here a simpler proof. It is well known that a path λ with finite rate function may be obtained from the hydrodynamical path through an external field. More precisely, if I T (λ) <, there exists H such that I T (λ) = 2 dt σ(λ t ) [ H t ] 2 dx, where σ is the mobility of the system and H is related to λ by the equation t λ (/2) D(λ) λ = [σ(λ) H t ] (..2) H(t, ) = at the boundary. This is an elliptic equation for the unknown function H for each t. ote that the left hand side of the first equation is the hydrodynamical equation. Instead of approximating λ by a sequence of smooth trajectories, we show that approximating H by a sequence of smooth functions, the corresponding smooth solutions of (..2) converge in the sense (..) to λ. This approach, closer to the original one, simplifies considerably the proof of the hydrodynamical large deviations. The third contribuition of this work is the proof of a large deviation principle of the empirical density under the invariant measure. More precisely, we prove that the quasi potential of the dynamical rate function is the large deviation functional of the stationary state. We follow closely the approach given in [8]. In fact, the arguments presented in there can be adapted modulo technical dificulties to our context. However there is a case not considered in the proof of the upper bound in [8], which we describe in detail in the following. For a fixed closed set C in the weak topology not containing the stationary density ρ, small neighborhoods V δ (which depends on a parameter δ > ) of ρ are considered. By following the Freidlin and Wentzell strategy, the proof of the upper bound is reduced to prove that the minimal quasi-potential of densities in C can be estimated from above by the minimal dynamical rate function of trajectories which start at V δ and touch C before a time T = T δ. At this moment, in [8] it is supposed that the time T = T δ is fixed and then, by a direct aplication of the dynamical large deviation upper bound, the desired result is obtained. The same argument still works if we assume the existence of a sequence of parameters δ n with the sequence of times T δn bounded. The problem here, is that such bounded sequence doesn t necessarily exist. Moreover, by the construction of such times T δ, it is expected that T δ as δ. In our context, to solve this missing case, we first prove that long trajectories which have their dynamical rate functions uniformly bounded has to be close in some moment to the stationary density ρ in the L 2 metric, and then we prove that the quasi potential is continuous at the stationary density ρ in the L 2 topology. 2

8 In this way, we fullfill the gap in [8] described above and extend their result for a broader class of models. Finally, as a consequence of these facts we obtain a direct proof of the lower semicontinuity of the quasi potential. In the context of one dimensional boundary driven SSEP, the lower semicontinuity of the quasi potential was obtained indirectly by using its exact formulation given in [, 4]. The organization of this thesis is as follows.. otations and Results Here we introduce the boundary driven gradient symmetric exclusion processes (Section.) and establish some notations in order to describe in detail the three results mentionated above: hydrostatics (Section.2), dynamical large deviations (Section.3), and statical large deviations (Section.4). 2. Hydrodynamics and Hydrostatics In this chapter we prove hydrostatics based on the hydrodynamic behaviour of the system and on the fact that the stationary profile is a global attractor of the hydrodynamic equation. More presicely, in Section 2. we present a proof for the hydrostatics and Fick s law by supposing the hydrodynamic behavior. A proof for the hydrodynamic behavior can be found in [4] but for the sake of completeness we present in Section 2.2 a detailed proof based on the entropy method. 3. Dynamical Large Deviations Here we obtain a dynamical large deviation principle for the empirical measure. We start by investigating the dynamical rate function I T ( γ) in Section 3.. The main result obtained here is the fact that the dynamical rate function has compact level sets. Then, in Section 3.2, we prove I T ( γ) density, which means that any trajectory λ t, t T, with finite rate function, I T (λ γ) <, can be approximated by a sequence of smooth trajectories λ n : n } such that λ n λ and I T (λ n γ) I T (λ γ). This is fundamental for the obtention of the lower bound. In Section 3.3, we prove large deviation upper and lower bound. The last one is obtained by the usual arguments (cf. Chapter in [6]) and the I T density proved in the last section. To prove the upper bound,we have to take care of some additional technical dificulties, the first one is the fact that the invariant measure is not explicitly known which difficulties the obtention of superexponential estimates, the second one is the necessity of energy estimates in order to prove that trajectories with infinite energy are negligible in the context of large deviations, and the last one is that we are working with the empirical measure instead of (as usual) the empirical density. 4. Statical Large Deviations In this chapter we prove a large deviation principle for the stationary measure. More precisely, Following the Freidling and Wentzell [5] strategy and more closely the article [8], we prove that the large deviation functional for 3

9 the stationary measure is given by the quasi potential of the dynamical rate function. In Section 4. we introduce the functional I T closely related to the dynamical rate function I T ( γ) and prove that trajectories which stays a long time far away from the stationary state ρ pays a nonnegligible cost. In Section 4.2 we study some properties of the quasi potential. The first main result obtained here is the continuity of the quasi potential at the stationary state ρ in the L 2 topology. The second one is a direct proof of the lower semicontinuity of the quasi potential. Finally, in Section 4.3 we prove large deviations lower and upper bounds. The first one is an inmediate consequence of the hydrostatic result and the dynamical large deviation lower bound. To prove the upper bound, we proceed as in [8] and fulfill the gap in there mentioned above by using the results developing in the previous sections. 5. Weak Solutions Finally, we study weak solutions of the hydrodynamic equation, of its stationary solutions and of the equation with external field (..2). These results are essential in the derivation of many of the results of the previous chapters. However, we have postponed their proofs until here because they are naturally expected for weak solutions of quasilinear parabolic equations. In Section 5. we establish existence and uniqueness of weak solutions as well as monotonicity and uniformly infinitely propagation of speed. In Section 5.2 we establish energy estimates for weak solutions, which is one of the mai ingredients in the proof of the results in Chapter 4. 4

