Large Deviations from a Hydrodynamic Scaling Limit for a Nongradient System Jeremy Quastel

MSC 1991 Classification Numbers: 60K35 (60F10, 82C21). Running Title: Large deviations for nongradient systems. Key Words: Large deviations, hydrodynamic limit, Ginzburg-Landau model, nongradient system.

Abstract. The hydrodynamic limit appears as a law of large numbers for rescaled density profiles of a large stochastic system. We study the large deviations from this scaling limit for a particular nongradient system, the nongradient version of the Ginzburg-Landau model.

I. Introduction

This paper is an extension of the basic paper of Donsker and Varadhan [DV], on the large deviations from the hydrodynamic scaling limit of the Ginzburg-Landau model, to the nongradient version of that model considered in [V2]. The setup in [V2] is as follows: Let S denote the unit circle 0 ≤ θ ≤ 1 with 1 ≡ 0, and, for an integer N, let S_{N} denote the periodic lattice {i/N}_{i=1,...,N}. The neighbours of a site i/N are i + 1/N and i − 1/N with addition modulo 1. To each site i/N is assigned a variable x_{i} ∈ R I which is thought of as the charge at i/N. The vector x ∈ R I ^{N} evolves according to the following system of stochastic differential equations,

\begin{matrix}dx_{i} = dz_{i−1,i} − dz_{i,i+1} \\ dz_{i,i+1} = \frac{N^{2}}{2} W_{i,i+1}dt + N\sqrt{a(x_{i}, x_{i+1})}dβ_{i}, \end{matrix}  (1.1)

where

W_{i,i+1} = a(x_{i}, x_{i+1})(φ′(x_{i}) − φ′(x_{i+1})) − \frac{∂a}{∂x}(x_{i}, x_{i+1}) + \frac{∂a}{∂y}(x_{i}, x_{i+1}),  (1.2)

a(x, y) is a function on R I ^{2} which is assumed to have bounded continuous first derivatives and satisfy

1/a^{∗} < a(x, y) < a^{∗}  (1.3)

for some a^{∗} < ∞, and φ ∈ C^{2} is assumed to satisfy

∫_{−∞}^{∞} e^{−φ(x)}dx = 1  (1.4)

∫_{−∞}^{∞} e^{λx−φ(x)}dx = M(λ) < ∞ for all λ ∈ R I  (1.5)
∫_{−∞}^{∞} e^{σ|φ′(x)|−φ(x)}dx < ∞ for all σ > 0,  (1.6)

and 1/K < φ′′ < K for some K < ∞. The β_{i} are independent Brownian motions and the N^{2} represents the diffusion scaling of time, corresponding to the lattice spacing of 1/N.

When a ≡ 1 the system is the standard Ginzburg-Landau model (see [GPV],[DV]) which is a gradient system, meaning that the microscopic current, W_{i,i+1}, can be expressed as the gradient τ^{i+1}g − τ^{i}g of a local function g(x_{−l}, ..., x_{l}). For such systems there is no net effect of fluctuations on the diffusion coefficient. While this simplifies the proofs considerably, it is not a natural condition. Many models, and in particular most stochastic lattice gases, do not satisfy the gradient condition. The Onsager coefficient a is introduced into the model to break the gradient structure.

The dynamics (1.1) can alternatively be described by the generator,

L_{N} = \frac{N^{2}}{2} ∑_{i=1}^{N} Φ_{N}^{−1}D^{ii+1}Φ_{N}a(x_{i}, x_{i+1})D^{ii+1},  (1.7)

where D^{ii+1} = \frac{∂}{∂x_{i}} − \frac{∂}{∂x_{i+1}} , or, since L_{N} is symmetric with respect to Φ_{N}(x) = exp − ∑_{i=1}^{N} φ(x_{i}), by the Dirichlet form,

D_{N}(f) = \frac{1}{2} ∫_{R I ^{N}} ∑_{i=1}^{N} a(x_{i}, x_{i+1}) (D^{ii+1}f)^{2} Φ_{N}(x)dx.  (1.8)

As initial distribution for x we take f_{N}^{0}Φ_{N} where

f_{N}^{0}(x) = exp ∑_{i=1}^{N} h′(m_{0}(i/N))x_{i} − log M(h′(m_{0}(i/N)))  (1.9)

where m_{0}(θ) is a given smooth function on S and

h(m) = sup_{λ}{λm − log M(λ)}.  (1.10)

Corresponding to the initial distribution (1.9), the evolution (1.1), and T < ∞, we have a measure Q_{N} on C([0, T] → R I ^{N}). Now for each x ∈ R I ^{N} define a measure on S by

μ_{N}(dθ) = \frac{1}{N} ∑_{i=1}^{N} x_{i}δ_{i/N},  (1.11)

where δ_{θ} represents the measure giving mass 1 to θ in S. The probability measure Q_{N} is carried by this mapping into a probability measure Q̂_{N} on X = C([0, T] → M[S]) where M[S] denotes the set of signed measures on S. In [V2] it is shown that under these conditions

Q̂_{N} ⇒ δ_{m(t,θ)dθ},  (1.12)

where δ_{m(t,θ)dθ} denotes the probability measure on X whose mass is concentrated on the unique weak solution of the nonlinear diffusion equation

\begin{aligned}\frac{∂m}{∂t} & = \frac{1}{2} \frac{∂}{∂θ} (â(m(t, θ))\frac{∂}{∂θ} h′(m(t, θ))) \\ & m(0, θ) = m_{0}(θ). \end{aligned}  (1.13)

The function â(m) is given by a variational formula which we now describe. Consider Ω = ∏_{−∞}^{∞} R I with product measure π_{m} = ∏_{i=−∞}^{∞} exp{h′(m)x_{i} − φ(x_{i}) − log M(h′(m))}dx_{i}. By a local function we mean a function on Ω which depends on only finitely many coordinates. Let T be the shift operator on Ω defined by (T x)_{i} = x_{i+1}, and for functions g on Ω, let τ^{i}g(x) = g(T^{i}x). For a local function g we can write the formal sum

ξ_{g} = ∑_{i=−∞}^{∞} τ^{i}g.  (1.14)

Although this is purely formal, the derivative D^{12}ξ_{g} is well defined. The variational formula for â(m) is

â(m) = inf_{g} E^{π_{m}}[a(x_{1}, x_{2}) (1 − D^{12}ξ_{g})^{2}]  (1.15)

where the infimum is taken over all smooth local functions g.