10 Chapter otations and Results. Boundary Driven Exclusion Process Fix a positive integer d 2. Denote by the open set (, ) T d, where T k is the k-dimensional torus [, ) k, and by Γ the boundary of : Γ = (u,..., u d ) [, ] T d : u = ±}. For an open subset Λ of R T d, C m (Λ), m +, stands for the space of m-continuously differentiable real functions defined on Λ. Fix a positive function b : Γ R +. Assume that there exists a neighbourhood V of and a smooth function β : V (, ) in C 2 (V ) such that β is bounded below by a strictly positive constant, bounded above by a constant smaller than and such that the restriction of β to Γ is equal to b. For an integer, denote by T d =,..., } d, the discrete (d )-dimensional torus of length. Let = +,..., } T d be the cylinder in Z d of length 2 and basis T d and let Γ = (x,..., x d ) Z T d x = ±( )} be the boundary of. The elements of are denoted by letters x, y and the elements of by the letters u, v. We consider boundary driven symmetric exclusion processes on. A configuration is described as an element η in X =, }, where η(x) = (resp. η(x) = ) if site x is occupied (resp. vacant) for the configuration η. At the boundary, particles are created and removed in order for the local density to agree with the given density profile b. The infinitesimal generator of this Markov process can be decomposed in two pieces: L = L, + L,b, where L, corresponds to the bulk dynamics and L,b to the boundary dynamics. The action of the generator L, on functions f : X R is given by ( L, f ) d (η) = r x,x+ei (η) [ f(η x,x+e i ) f(η) ], i= x where (e,..., e d ) stands for the canonical basis of R d and where the second sum is performed over all x Z d such that x, x + e i. For x, y, η x,y is the configuration obtained from η by exchanging the occupations variables 5

11 η(x) and η(y): η(y) if z = x, η x,y (z) = η(x) if z = y, η(z) if z x, y. For a > /2, the rate functions r x,x+ei (η) are given by r x,x+ei (η) = + a η(x e i ) + η(x + 2e i ) } if x e i, x + 2e i belongs to. At the boundary, the rates are defined as follows. Let ˇx = (x 2,, x d ) T d. Then, r ( +,ˇx),( +2,ˇx) (η) = + a η( + 3, ˇx) + b(, ˇx/) }, r ( 2,ˇx),(,ˇx) (η) = + a η( 3, ˇx) + b(, ˇx/) }. The non-conservative boundary dynamics can be described as follows. For any function f : X R, (L,b f) (η) = C b [ (x, η) f(η x ) f(η) ], x Γ where η x is the configuration obtained from η by flipping the occupation variable at site x: η x η(z) if z x (z) = η(x) if z = x and the rates C b (x, ) are chosen in order for the Bernoulli measure with density b( ) to be reversible for the flipping dynamics restricted to this site: C b( ( +, ˇx), η ) = η( +, ˇx) [ b(, ˇx/) ] + [ η( +, ˇx) ] b(, ˇx/), C b( (, ˇx), η ) = η(, ˇx) [ b(, ˇx/) ] + [ η(, ˇx) ] b(, ˇx/), where ˇx = (x 2,, x d ) T d, as above. Denote by η t = ηt : t } the Markov process associated to the generator L speeded up by 2. For a smooth function ρ : (, ), let νρ( ) be the Bernoulli product measure on X with marginals given by νρ( ) (η(x) = ) = ρ(x/). It is easy to see that the Bernoulli product measure associated to any constant function is invariant for the process with generator L,. Moreover, if b( ) b for some constant b then the Bernoulli product measure associated to the constant density b is reversible for the full dynamics L..2 Hydrostatics Denote by µ ss the unique stationary state of the irreducible Markov process η t : t }. We examine in Section 2. the asymptotic behavior of the empirical measure under the stationary state µ ss. 6

12 Let M = M() be the space of positive measures on with total mass bounded by 2 endowed with the weak topology. For each configuration η, denote by π = π (η) the positive measure obtained by assigning mass d to each particle of η : π = d x η(x) δ x/, where δ u is the Dirac measure concentrated on u. For a measure ϑ in M and a continuous function G : R, denote by ϑ, G the integral of G with respect to ϑ: ϑ, G = G(u) ϑ(du). To define rigorously the quasi-linear elliptic problem the empirical measure is expected to solve, we need to introduce some Sobolev spaces. Let L 2 () be the Hilbert space of functions G : C such that G(u) 2 du < equipped with the inner product G, J 2 = G(u) J(u) du, where, for z C, z is the complex conjugate of z and z 2 = z z. The norm of L 2 () is denoted by 2. Let H () be the Sobolev space of functions G with generalized derivatives u G,..., ud G in L 2 (). H () endowed with the scalar product,,2, defined by d G, J,2 = G, J 2 + uj G, uj J 2, is a Hilbert space. The corresponding norm is denoted by,2. For each G in H () we denote by G its generalized gradient: G = ( u G,..., ud G). Let = [, ] T d and denote by C m () (resp. Cc m ()), m +, the space of m-continuously differentiable real functions defined on which vanish at the boundary Γ (resp. with compact support in ). Let ϕ : [, ] R + be given by ϕ(r) = r(+ar) and let be the Euclidean norm: (v,..., v d ) 2 =. A function ρ : [, ] is said to be a weak solution of the elliptic i d v2 i boundary value problem if (S) ρ belongs to H (): j= ϕ(ρ) = on, ρ = b on Γ, ( ) (S2) For every function G in C 2, ( ) ( ) G (u) ϕ ρ(u) du = ρ(u) 2 du <. Γ ϕ(b(u)) n (u) ( u G)(u)dS, (.2.) where n=(n,..., n d ) stands for the outward unit normal vector to the boundary surface Γ and ds for an element of surface on Γ. 7