In the present article we are concerned with the large deviations from the scaling limit described above. Let ⟨f, g⟩ be the usual inner product on L^{2}(S, dθ). Define, for functions m(θ), f(θ) on S,

I_{static}(m(·)) = ∫_{S} (h(m(θ)) − h(m_{0}(θ)) − h′(m_{0}(θ)) (m(θ) − m_{0}(θ))) dθ,  (1.16)

∥f∥_{−1,â(m(·))}^{2} = sup_{J∈H_{1}(S)}\{2 ⟨J, f⟩ − ∫_{S} â(m(θ))(\frac{∂J}{∂θ} (θ))^{2}dθ\},  (1.17)

and, for a function m(t, θ) on [0, T] × S,

I_{dynamic}(m(·, ·)) = \frac{1}{2} ∫_{0}^{T} ∥\frac{∂m}{∂t} − \frac{1}{2} \frac{∂}{∂θ} (â(m(t, θ))\frac{∂}{∂θ} h′(m(t, θ)))∥_{−1,â(m(t,·))}^{2} dt. (1.18)

We will need as well the form I_{0}(μ(·, ·)) which is defined as follows: I_{0}(μ(·, ·)) = ∞ unless μ(t, dθ) = m(t, θ)dθ for each t ∈ [0, T], in which case

I_{0}(μ(·, ·)) = ∫_{0}^{T} ∫_{S} â(m)(\frac{∂}{∂θ} (h′(m)))^{2}dθdt.  (1.19)

We can now describe the rate function I on X. If I_{0}(μ(·, ·)) = ∞ then I(μ(·, ·)) = ∞. Otherwise μ(t, dθ) = m(t, θ)dθ for each t ∈ [0, T] and

I(μ(·, ·)) = I_{static}(m(0, ·)) + I_{dynamic}(m(·, ·)).  (1.20)

The main result of this paper is that Q_{N} has the large deviations property with rate function I. In particular, let M^{l}[S] be the set of signed measures on S with total variation at most l. M^{l}[S] with the topology of weak convergence is metrizable. X^{l} = C([0, T] → M^{l}[S]) can therefore be given the topology of uniform convergence. X = ⋃ X^{l} is given the direct limit topology. For any closed set C ⊂ X,

lim sup_{N→∞} \frac{1}{N} log Q̂_{N}(C) ≤ − inf_{μ(·,·)∈C} I(μ(·, ·)),  (1.21)

and, for any open set U ⊂ X,

lim inf_{N→∞} \frac{1}{N} log Q̂_{N}(U) ≥ − inf_{μ(·,·)∈U} I(μ(·, ·)).  (1.22)

II. Upper bound

To prove the upper bound (1.21) we will produce a family of functionals, F̂_{J,G,ϵ_{1},ϵ_{2}}(μ(·, ·)), depending on certain parameters J, G, ϵ_{1}, ϵ_{2}, for which we can prove

Lemma 2.1. For each fixed J and G,

lim_{ϵ_{2}→0} lim sup_{ϵ_{1}→0} lim sup_{N→∞} \frac{1}{N} log EQ̂_{N} [exp NF̂_{J,G,ϵ_{1},ϵ_{2}}] ≤ 0,  (2.1)

and

Lemma 2.2. For each μ(·, ·) ∈ X for which I_{0}(μ(·, ·)) < ∞,
I(μ(·, ·)) ≤ lim_{ϵ_{2}→0} lim sup_{ϵ_{1}→0} sup_{J,G} F̂_{J,G,ϵ_{1},ϵ_{2}}(μ(·, ·)).  (2.2)

In addition we will show Lemma 2.3. If I_{0}(μ(·, ·)) ≡ ∞ on a compact K ⊂ X then,

lim sup_{N→∞} \frac{1}{N} log Q̂_{N}(K) = −∞.  (2.3)

The upper bound for compact sets will follow by the exponential Chebyshev inequality. These are essentially standard arguments in large deviations (see [V1],[DV]). To extend this to arbitrary closed sets we need the following exponential tightness estimates whose proofs can be found in [V2].

Lemma 2.4. For each T < ∞, ϵ > 0, and smooth function J on S,

lim_{l→∞} lim sup_{N→∞} \frac{1}{N} log Q̂_{N}( sup_{0≤t≤T} ∥μ(t)∥ ≥ l) = −∞,  (2.4)

lim_{δ↓0} lim sup_{N→∞} \frac{1}{N} log Q̂_{N}( sup_{{0≤s<t≤T \atop |s−t|<δ}} |⟨J, μ(t)⟩ − ⟨J, μ(s)⟩| ≥ ϵ) = −∞.  (2.5)