13 We prove in Section 5. existence and uniqueness of weak solutions of (.2.). The first main result of this work establishes a law of large number for the empirical measure under µ ss. Denote by E µ the expectation with respect to a probability measure µ. Theorem.2.. For any continuous function G : R, [ ] π, G G(u) ρ(u)du =, Eµ ss where ρ(u) is the unique weak solution of (.2.). Denote by Γ, Γ + the left and right boundary of : Γ ± = (u,..., u d ) u = ±} and denote by W x,x+ei, x, x+e i, the instantaneous current over the bond (x, x + e i ). This is the rate at which a particle jumps from x to x + e i minus the rate at which a particle jumps from x + e i to x. A simple computation shows that W x,x+ei (η) = τ x+ei h i (η) τ x h i (η), provided x e i and x + 2e i belong to. Here, h i (η) = η() + aη()[η( e i ) + η(e i )] η( e i )η(e i )}. Furthermore, if x =, W x e,x = η(x e ) η(x) } + a η(x 2e ) + a b((x + e )/) } and if x = +, W x,x+e = η(x) η(x + e ) } + a η(x + 2e ) + a b((x e )/) }. Theorem.2.2. (Fick s law) Fix < u <. Then, [ ] 2 Eµ ss d W ([u],y),([u]+,y) y T d = ϕ(b(v)) S(dv) ϕ(b(v)) S(dv). Γ Γ + Remark.2.3. We could have considered different bulk dynamics. The important feature used here to avoid painful arguments is that the process is gradient, which means that the currents can be written as the difference of a local function and its translation..3 Dynamical Large Deviations Fix T >. Let M be the subset of M of all absolutely continuous measures with respect to the Lebesgue measure with positive density bounded by : M = ϑ M : ϑ(du) = ρ(u)du and ρ(u) a.e. }, 8

14 and let D([, T ], M) be the set of right continuous with left its trajectories π : [, T ] M, endowed with the Skorohod topology. M is a closed subset of M and D([, T ], M ) is a closed subset of D([, T ], M). Let T = (, T ) and T = [, T ]. For m, n +, denote by C m,n ( T ) the space of functions G = G t (u) : T R with m continuous derivatives in time and n continuous derivatives in space. We also denote by C m,n ( T ) (resp. Cc ( T )) the set of functions in C m,n ( T ) (resp. C, ( T )) which vanish at [, T ] Γ (resp. with compact support in T ). Let the energy Q T : D([, T ], M) [, + ] be given by Q T (π) = d sup i= G Cc ( T ) 2 } π t, ui G t dt dt G(t, u) 2 du. For each G C,2 ( T ) and each measurable function γ : [, ], let Ĵ G = ĴG,γ,T : D([, T ], M ) R be the functional given by Ĵ G (π) = π T, G T γ, G ϕ(ρ t ), G t dt + π t, t G t dt dt dt ϕ(b) u G ds Γ 2 Γ + ϕ(b) u G ds σ(ρ t ), G t 2 dt, where σ(r) = 2r( r)( + 2ar) is the mobility and π t (du) = ρ t (u)du. Define J G = J G,γ,T : D([, T ], M) R by J G (π) = ĴG (π) if π D([, T ], M ), + otherwise. We define the rate functional I T ( γ) : D([, T ], M) [, + ] as sup JG (π) } if Q T (π) <, I T (π γ) = G C,2 ( T ) + otherwise. We are no ready to state our second main result. Theorem.3.. Fix T > and a measurable function ρ : [, ]. Consider a sequence η of configurations in X associated to ρ in the sense that: π (η ), G = G(u)ρ (u) du for every continuous function G : R. Then, the measure Q η = P η (π ) on D([, T ], M) satisfies a large deviation principle with speed d and rate function I T ( ρ ). amely, for each closed set C D([, T ], M), d log Q η (C) inf I T (π ρ ) π C 9

15 and for each open set O D([, T ], M), d log Q η (O) inf I T (π ρ ). π O Moreover, the rate function I T ( ρ ) is lower semicontinuous and has compact level sets..4 Statical Large Deviations Let us introduce P = µ ss (π ), which is a probability measure on M and describes the behavior of the empirical measure under the invariant measure. Let ρ : [, ] be the weak solution of (.2.). Following [3], [5], we define V : M [, + ] as the quasi potential for the dynamical rate function I T ( ρ). V (ϑ) = inf I T (π ρ) : T >, π D([, T ], M) and π T = ϑ}. It is clear that for the measure ϑ(du) = ρ(u)du we have that V (ϑ) =. We will prove in Section 3. that if I T (π ρ) is finite then π belongs to C([, T ], M ). Therefore we may restrict the infimum in the definition of V (ϑ) to paths in C([, T ], M ) and if V (ϑ) is finite, ϑ belongs to M. Reciprocally, we will see in Section 4.2 that V is bounded on M. The last main result of this work establishes a large deviation principle for the invariant measure. Theorem.4.. The measure P satisfies a large deviation principle on M with speed d and lower semicontinuous rate function V. amely, for each closed set C M and each open set O M, d log P (C) inf ϑ C V (ϑ), d log P (O) inf ϑ O V (ϑ).

16 Chapter 2 Hydrodynamics and Hydrostatics 2. Hydrodynamics, Hydrostatics and Fick s Law We prove in this section Theorem.2.. The idea is to couple three copies of the process, the first one starting from the configuration with all sites empty, the second one starting from the stationary state and the third one from the configuration with all sites occupied. The hydrodynamic it states that the empirical measure of the first and third copies converge to the solution of the initial boundary value problem (2..) with initial condition equal to and. Denote these solutions by ρ t, ρ t, respectively. In turn, the empirical measure of the second copy converges to the solution of the same boundary value problem, denoted by ρ t, with an unknown initial condition. Since all solutions are bounded below by ρ and bounded above by ρ, and since ρ j converges to a profile ρ as t, ρ t also converges to this profile. However, since the second copy starts from the stationary state, the distribution of its empirical measure is independent of time. Hence, as ρ t converges to ρ, ρ = ρ. As we shall see in the proof, this argument does not require attractiveness of the underlying interacting particle system. This approach has been followed in [2] to prove hydrostatics for interacting particles systems with Kac interaction and random potential. We first describe the hydrodynamic behavior. Fix T > and a profile ρ : [, ]. A measurable function ρ : T [, ] is said to be a weak solution of the initial boundary value problem in the layer [, T ] if t ρ = ϕ ( ρ ), ρ(, ) = ρ ( ), ρ(t, ) Γ = b( ) for t T, (H) ρ belongs to L ( 2 [, T ], H () ) : ( ) ds ρ(s, u) 2 du < ; (2..)

17 (H2) For every function G = G t (u) in C,2 ( T ), GT (u)ρ(t, u) G (u)ρ (u) } du ds = ds du ( G s )(u)ϕ ( ρ(s, u) ) ds Γ du ( s G s )(u)ρ(s, u) ϕ(b(u))n (u)( u G s (u))ds, where n=(n,..., n d ) stands for the outward unit normal vector to the boundary surface Γ and ds for an element of surface on Γ. We prove in Section 5. existence and uniqueness of weak solutions of (2..). For a measure µ on X, denote by P µ = P µ the probability measure on the path space D(R +, X ) corresponding to the Markov process η t : t } with generator 2 L starting from µ, and by E µ expectation with respect to P µ. Recall the definition of the empirical measure π and let πt = π (η t ): πt = d η t (x) δ x/. x Theorem 2... Fix a profile ρ : (, ). Let µ be a sequence of measures on X associated to ρ in the sense that : } π, G G(u)ρ (u) du > δ =, (2..2) µ for every continuous function G : R and every δ >. Then, for every t >, } P µ πt, G G(u)ρ(t, u) du > δ =, where ρ(t, u) is the unique weak solution of (2..). A proof of this result can be found in [4]. Denote by Q ss the probability measure on the Skorohod space D([, T ], M) induced by the stationary measure µ ss and the process π (η t ) : t T }. ote that, in contrast with the usual set-up of hydrodynamics, we do not know that the empirical measure at time converges. We can not prove, in particular, that the sequence Q ss converges, but only that this sequence is tight and that all it points are concentrated on weak solution of the hydrodynamic equation for some unknown initial profile. We first show that the sequence of probability measures Q ss : } is weakly relatively compact: Proposition The sequence Q ss, } is tight and all its it points Q ss are concentrated on absolutely continuous paths π(t, du) = ρ(t, u)du whose density ρ is positive and bounded above by : } Q ss π : π(t, du) = ρ(t, u)du, for t T =, } π : ρ(t, u), for (t, u) T =. Q ss 2

18 The proof of this statement is similar to the one of Proposition 3.2 in [9]. Actually, the proof is even simpler because the model considered here is gradient. The next two propositions show that all it points of the sequence Q ss : } are concentrated on absolutely continuous measures π(t, du) = ρ(t, u)du whose density ρ are weak solution of (2..) in the layer [, T ]. Denote by A T D ( [, T ], M ) the set of trajectories π(t, du) = ρ(t, u)du : t T } whose density ρ satisfies condition (H2). Proposition All it points Q ss of the sequence Q ss, > } are concentrated on paths π(t, du) = ρ(t, u)du in A T : Q ssa T } =. The proof of this proposition is similar to the one of Proposition 3.3 in [9]. ext result implies that every it point Q ss of the sequence Q ss, > } is concentrated on paths whose density ρ belongs to L 2 ([, T ], H ()) : Proposition Let Q ss be a it point of the sequence Q ss, > }. Then, [ ( E Q ss ds ρ(s, u) du) ] 2 <. The proof of this proposition is similar to the one of Lemma A.. in [7]. We are now ready to prove the first main result of this work. Proof of Theorem.2.. Fix a continuous function G : R. We claim that [ π, G ρ(u)du, G ] =. Eµ ss ote that the expectations are bounded. Consider a subsequence k along which the left hand side converges. It is enough to prove that the it vanishes. Fix T >. Since µ ss is stationary, by definition of Q k ss, [ E µ k ss π, G ρ(u)du, G ] [ = Q k ss π T, G ρ(u)du, G ]. By Proposition 2..2, there is a it point Q ss of Q k ss : k }. Since the expression inside the expectation is bounded, by Propositions 2..3 and 2..4, k Q k ss [ π T, G ρ(u)du, G ] = Q ss [ π T, G ρ(u)du, G S T }] G Qss[ ρ(t, ) ρ( ) ] S T }, where stands for the L () norm and where S T stands for the subset of D([, T ], M ) consisting on those trajectories π(t, du) = ρ(t, u)du : t T } whose density ρ is a weak solution of (2..). Denote by ρ (, ) (resp. ρ (, )) the weak solution of the boundary value problem (2..) with initial condition ρ(, ) (resp. ρ(, ) ). By Lemma 5..4, each profile ρ in A T, including the stationary profile ρ, is bounded below by ρ and above by ρ. Therefore k k Eµ ss [ π, G ρ(u)du, G ] G ρ (T, ) ρ (T, ). ote that the left hand side does not depend on T. To conclude the proof it remains to let T and to apply Lemma