We now describe the functionals F̂_{J,G,ϵ_{1},ϵ_{2}}(μ(·, ·)). Let J(t, θ) be a smooth test function on [0, T] × S, and G(θ) a smooth test function on S. For ϵ_{1} > 0, let m_{θ}^{ϵ_{1}} be the average density in an ϵ_{1} neighbourhood of θ: m_{θ}^{ϵ_{1}} = (2ϵ_{1})^{−1}μ([θ − ϵ_{1}, θ + ϵ_{1}]). For ϵ_{1}, ϵ_{2} > 0 let

\begin{aligned}F̂_{J,G,ϵ_{1},ϵ_{2}}(μ(·, ·)) = & ∫_{S} log M(h′(m_{0}(θ))) − log M(G(θ) + h′(m_{0}(θ)))dθ \\ & + ∫_{S} J(T, θ)μ(T, dθ) \\ & − ∫_{S}(J(0, θ) − G(θ))μ(0, dθ) \\ & − ∫_{0}^{T} ∫_{S} \frac{∂J}{∂t} (t, θ)μ(t, dθ)dt \\ & − \frac{1}{2} ∫_{0}^{T} ∫_{S} \frac{∂J}{∂θ} (t, θ)â(m_{θ}^{ϵ_{1}})(\frac{h′(m_{θ−ϵ_{2}}^{ϵ_{1}}) − h′(m_{θ+ϵ_{2}}^{ϵ_{1}})}{2ϵ_{2}})dθdt \\ & − \frac{1}{2} ∫_{0}^{T} ∫_{S}(\frac{∂J}{∂θ} (t, θ))^{2}â(m_{θ}^{ϵ_{1}})dθdt. \end{aligned} (2.6)

We will now begin the proof of Lemma 2.1 which depends on several preliminary lemmas. Before we start it is worth making a remark. Let P_{N} be the equilibrium process, by which we mean the measure on C([0, T] → R I ^{N}) corresponding to taking the invariant measure, Φ_{N}(x)dx, as initial distribution. Suppose that for some sequence of functionals F_{N} we can prove a superexponential estimate which says that for any α > 0,

lim sup_{N→∞} \frac{1}{N} log E^{P_{N}} [exp αNF_{N}] ≤ 0.  (2.7)

Then, by Hölder’s inequality, this estimate extends to Q_{N},

lim sup_{N→∞} \frac{1}{N} log E^{Q_{N}} [exp αNF_{N}] ≤ lim sup_{γ↓0} lim sup_{N→∞} \frac{1}{(1 + γ)N} log E^{P_{N}} [(f_{N}^{0})^{1+γ}]

+ lim sup_{γ↓0} lim sup_{N→∞} \frac{γ}{(1 + γ)N} log E^{P_{N}} [exp \frac{(1 + γ)}{γ}αNF_{N}]

≤ 0.

(2.8)

Now let g(x_{−l}, ..., x_{l}, m) be a smooth function on R I ^{2l+2} with bounded first derivatives, and for ξ_{g} = ∑_{i=−∞}^{∞} τ^{i}g let

A(g, m) = E^{π_{m}}[a(x_{1}, x_{2}) (1 − D^{12}ξ_{g})^{2}] − â(m).  (2.9)

For k > l let

\begin{aligned}V^{1} = & \frac{1}{2} ∑_{i=1}^{N} J(t, i/N)[W_{i,i+1} − (L̂g)(x_{i−l}, ..., x_{i+l}, m_{i/N}^{k/N})] \\ & + \frac{1}{2N} ∑_{i=1}^{N} J(t, i/N)â(m_{i/N}^{ϵ_{1}})(\frac{h′(m_{i/N+ϵ_{2}}^{ϵ_{1}}) − h′(m_{i/N−ϵ_{2}}^{ϵ_{1}})}{2ϵ_{2}}) \\ & − \frac{α}{4N} ∑_{i=1}^{N}(J(t, i/N))^{2}A(g, m_{i/N}^{ϵ_{1}}), \end{aligned}  (2.10)

Here L̂g represents the generator, without the N^{2} scaling, acting only on the variables x_{−l} through x_{l}.

The main point in our proof is the following estimate which is due to Varadhan. Lemma 2.5. For each α > 0, g, and J,

lim_{ϵ_{2}→0} lim sup_{ϵ_{1}→0} lim sup_{k→∞} lim sup_{N→∞} \frac{1}{N} log E^{P_{N}} [exp αN ∫_{0}^{T} V^{1}(t, x(t))dt] ≤ 0.  (2.11)

Proof. This is Theorem 7.1 of [V2].

Lemma 2.6. For any ϵ > 0 and C < ∞ we can find a smooth function g(x_{−l}, ..., x_{l}, m) on R I ^{2l+2} with bounded first derivatives such that

sup_{|m|≤C} A(g, m) ≤ ϵ  (2.12)

and

sup_{m} A(g, m) ≤ a^{∗}  (2.13)

Proof. Let G_{l,B} be the set of smooth g of the form g(x_{−l}, ..., x_{l}) with ∥g∥_{∞} ≤ B and ∥\frac{∂g}{∂x_{i}} ∥_{∞} ≤ B for i = −l, ..., l; and let

A_{l,B}(m) = inf_{g∈G_{l,B}} A(g, m).  (2.14)

A_{l,B}(m) is upper semi-continuous in m, nonincreasing in l and B, and, for each m, lim_{l→∞} lim_{B→∞} A_{l,B}(m) = 0. Therefore lim_{l→∞} lim_{B→∞} sup_{|m|≤C+1} A_{l,B}(m) = 0. Thus we can find l, B, and, for each |m| ≤ C + 1, g(m) ∈ G_{l,B} so that A(g(m), m) ≤ ϵ/4. For |m| > C + 1 take g(m) = 0. Of course this g is not regular in m, so we will smooth it out. Let B_{M} = {x_{−2l+1}^{2} + ⋯ + x_{2l+3}^{2} ≤ M} and let M be large enough so that for all g ∈ G_{l,B},

sup_{|m|≤C} E^{π_{m}}[a(x_{1}, x_{2}) (1 − D^{12}ξ_{g})^{2} 11_{B_{M}^{C}} ] ≤ \frac{ϵ}{4},  (2.15)

and let 1 > δ > 0 be small enough that if |m| ≤ C+1 and |m′−m| ≤ δ then |â(m′)−â(m)| ≤ ϵ/4 and

sup_{B_{M}}\{\frac{dπ_{m′}}{dπ_{m}} |_{F_{−2l+1,...,2l+3}}\} ≤ 1 + \frac{ϵ}{4a^{∗}} .  (2.16)