19 Fick s law, announced in Theorem.2.2, follows from the hydrostatics and elementary computations presented in the Proof of Theorem 2.2 in [7]. The arguments here are even simpler and explicit since the process is gradient. In the next section we will show that Propositions 2..2, 2..3 and 2..4 holds for any sequence of probability measures µ on X in the place of the stationary ones µ ss. Furthermore, if the sequence µ satisfies (2..2) with some profile ρ : [, ] then all it points Q of Q µ are concentrated on paths π with π(, du) = ρ (u)du: Q π : π(, du) = ρ (u)du } =. From these facts and the uniqueness of weak solutions of (2..) we may obtain the next result. Theorem Under the conditions in Theorem 2.., the sequence of probability measures Q µ converges weakly to the measure Q that is concentrated on the absolutely continuous path π(t, du) = ρ(t, u)du whose density ρ(, ) is the unique weak solution of the hydrodynamic equation (2..). Theorem 2.. follows from this last result by standard arguments (cf. Section 4.2 in [6]). 2.2 Proofs of Propositions 2..2, 2..3 and 2..4 Fix T > and a sequence µ of measures on X. Denote by Q the probability measure on the path space D([, T ], M) induced by the process π (η t ) : t T } and with initial distribution µ. Fix a it point Q of the sequence Q and assume, without loss of generality, that Q converges to Q. G, t where For a function G in C,2 ( T ), consider the martingales Mt G = M G, defined by Mt G = πt t, G t π, G ds ( s + 2 L ) πs, G s, G t = ( M G t ) 2 t ds A G, s, A G, s = 2 L π s, G s 2 2 π s, G s 2 L π s, G s. t, G t = A simple computation give us that A G, s is bounded above by C(G) d. Therefore, by Doob s and Chebychev s inequalities, for every δ >, P µ sup t T M G t > δ } = (2.2.) Denote by Γ, resp. Γ+, the left, resp. right, boundary of : Γ ± = (x,, x d ) Γ : x = ±( )}. 4

20 For each x in Γ ±, let ˆx = x±e. After two summations by parts we may rewrite the part inside the integral term of the martingale Mt G as π s, s G s + d + d d x Γ x Γ + + O G ( ), d ( ) i G s (x/)τx h i (η s ) i= x ( G s ) (x/)[ϕ(b(ˆx/)) + av (x, η s )] ( G s ) (x/)[ϕ(b(ˆx/)) + av + (x, η s )] (2.2.2) where i G stands for the discrete second partial derivative in the i-th direction, ( i G ) (x/) = 2 [G(x + e i /) + G(x e i /) 2G(x/)], and V ± (x, η) = [η(x) + b(ˆx/)][η(x e ) b(ˆx/)]. Proof of Proposition In order to prove tightness for the sequence Q, we just need to prove tightness of the real process πt, G for any function G in C 2 (). Moreover, by approximations of G in L () and since there is at most one particle per site, we may assume that G belongs C(). 2 In that case, tightness for πt, G follows from (2.2.), (2.2.2) and the fact that the total mass of the empirical measure πt is bounded by 2. The other two statements follows from the fact that there is at most one particle per site (cf. Section 4.2 in [6]). Fix here and throughout the rest of the section a real number α in (, ) and a function β as in the beginning of Section. and such that there is a θ > such that for all ǔ in T d : β(u, ǔ) = b(, ǔ) if u + θ, β(u, ǔ) = b(, ǔ) if θ u. (2.2.3) otice that, for large enough, νβ( ) is reversible with respect to the generator L,b. For a cylinder function Ψ, denote the expectation of Ψ with respect to the Bernoulli product measure να by Ψ(α): Ψ(α) = E ν α [Ψ] For each integer l > and each site x in, denote the empirical mean density on a box of size 2l + centered at x by η l (x): η l (x) = Λ l (x) y Λ l (x) η(y), where Λ l (x) = Λ,l (x) = y : y x l}. 5

21 For each cylinder function Ψ and each ε >, let V,ε(η) Ψ = d τ x+y Ψ(η) Λ ε (x) Ψ(η ε (x)), x y Λ ε (x) where the sum is carried over all x for which the support of τ x+y Ψ is contained in for every y in Λ ε (x). For a continuous function H : [, T ] Γ R, let V ±,H = ds d x Γ ± V ± (x, η s )H(s, ˆx/). Proposition 2..3 follows in the ususal way from (2.2.2) and the next replacement Lemma (cf. Section 5. in [6]). Lemma Let Ψ be a cylinder function and H : [, T ] Γ R a continuous function. For every δ >, [ ] ds V,ε(η Ψ s ) > δ = (2.2.4) P µ ε P µ [ V ±,H > δ] = (2.2.5) For probability measures µ, ν in X, denote by H(µ ν) the entropy of µ with respect to ν. Since there are at most one particle per site, there exists a constant C = C(β) > such that ( H µ ) ν β( ) C d (2.2.6) for any probability measure µ on X (cf. comments following Remark in [6]). For the proof of (2.2.4) we need to establish an estimate on the entropy production. Denote by S t the semigroup associated with the infinitesimal generator 2 L and let µ t = µ S t. Let also f t, resp. g t, be the density of µ t with respect to ν β( ), resp. ν α ). otice that t f t = 2 L f t, (2.2.7) where L is the adjoint of L in L 2( ν β( )). For a density f with respect to a probability measure µ on X, let D ( ) d f, µ = Dx,x+e ( ) i f, µ, i= where the second sum is performed over all x such that x, x + e i belong to and Dx,x+e ( ) i f, µ = r x,x+ei (η) ( f(η 2 x,x+ei ) f(η) ) 2 µ(dη). Denote by D β ( ) the Dirichlet form of the generator L,b with respect to its reversible probabilty measure ν β( ) and let H (t) = H ( µ t ν β( )). x 6

22 Lemma There exists a positive constant C = C(β) such that t H (t) 2 D ( g t, να ) + C d Proof. By (2.2.7) and the explicit formula for the entropy, t H (t) = 2 ft L log ft νβ( ) (dη) Since νβ( ) is reversible with respect to L,b, standard estimates gives that the piece of the right-hand side ( of ) the last equation corresponding to L,b is bounded above by 2 2 Dβ f t. Hence, in order to conclude the proof, we just need to show that there exists a constant C = C(β) > such that, for any sites x, y = x + e i in, ft L x,y log ft νβ( ) (dη) D x,y(gt, να ) + C 2, (2.2.8) where L x,y is the piece of the generator L, that corresponds to jumps between x and y. Fix then x, y = x + e i in. By the definitions of ft and gt, ft L x,y log ft νβ( ) (dη) = gt L x,y log gt να (dη) + g t L x,y log ( ) ν α (η) νβ( ) (η) να (dη). (2.2.9) Since the product measure να is invariant for the generator L x,y, by standard estimates, the first ) term of the right-hand side of (2.2.9) is bounded above by 2Dx,y( g t, να. On the other side, since να and νβ( ) are product measures, we may compute the second term on the right-hand side of (2.2.9). It is equal to [ Φ(y/) Φ(x/) ] [η(y) η(x)]r x,y (η)g t (η)ν α (dη) = [ Φ(y/) Φ(x/) ] η(x)r x,y (η)[g t (η x,y ) g t (η)]ν α (dη), where Φ = log( β β ). By the elementary inequality 2ab Aa2 + A b 2, the previous expression is bounded above by [ ] 2 Φ(y/) Φ(x/) 2 +D x,y(g t, ν α ). ( 2ν η(x)r x,y (η) gt (η x,y ) + gt (η)) α (dη) This and the fact that g t is a density with respect to ν α permit us to deduce (2.2.8). The proof of (2.2.5) requires the following estimate. 7

23 Lemma There exists a positive constant C = C(β) such that if f is a density with respect to ν β( ), then L f, f ( ) ν β( ) 2 D f, ν β( ) D β (f) + C d 2. Proof. It is enough to show that there is a constant C = C(β) such that Lx,y f, f ν β( ) 2 D ( ) x,y f, ν β( ) + C 2, (2.2.) for any x, y = x + e i in. Fix then x, y = x + e i in. Lx,y f, f = r ν x,y (η) ( f(η β( ) 2 x,y ) f(η) ) 2 ν β (dη) + [ ν β (η x,y ] ) r x,y (η)f(η) 2 νβ( ) (η) νβ (dη) = Dx,y ( ) f, ν β [ + 4 r x,y (η)[f(η x,y ) f(η)] ν β (ηx,y ) ν β (η) ] ν β (dη). otice that, for some constant C = C (β), ν β (ηx,y ) νβ (η) C B(y/) B(x/), (2.2.) where B = β β. Hence, by the elementary inequality 2ab Aa2 + A b 2, the left hand side in (2.2.) is bounded above by 2 C r x,y (η) ( f(η 6 x,y ) + f(η) ) 2 [B(x/) B(y/)] 2 νβ (dη) 2 D x,y( f, ν β( ) ). From this fact and since f is a density with respect to νβ( ) we obtain (2.2.). Proof of Lemma By (2.2.6) and Lemma 2.2.2, the proof of (2.2.4) may be reduced (cf. Section 5.3 in [6]) to show that for every positive constant C, V,ε(η)g(η)ν Ψ α (dη) =, (2.2.2) sup ε g where ( the) supremum is carried over all densities g with respect to να such that D g, ν α C d 2. Moreover, since V,ε Ψ is bounded, we may replace g by its conditional expectation g ε given η(x) : x 2ε } in the left hand side of (2.2.2). In that case, this it may be estimated by the one of the periodic case. Hence, (2.2.2) follows from Lemma in [6]. 8

24 We turn now to the proof of (2.2.5). Fix A >. By the entropy inequality and (2.2.6), E µ [ V,H ] C A + [ [ log Eν A d exp A d V β( ),H }]]. Thus we just need to show that for some constant C = C(β) >, Since e x e x + e x and d log E [ ν exp A d V β( ),H }] C. (2.2.3) d log(a + b ) max d log a, } d log b, we may remove the absolute value in (2.2.3), provided our estimates remain in force if we replace H with H. Let V ± H (x, η, s) = V ± (x, η)h(s, ˆx/). By the Feynman-Kac formula, the left hand side of (2.2.3), without the absolute value, is bounded by T d λ s ds, where λ s A x Γ V H stands for the largest eigenvalue of the νβ( ) -reversible operator 2 L sym + (x, η, s) and Lsym is the symmetric part of the operator L in L 2( νβ( )). By the variational formula for the largest eigenvalue, for each s [, T ], d λ s is equal to } A sup f d V H (x, η, s), f + L f, f ν d 2, β( ) x Γ where the supremum is carried over all densities f with respect to νβ( ). By Lemma 2.2.3, for a constant C = C (β) >, the expression inside braces is less than or equal to C + A d V H (x, η, s), f ν ( ( ) ( ) )} β( ) A 2 D f, ν β( ) + D b f, ν β( ). x Γ In this last expression, for some positive constant C 2 = C 2 (b), the part inside braces is bounded above by V H (x, η, s), f ν C [ 2 ( f(η ) x,x+e ) 2 f(η) β( ) A x Γ ν β( ) + ( f(η x ) f(η) ) ]} 2 ν β (dη) 9