Then for |m| ≤ C and |m′ − m| ≤ δ,

\begin{aligned}A(g(m), m′) & = E^{π_{m′}} [a(x_{1}, x_{2}) (1 − D^{12}ξ_{g(m)})^{2}] − â(m′) \\ & ≤ E^{π_{m′}} [a(x_{1}, x_{2}) (1 − D^{12}ξ_{g(m)})^{2} 11_{B_{M}} ] + \frac{ϵ}{4} − â(m′) \\ & ≤ (1 + \frac{ϵ}{4a^{∗}} ) E^{π_{m}}[a(x_{1}, x_{2}) (1 − D^{12}ξ_{g(m)})^{2} 11_{B_{M}} ] + \frac{ϵ}{4} − â(m′) \\ & ≤ ϵ. \end{aligned} (2.17)

Let φ be a smoothing kernel with support in (−δ, δ) and let

g̃(x_{−l}, ..., x_{l}, m) = ∫ φ(m′ − m)g(x_{−l}, ..., x_{l}, m′)dm′.  (2.18)

Clearly \frac{∂g̃}{∂m} is bounded and by (2.17) and convexity,

A(g̃(m), m) ≤( ∫ φ(m′ − m)(A(g(m′), m))^{1/2}dm′)^{2}≤ ϵ,  (2.19)

for |m| ≤ C. Finally, if g = 0, A(g, m) = E^{π_{m}}[a(x_{1}, x_{2})] − â(m) ≤ a^{∗}, so by (2.19), for |m| > C, A(g̃(m), m) ≤ a^{∗}.

The next lemma allows us to control the difference between the corrector term which appears in (2.10),

∑_{i=1}^{N} J(t, i/N)(L̂g)(x_{i−l}, ..., x_{i+l}, m_{i/N}^{k/N}) = ∑_{i=1}^{N} J(t, i/N) ∑_{j=i−k}^{i+k−1} L^{j,j+1}g(x_{i−l}, ..., x_{i+l}, m_{i/N}^{k/N}),

(2.20)

where L^{i,i+1} = Φ_{N}^{−1}D^{ii+1}Φ_{N}a(x_{i}, x_{i+1})D^{ii+1}, and the ‘derivative’ term,

∑_{i=1}^{N} J(t, i/N)(\frac{L_{N}}{N^{2}} g)(x_{i−l}, ..., x_{i+l}, m_{i/N}^{k/N}).  (2.21)

The difference is in action of the generator on the local averages. It is given by

V^{2} = ∑_{i=1}^{N} J(t, i/N)(L^{i+k,i+k+1} + L^{i−k−1,i−k})g(x_{i−l}, ..., x_{i+l}, m_{i/N}^{k/N}).  (2.22)

Lemma 2.7. For any α > 0,

lim sup_{k→∞} lim sup_{N→∞} \frac{1}{N} log E^{P_{N}} [exp αN ∫_{0}^{T} V^{2}(t, x(t))dt] ≤ 0.  (2.23)

Proof. Let g^{i} denote g(x_{i−l}, ..., x_{i+l}, m_{i/N}^{k/N}). By Schwarz’s inequality, for any f,

\begin{aligned}∫ & ∑_{i=1}^{N} L^{i+k,i+k+1}g^{i}fΦ_{N}dx \\ & = − ∫ ∑_{i=1}^{N} a(x_{i+k}, x_{i+k+1})D^{i+k,i+k+1}g^{i}D^{i+k,i+k+1}fΦ_{N}dx \\ & = − ∫ ∑_{i=1}^{N} a(x_{i+k}, x_{i+k+1})D^{i+k,i+k+1}g^{i}\sqrt{f} \frac{D^{i+k,i+k+1}f}{\sqrt{f}} Φ_{N}dx \\ & ≤ (∫ ∑_{i=1}^{N} a(x_{i+k}, x_{i+k+1}) (D^{i+k,i+k+1}g^{i})^{2} fΦ_{N}dx)^{1/2}(2D_{N}(\sqrt{f}))^{1/2}. \end{aligned}  (2.24)

There is of course an analogous estimate for ∫ ∑_{i} L^{i−k−1,i−k}g^{i}fΦ_{N}dx. Also

D^{i+k,i+k+1}g^{i} = −D^{i−k−1,i−k}g^{i} = \frac{1}{2k} \frac{∂g}{∂m}(x_{i−l}, ..., x_{i+l}, m_{i/N}^{k/N}).  (2.25)

By spectral theorem and (2.24) and (2.25),

\frac{1}{N} log E^{P_{N}} [exp αN ∫_{0}^{T} ∑_{i=1}^{N} J(t, i/N)(L^{i+k,i+k+1} + L^{i−k−1,i−k})g^{i}dt]

≤ ∫_{0}^{T} \frac{1}{N} sup_{f}\{αN∫ ∑_{i=1}^{N} J(t, i/N)(L^{i+k,i+k+1}+ L^{i−k−1,i−k})g^{i}fΦ_{N}dx − N^{2}D_{N}(\sqrt{f})\}dt

≤ \frac{C}{N} sup_{f}\{α^{2}∫ ∑_{i=1}^{N} a(x_{i+k}, x_{i+k+1})[(D^{i+k,i+k+1}g^{i})^{2} + (D^{i−k−1,i−k}g^{i})^{2}]fΦ_{N}dx\}

≤ C′α^{2}k^{−2} ∥∂g/∂m∥_{∞}^{2} .