25 Hence, in order to prove (2.2.5), it is enough to show that the part inside braces in the last expression is bounded above by some positive constant c, not depending on s, f or x, which converges to as. Fix such s, f and x, and denote by f x the conditional expectation of f given η(x), η(x + e )}. Since V H,x,s (η) = V H (x, η, s) depends on the configuration η only through η(x), η(x + e )}, the part inside braces in the last expression is bounded above by V H,x,s, f x ν C [ 2 ( fx (η β( ) A x,x+e ) fx (η) ) 2 + ( fx (η x ) f x (η) ) ] 2 ν β (dη). Let Λ x =, } x,x+e} and denote by ˆf x the restriction of f x to Λ x. ote that, for large enough, the restriction of νβ( ) to Λ x is the Bernoulli product measure associated to the constant function b x = b(ˆx/). Hence, for a constant C 3 = C 3 (b) >, the last expression is bounded above by V H,x,s, ˆf x ν b C ( ) 3 x A V ar ν ˆf bx x. which, by the elementary inequality 2ab Aa 2 +A b 2 and since E ν bx (V H,x,s ) =, is bounded by A E 4C ν bx [(V H,x,s ) 2( ) ] 2 ˆf x + E ˆf x. 3 Since ˆf x is a density with respect to νb expression is bounded by which concludes the proof. c = 4A H 2 C 3, and V H,x,s 2 H, the previous For each function G in Cc ( T ), each integer i d and C >, let : D([, T ], M) R be the functional given by Q G,i,C T Q G,i,C T T (π) = πs, ui G s ds C ds du G(s, u) 2. Recall from Section.3 that the energy Q T (π) was defined as Q T (π) = d Q i T (π), i= where Q i T (π) = sup G C c ( T ) 2 } π t, ui G t dt dt G(t, u) 2 du. 2

26 otice that sup Q G,i,C G Cc ( T (π) } = Qi T (π) T ) 4C. (2.2.4) The next result is the key ingredient in the proof of Proposition (2..4). Lemma There exists a constant C = C (β) > such that for every integer i d and every function G in C c ( T ), [ }] d log E ν exp d Q G,i,C β( ) T (π ) C Proof. By the Feynman-Kac formula, [ d log E ν exp β( ) is bounded above by ds }] (η s (x) η s (x e i ))G(s, x/) x T d λ s ds, where λ s stands for the largest eigenvalue of the νβ( ) -reversible operator 2 L sym + x [η(x) η(x e i )]G(s, x/). By the variational formula for the largest eigenvalue, for each s [, t], λ s is equal to sup (η(x) η(x e i ))G(s, x/), f + 2 } L f, f, ν f β( ) x where the supremum is carried over all densities f with respect to ν β( ). By Lemma 2.2.3, for a constant C = C(β) >, the expression inside braces is bounded above by C d 2 2 D (f, νβ( ) ) + G(s, x/) x ν β( ) } [η(x) η(x e i )]f(η)νβ (dη), By the elementary inequality 2ab Aa 2 + A b 2, the part inside braces in the last expression is bounded above by G(s, x/) 2 f(η x ei,x )νβ (dη) G(s, x/) 2 [ ( η(x)f(η x ei,x ) ν β (ηx ei,x )] ) 2 νβ (η) νβ (dη) η(x) ( f(η x e i,x ) + f(η) ) 2 ν r β (dη), x ei,x(η) which is bounded above by C G(s, x/) 2 + C, by some positive constant C = C (β), because of (2.2.) and the fact that f is a density with respect to ν β( ). Thus, C = C + C satisfies the statement of the Lemma. 2

27 It is well known that a trajectory π(t, du) = ρ(t, u)du in D([, T ], M ) has finite energy, Q T (π) <, if and only if its density ρ belongs to L 2 ([, T ], H ()), in which case, Q T (π) = dt du ρ t (u) 2 <. Proof of Proposition Fix a constant C > satisfying the statement of Lemma Let G k : k } be a sequence of smooth functions dense in L 2 ([, T ], H ()) and i d an integer. By the entropy inequality and (2.2.6), there is a constant C = C(β) > such that is bounded above by E µ [ C + d log E ν β( ) max k r [ exp Q G k,i,c T (π ) }] d max k r Q G k,i,c T (π ) }}]. Hence, Lemma together with the facts that e maxx,...,x n } e x + + e x n and that } d log(a + b ) max d log a, d log b, imply E Q [ max k r Q G k } ] i,c = [ µ max Q G k i,c k r (π ) }] C + C. This together with (2.2.4) and the monotone convergence theorem prove the desired result. 22

28 Chapter 3 Dynamical Large Deviations In this chapter, we investigate the large deviations from the hydrodynamic it. 3. The Dynamical Rate Function We examine in this section the rate function I T ( γ). The main result, presented in Theorem 3..7 below, states that I T ( γ) has compact level sets. The proof relies on two ingredients. The first one, stated in Lemma 3..2, is an estimate of the energy and of the H norm of the time derivative of the density of a trajectory in terms of the rate function. The second one, stated in Lemma 3..6, establishes that sequences of trajectories, with rate function uniformly bounded, whose densities converges weakly in L 2 converge in fact strongly. We start by introducing some Sobolev spaces. Recall that we denote by Cc () the set of infinitely differentiable functions G : R, with compact support in. Recall from Section.2 the definition of the Sobolev space H () and of the norm,2. Denote by H () the closure of Cc () in H (). Since is bounded, by Poincaré s inequality, there exists a finite constant C such that for all G H () This implies that, in H () d G 2 2 C u G 2 2 C uj G, uj G 2. G,2, = j= d uj G, uj G 2 j= is a norm equivalent to the norm,2. Moreover, H () is a Hilbert space with inner product given by /2 G, J,2, = d uj G, uj J 2. j= To assign boundary values along the boundary Γ of to any function G in H (), recall, from the trace Theorem ([26], Theorem 2.A.(e)), that there 23

29 exists a continuous linear operator B : H () L 2 (Γ), called trace, such that BG = G Γ if G H () C(). Moreover, the space H () is the space of functions G in H () with zero trace ([26], Appendix (48b)): H () = G H () : BG = }. Since C () is dense in H () ([26], Corollary 2.5.(a)), for functions F, G in H (), the product F G has generalized derivatives ui (F G) = F ui G + G ui F in L () and F (u) u G(u) du + G(u) u F (u) du (3..) = BF (u) BG(u) du Γ + BF (u) BG(u) du. Γ Moreover, if G H () and f C (R) is such that f is bounded then f G belongs to H () with generalized derivatives ui (f G) = (f G) ui G and trace B(f G) = f (BG). Finally, denote by H () the dual of H (). H () is a Banach space with norm given by } v 2 = sup 2 v, G, G(u) 2 du, () G C c where v, G, stands for the values of the linear form v at G. For each function G in Cc ( T ) and each integer i d, let Q G,i T : D([, T ], M ) R be the functional given by Q G,i T (π) = 2 π t, ui G t dt dt du G(t, u) 2, and recall, from Section.3, that the energy Q(π) was defined as Q T (π) = d i= Q i T (π) with Q i T (π) = sup Q G,i T (π). G Cc ( T ) The functional Q G,i T is convex and continuous in the Skorohod topology. Therefore Q i T and Q T are convex and lower semicontinuous. Furthermore, it is well known that a trajectory π(t, du) = ρ(t, u)du in D([, T ], M ) has finite energy, Q T (π) <, if and only if its density ρ belongs to L 2 ([, T ], H ()), in which case, Q T (π) = dt du ρ t (u) 2 <. Let D γ = D γ,b be the subset of C([, T ], M ) consisting of all paths π(t, du) = ρ(t, u)du with initial profile ρ(, ) = γ( ), finite energy Q T (π) (in which case ρ t belongs to H () for almost all t T and so B(ρ t ) is well defined for those t) and such that B(ρ t ) = b for almost all t in [, T ]. Lemma 3... Let π be a trajectory in D([, T ], M) such that I T (π γ) <. Then π belongs to D γ. 24

30 Proof. Fix a path π in D([, T ], M) with finite rate function, I T (π γ) <. By definition of I T, π belongs to D([, T ], M ). Denote its density by ρ: π(t, du) = ρ(t, u)du. The proof that ρ(, ) = γ( ) is similar to the one of Lemma 3.5 in [4] and is therefore omitted. To prove that B(ρ t ) = b for almost all t [, T ], since the function ϕ : [, ] [, + a] is a C diffeomorphism and since B(ϕ ρ t ) = ϕ(bρ t ) (for those t such that ρ t belongs to H ()), it is enough to show that B(ϕ ρ t ) = ϕ(b) for almost all t [, T ]. To this end, we just need to show that, for any function H ± C,2 ([, T ] Γ ± ), dt du B(ϕ(ρ t ))(u) ϕ(b(u)) } H ± (t, u) =. (3..2) Γ ± Fix a function H C,2 ([, T ] Γ ). For each < θ <, let h θ : [, ] R be the function given by r + if r + θ, θr h θ (r) = θ if + θ r, if r, and define the function G θ : T R as G(t, (u, ǔ)) = h θ (u )H(t, (, ǔ)) for all ǔ T d. Of course, G θ can be approximated by functions in C,2 ( T ). From the integration by parts formula (3..) and the definition of J Gθ, we obtain that J G θ (π) = dt du B(ϕ(ρ t ))(u) ϕ(b(u)) } H(t, u), θ Γ which proves (3..2) because I T (π γ) <. We deal now with the continuity of π. We claim that there exists a positive constant C such that, for any g C c (), and any s < r < T, π r, g π s, g C (r s) /2 I T (π γ) + g 2,2, + (r s) /2 g }. (3..3) Indeed, for each δ >, let ψ δ : [, T ] R be the function given by if t s or r + δ t T, (r s) /2 ψ δ t s (t) = δ if s t s + δ, if s + δ t r, t r δ if r t r + δ, and let G δ (t, u) = ψ δ (t)g(u). Of course, G δ can be approximated by functions in C,2 ( T ) and then (r s) /2 J G δ(π) = π r, g π s, g δ 2(r s) /2 25 r s r s dt ϕ(ρ t ), g dt σ(ρ t ), g 2.

31 To conclude the proof, it remains to observe that the left hand side is bounded by (r s) /2 I T (π γ), and to note that ϕ, σ are positive and bounded above on [, ] by some positive constant. Denote by L 2 ([, T ], H ()) the dual of L 2 ([, T ], H ()). By Proposition 23.7 in [26], L 2 ([, T ], H ()) corresponds to L 2 ([, T ], H ()), i.e., for each v in L 2 ([, T ], H ()), there exists a unique v t : t T } in L 2 ([, T ], H ()) such that for any G in L 2 ([, T ], H ()), v, G, = v t, G t, dt, (3..4) where the left hand side stands for the value of the linear functional v at G. Moreover, if we denote by v the norm of v, v 2 = v t 2 dt. Fix a path π(t, du) = ρ(t, u)du in D γ and suppose that } sup 2 ρ t, t H t dt dt du H t 2 H Cc ( T ) <. (3..5) In this case t ρ : C c ( T ) R defined by t ρ(h) = ρ t, t H t dt can be extended to a bounded linear operator t ρ : L 2 ([, T ], H ()) R. It belongs therefore to L 2 ([, T ], H ()) = L 2 ([, T ], H ()). In particular, there exists v t : t T } in L 2 ([, T ], H ()), which we denote by v t = t ρ t, such that for any H in L 2 ([, T ], H ()), Moreover, t ρ, H, = t ρ t, H t, dt. t ρ 2 = = sup H C c ( T ) t ρ t 2 dt 2 } ρ t, t H t dt dt du H t 2. Let W be the set of paths π(t, du) = ρ(t, u)du in D γ such that (3..5) holds, i.e., such that t ρ belongs to L 2 ( [, T ], H () ). For G in L 2 ( [, T ], H () ), let J G : W R be the functional given by J G (π) = t ρ, G, + 2 dt du G t (u) (ϕ(ρ t (u))) dt du σ(ρ t (u)) G t (u) 2. 26