(2.26)

This completes the proof of Lemma 2.7.

Proof of Lemma 2.1. For a smooth function g(x_{−l}, ..., x_{l}, m) on R I ^{2l+2} with bounded first derivatives, let

ξ^{N}(t) = ∑_{i=1}^{N} \frac{∂J}{∂θ} (t, i/N)g(x_{i−l}(t), ..., x_{i+l}(t), m_{i/N}^{k/N}(t)),  (2.27)

and let

F^{1}(x(·))= ∫_{S} log M(h′(m_{0}(θ))) − log M(G(θ) + h′(m_{0}(θ)))dθ

+ \frac{1}{N} ∑_{i=1}^{N} J(T, i/N)x_{i}(T) − \frac{1}{N^{2}} ξ^{N}(T)

− \frac{1}{N} ∑_{i=1}^{N}(J(0, i/N) − G(i/N))x_{i}(0) + \frac{1}{N^{2}} ξ^{N}(0)  (2.28)

− ∫_{0}^{T} \frac{1}{N} ∑_{i=1}^{N} (\frac{∂J}{∂t} (t, i/N)x_{i}(t)) dt

+ ∫_{0}^{T} \frac{1}{N^{2}} ∑_{i=1}^{N} (\frac{∂^{2}J}{∂t∂θ} (t, i/N)g(x_{i−l}(t), ..., x_{i+l}(t), m_{i/N}^{k/N}(t))) dt

− ∫_{0}^{T} \frac{N}{2} ∑_{i=1}^{N}(J(t, i + 1/N) − J(t, i/N))W_{i.i+1} − \frac{L_{N}}{N^{2}} ξ^{N}(t)dt

− ∫_{0}^{T} \frac{1}{2N} ∑_{i=1}^{N} a(x_{i}, x_{i+1})(N(J(t, i + 1/N) − J(t, i/N)) + D^{ii+1}ξ^{N})^{2}dt.

The functional exp NF^{1}(T) is a martingale under Q_{N} so

lim sup_{N→∞} \frac{1}{N} log E^{Q_{N}} [exp NF^{1}(T)] = lim sup_{N→∞} \frac{1}{N} log E^{Q_{N}} [exp NF^{1}(0)]

= − ∫_{S} log \frac{M(G(θ) + h′(m_{0}(θ)))}{M(h′(m_{0}(θ)))}dθ + lim sup_{N→∞} \frac{1}{N} log E^{f_{N}^{0} Φ_{N}} [exp ∑_{i=1}^{N} G(i/N)x_{i}]

= 0.

(2.29)

By bounding the last term in (2.28) in terms of the bound on the derivatives of g it is not difficult to show that

lim sup_{γ↓0} lim sup_{N→∞} \frac{1}{N} log E^{Q_{N}} [exp(1 + γ)NF^{1}] ≤ 0.  (2.30)

Consider the functional F̂(μ(·, ·)) defined in (2.6). For each N we have a mapping x ↦ μ given by (1.11). F̂(μ(·, ·)) pulls back through this map to a function F (x(·)) on C ([0, T] → R I ^{N}). By Hölder’s inequality, for any γ > 0,

\frac{1}{N} log E^{Q_{N}} [exp NF] ≤\frac{1}{(1 + γ)N} log E^{Q_{N}} [exp(1 + γ)NF^{1}]

+ \frac{γ}{(1 + γ)N} log E^{Q_{N}} [exp \frac{1 + γ}{γ} N(F − F^{1})].

(2.31)

To prove Lemma 2.1 it therefore suffices to show that for any α > 0,

inf_{g} lim_{ϵ_{2}→0} lim sup_{ϵ_{1}→0} lim sup_{k→∞} lim sup_{N→∞} \frac{1}{N} log E^{Q_{N}} [exp αN(F − F^{1})] ≤ 0.  (2.32)

We break it up as follows: F − F^{1} = ∫_{0}^{T} (V^{3} + V^{4} + V^{5}) dt + o(1) where

V^{3} = \frac{1}{2} ∑_{i=1}^{N}(N(J(t, i + 1/N) − J(t, i/N)) − \frac{∂J}{∂θ} (t, i/N))W_{i,i+1}
V^{4} = \frac{1}{2} ∑_{i=1}^{N} \frac{∂J}{∂θ} (t, i/N)W_{i,i+1} − \frac{L_{N}}{N^{2}} ξ^{N}  (2.33)

+ \frac{1}{2N} ∑_{i=1}^{N} \frac{∂J}{∂θ} (t, i/N)â(m_{i/N}^{ϵ_{1}})(\frac{h′(m_{i/N+ϵ_{2}}^{ϵ_{1}}) − h′(m_{i/N−ϵ_{2}}^{ϵ_{1}})}{2ϵ_{2}})

V^{5} = \frac{1}{2N} ∑_{i=1}^{N} a(x_{i}, x_{i+1})(\frac{∂J}{∂θ} (t, i/N) + D^{ii+1}ξ^{N})^{2} − \frac{1}{2N} ∑_{i=1}^{N} â(m_{i/N}^{ϵ_{1}})(\frac{∂J}{∂θ} (t, i/N))^{2}.

By (3.8) of [V2] there is a constant C < ∞ such that, for any α > 0,

\frac{1}{N} log E^{P_{N}}[exp αN∫_{0}^{T}V^{3}(x(t), t)dt] ≤ Cα^{2}∫_{0}^{T} \frac{1}{N} ∑_{i=1}^{N}(N(J(t, \frac{i+1}{N})−J(t, \frac{i}{N})) − \frac{∂J}{∂θ} (t, \frac{i}{N}))^{2}dt,

(2.34)

the latter term being o(1). Now let α > 0 and ϵ > 0 and choose C < ∞ so that

lim sup_{ϵ_{1}→0} lim sup_{N→∞} \frac{1}{N} log E^{P_{N}} [exp 2α^{2} ∫_{0}^{T} ∑_{i=1}^{N}(\frac{∂J}{∂θ} (t, i/N))^{2}a^{∗}11_{\{|m_{i/N}^{ϵ_{1}} |>C\}}dt] ≤ ϵ, (2.35)

and, for this C, choose g(m) as in Lemma 2.6 so that A(g(m), m) ≤ a^{∗} and

sup_{|m|≤C} A(g(m), m) ≤ \frac{1}{4T} ϵα^{−2}∥\frac{∂J}{∂θ} ∥_{∞}^{−2}.  (2.36)

By Schwarz’s inequality,

\frac{1}{N} log E^{P_{N}} [exp α^{2} ∫_{0}^{T} ∑_{i=1}^{N}(\frac{∂J}{∂θ} (t, i/N))^{2}A(g(m_{i/N}^{ϵ_{1}}), m_{i/N}^{ϵ_{1}})dt]

≤ \frac{1}{2N} log E^{P_{N}} [exp \frac{α^{2}}{2T} ∫_{0}^{T} ∑_{i=1}^{N}(\frac{∂J}{∂θ} (t, i/N))^{2}ϵα^{−2}∥\frac{∂J}{∂θ} ∥_{∞}^{−2}dt]

+ \frac{1}{2N} log E^{P_{N}} [exp 2α^{2} ∫_{0}^{T} ∑_{i=1}^{N}(\frac{∂J}{∂θ} (t, i/N))^{2}a^{∗}11_{\{|m_{i/N}^{ϵ_{1}} |>C\}}dt],

(2.37)

the latter terms each being less than ϵ/2. Together with Lemmas 2.5 and 2.7 this shows that

lim_{ϵ_{2}→0} lim sup_{ϵ_{1}→0} lim sup_{k→∞} lim sup_{N→∞} \frac{1}{N} log E^{P_{N}} [exp αN ∫_{0}^{T} V^{4}(x(t), t)dt] ≤ ϵ.  (2.38)

To deal with V^{5} we use Theorem 4.1 of [GPV] which says that if ν(x_{−k}, ..., x_{k}) is bounded and continuous, and J(θ) is smooth, then, for any α > 0,

lim sup_{ϵ_{1}→0} lim sup_{k→∞} lim sup_{N→∞} \frac{1}{N} log E^{P_{N}} [exp α ∫_{0}^{T} |∑_{i=1}^{N} J(i/N) (ν^{i} − E^{π_{m{ϵ_{1} \atop i/N}}} [ν^{i}])|dt] ≤ 0,

(2.39)

where ν^{i} = ν(x_{i−k}, ..., x_{i+k}). It is true that the theorem was only proved in [GPV] for the case a ≡ 1 but the proof extends to the general case by means of the inequality

D_{N}(f) ≥ \frac{1}{a^{∗}} ∫ ∑_{i=1}^{N} (D^{ii+1}f)^{2} Φ_{N}dx.  (2.40)

From (2.37) and (2.39) it is easy to see that

lim_{ϵ_{2}→0} lim sup_{ϵ_{1}→0} lim sup_{k→∞} lim sup_{N→∞} \frac{1}{N} log E^{P_{N}} [exp αN ∫_{0}^{T} V^{5}(x(t), t)dt] ≤ ϵ.  (2.41)

To complete the proof of Lemma 2.1 we remark that (2.32) follows from (2.34), (2.38), and (2.41) by another application of Hölder’s inequality.

Proof of Lemma 2.2. First note that

I_{static}(m(0, ·)) =sup_{G}{∫_{S}G(θ)m(0, θ)dθ +∫_{S}logM(h′(m_{0}(θ)))−logM(G(θ)+h′(m_{0}(θ)))dθ}.

(2.42)

By assumption I_{0}(μ(·, ·)) < ∞, and therefore m_{θ}^{ϵ_{1}}(t) → m(t, θ) for almost every t and θ. Suppose that I_{dynamic} (μ(·, ·)) ≥ M. Let ϵ > 0 and choose J(t, θ) so that

\begin{aligned}M − ϵ ≤ & ∫_{S} J(T, θ)μ(T, dθ) − ∫_{S} J(0, θ)μ(0, dθ) − ∫_{0}^{T} ∫_{S} \frac{∂J}{∂t} (t, θ)μ(t, dθ)dt \\ & + \frac{1}{2} ∫_{0}^{T} ∫_{S} \frac{∂J}{∂θ} (t, θ)â(m(t, θ))\frac{∂}{∂θ} h′(m(t, θ))dθdt \\ & − \frac{1}{2} ∫_{0}^{T} ∫_{S}(\frac{∂J}{∂θ} (t, θ))^{2}â(m(t, θ))dθdt. \end{aligned}  (2.43)

By the bounded convergence theorem, since â is bounded and continuous,

∫_{0}^{T} ∫_{S}(\frac{∂J}{∂θ} (t, θ))^{2}â(m_{θ}^{ϵ_{1}})dθdt → ∫_{0}^{T} ∫_{S}(\frac{∂J}{∂θ} (t, θ))^{2}â(m(t, θ))dθdt.  (2.44)

Since I_{0}(μ(·, ·)) < ∞ we have that

∫_{0}^{T} ∫_{S} \frac{∂J}{∂θ} â(m) (\frac{∂}{∂θ} h′(m) − \frac{h′(m_{θ+ϵ_{2}}^{ϵ_{1}}) − h′(m_{θ−ϵ_{2}}^{ϵ_{1}})}{2ϵ_{2}}) dθdt → 0  (2.45)

as ϵ_{1} is sent to zero, followed by ϵ_{2}. By a straightforward application of Schwarz’s inequality and the fact that â(m) is continuous and bounded away from both 0 and ∞ we see that as ϵ_{1}, then ϵ_{2} go to zero,

∫_{0}^{T} ∫_{S} \frac{∂J}{∂θ} (â(m) − â(m_{θ}^{ϵ_{1}}))(\frac{h′(m_{θ+ϵ_{2}}^{ϵ_{1}}) − h′(m_{θ−ϵ_{2}}^{ϵ_{1}})}{2ϵ_{2}})dθdt → 0.  (2.46)

Therefore, for ϵ_{1} and ϵ_{2} suficiently small,

\begin{aligned}M − 2ϵ ≤ & ∫_{S} J(T, θ)μ(T, dθ) − ∫_{S} J(0, θ)μ(0, dθ) − ∫_{0}^{T} ∫_{S} \frac{∂J}{∂t} (t, θ)μ(t, dθ)dt \\ & + \frac{1}{2} ∫_{0}^{T} ∫_{S} \frac{∂J}{∂θ} (t, θ)â(m_{θ}^{ϵ_{1}})(\frac{h′(m_{θ+ϵ_{2}}^{ϵ_{1}}) − h′(m_{θ−ϵ_{2}}^{ϵ_{1}})}{2ϵ_{2}})dθdt \\ & − \frac{1}{2} ∫_{0}^{T} ∫_{S}(\frac{∂J}{∂θ} (t, θ))^{2}â(m_{θ}^{ϵ_{1}})dθdt. \end{aligned} (2.47)

Comparing with (2.6) we see that Lemma 2.2 follows.

Proof of Lemma 2.3. For smooth test functions J(t, θ) on [0, T] × S and ϵ_{1}, ϵ_{2} > 0 consider the family of functionals given by

F̂_{J,ϵ_{1},ϵ_{2}}(μ(·, ·))=2∫_{0}^{T}∫_{S}J(t, θ)(\frac{h′(m_{θ+ϵ_{2}}^{ϵ_{1}}) − h′(m_{θ−ϵ_{2}}^{ϵ_{1}})}{2ϵ_{2}})dθdt − ∫_{0}^{T}∫_{S}J^{2}(t, θ)â^{−1}(m_{θ}^{ϵ_{1}})dθdt,

(2.48) so that for each μ(·, ·),

lim_{ϵ_{2}→0} lim sup_{ϵ_{1}→0} sup_{J} F̂_{J,ϵ_{1},ϵ_{2}}(μ(·, ·)) = I_{0}(μ(·, ·)),  (2.49)

in the sense that if the right hand side is infinite then the left hand side is as well. Let

\begin{matrix}F_{J,ϵ_{1},ϵ_{2},N}(x(·)) = & 2 ∫_{0}^{T} \frac{1}{N} ∑_{i=1}^{N} J(t, i/N)(\frac{h′(m_{i/N+ϵ_{2}}^{e_{1}}) − h′(m_{i/N−ϵ_{2}}^{ϵ_{1}})}{2ϵ_{2}})dt \\ − ∫_{0}^{T} \frac{1}{N} ∑_{i=1}^{N} J^{2}(t, i/N)â^{−1}(m_{i/N}^{ϵ_{1}})dt, \end{matrix}  (2.50)

so that F_{J,ϵ_{1},ϵ_{2},N} differs from the image of F̂_{J,ϵ_{1},ϵ_{2}} under the mapping x ↦ μ by o(1). To prove the lemma it is therefore enough to show that for each fixed J,

lim_{ϵ_{2}→0} lim sup_{ϵ_{1}→0} lim sup_{N→∞} \frac{1}{N} log E^{P_{N}} [exp NF_{J,ϵ_{1},ϵ_{2},N}] ≤ 0.  (2.51)

This follows from Theorems 4.1 and 6.7, and Lemmas 7.2, 7.3, and 7.4 of [V2].

III. Hydrodynamic limit for system with weak driving forces

We introduce a configuration dependent weak driving force into the system by changing the generator to

L_{N,H,f} = L_{N} + N/2 ∑_{i=1}^{N} H(t, i/N)a(x_{i}, x_{i+1}) (1 + D^{ii+1}ξ_{f}) D^{ii+1},  (3.1)

where H(t, θ) is a continuous function on [0, T]×S, f(t, θ, x_{−r}, ..., x_{r}) is a smooth function on [0, T] × S × R I ^{2r+1} with bounded first derivatives, and ξ_{f} = ∑_{i=1}^{N} f(t, θ, x_{i−r}, ..., x_{i+r}). The ξ_{f} term does not affect the hydrodynamics, but it allows us to obtain the correct lower bound in Section IV. Let m(θ) be a smooth function on S and let f_{N} = exp ∑_{i=1}^{N} h′(m(i/N))x_{i} − log M(h′(m(i/N))). Corresponding to the generator L_{N,H,f} and initial distribution f_{N}Φ_{N} we have probability measures Q_{N,H,f} on C([0, T] → R I ^{N}) and Q̂_{N,H,f} on X.

Theorem 3.1. For each continuous H and smooth local f,

Q̂_{N,H,f} ⇒ δ_{m(t,θ)dθ},  (3.2)

where δ_{m(t,θ)dθ} is the probability measure concentrated on the unique weak solution of

\begin{matrix}\frac{∂m}{∂t} = \frac{1}{2} \frac{∂}{∂θ} & (â(m) (\frac{∂}{∂θ} h′(m) + H)) \\ m(0, θ) & = m(θ) \end{matrix}  (3.3)

Proof. For dQ_{N,H,f}/dP_{N} we have the following Cameron–Martin–Girsanov formula,

\begin{matrix}\frac{dQ_{N,H,f}}{dP_{N}} = f_{N}(x(0)) exp\{ & ∑_{i=1}^{N} ∫_{0}^{T} H(t, i/N)(1 + D^{ii+1}ξ_{f})\sqrt{a (x_{i}, x_{i+1})}dβ_{i} \\ −\frac{1}{2} & ∑_{i=1}^{N} ∫_{0}^{T} H^{2}(t, i/N)a(x_{i}, x_{i+1})(1 + D^{ii+1}ξ_{f})^{2}dt\}. \end{matrix}  (3.4)

Using this formula it is quite easy to show that there exists C < ∞ so that for all γ > 0,

E^{P_{N}} [(\frac{dQ_{N,H,f}}{dP_{N}} )^{1+γ}] ≤ e^{Cγ(1+γ)N}.  (3.5)

Therefore, as discussed in (2.8), by Hölder’s inequality superexponential estimates for P_{N} extend to Q_{N,H,f}. In particular Q̂_{N,H,f} has a weak limit, Q̂_{H,f}. Let

F̂_{J,G,ϵ_{1},ϵ_{2}}^{H} = F̂_{J,G,ϵ_{1},ϵ_{2}} + \frac{1}{2} ∫_{0}^{T} ∫_{S} \frac{∂J}{∂θ} (t, θ)â(m_{θ}^{ϵ_{1}})H(t, θ)dθdt,  (3.6)

where F̂_{J,G,ϵ_{1},ϵ_{2}} is defined in (2.6). We claim that for each fixed J and G,

lim_{ϵ_{2}→0} lim sup_{ϵ_{1}→0} lim sup_{N→∞} \frac{1}{N} log EQ̂_{N,H,f} [exp NF̂_{J,G,ϵ_{1},ϵ_{2}}^{H}] ≤ 0.  (3.7)

To this end note that exp NF^{2}(T) is a martingale under Q_{N,H,f}, where

F^{2}(T) = F^{1}(T) + ∫_{0}^{T} \frac{1}{2} ∑_{i=1}^{N} \frac{∂J}{∂θ} (t, i/N)H(t, i/N)ν^{i}(x(t))dt + o(1).  (3.8)

Here F^{1} is defined in (2.28) and

ν^{i} = a(x_{i}, x_{i+1}) (1 + D^{ii+1}ξ_{f}) (1 + D^{ii+1}ξ_{g}) .  (3.9)

where ξ_{g} is given by

ξ_{g} = ∑_{i=1}^{N} g(x_{i−l}, ..., x_{i+l}, m_{i/N}^{k/N}).  (3.10)

Let

V^{6} = \frac{1}{N} ∑_{i=1}^{N} \frac{∂J}{∂θ} (t, i/N)H(t, i/N) (ν^{i} − â(m_{i/N}^{ϵ_{1}})) .  (3.11)

Let α > 0, ϵ > 0, and choose C < ∞ as in (2.35) and g(x_{−l}, ..., x_{l}, m) with l > r such that A(g(m), m) ≤ a^{∗}, and

sup_{|m|≤C} A(g(m), m) ≤ ϵα^{−1}∥\frac{∂J}{∂θ} ∥_{∞}^{−1}  (3.12)

Let g^{∗}(x_{−l}, ..., x_{l}, m) be the minimiser of A(g, m) among functions of x_{−l} through x_{l}. We can assume in addition that

sup_{|m|≤C} E^{π_{m}}[a(x_{1}, x_{2}) (D^{12}ξ_{g^{∗}} − D^{12}ξ_{g})^{2}] ≤ ϵ^{2}α^{−2}∥\frac{∂J}{∂θ} ∥_{∞}^{−2}  (3.13)

and that E^{π_{m}}[a(x_{1}, x_{2})(D^{12}ξ_{f})^{2}] and E^{π_{m}}[a(x_{1}, x_{2})(D^{12}ξ_{g})^{2}] are bounded by some M < ∞. Since g^{∗} is the minimiser we have

E^{π_{m}}[a(x_{1}, x_{2})(D^{12}ξ_{f} − D^{12}ξ_{g})(1 − D^{12}ξ_{g^{∗}})] = 0.  (3.14)

By Schwarz’s inequality, we have for |m| ≤ C,

E^{π_{m}}[ν^{i}] − â(m) ≤ ϵα^{−1}∥\frac{∂J}{∂θ} ∥_{∞}^{−1}(1 + \sqrt{4M})  (3.15)

and for |m| > C,

E^{π_{m}}[ν^{i}] − â(m) ≤ M.  (3.16)

Then

\begin{aligned}V^{6} ≤ & \frac{1}{N} ∑_{i=1}^{N} \frac{∂J}{∂θ} (t, i/N)H(t, i/N)(ν^{i} − E^{π_{m{ϵ_{1} \atop i/N}}} [ν^{i}]) \\ & + \frac{1}{N} ∑_{i=1}^{N} |\frac{∂J}{∂θ} (t, i/N)||H(t, i/N)|M11_{\{|m_{i/N}^{ϵ_{1}} |>C\}} \\ & + \frac{1}{2N} ∑_{i=1}^{N} |\frac{∂J}{∂θ} (t, i/N)||H(t, i/N)|ϵα^{−1}∥\frac{∂J}{∂θ} ∥_{∞}^{−1}(1 + \sqrt{4M}). \end{aligned}  (3.17)

So, by Schwarz’s inequality, if we take the limit superior as N → ∞, followed by K → ∞, then ϵ_{1} → 0, we get

lim sup_{{ϵ_{1}→0 \atop k→∞}} lim sup_{N→∞} \frac{1}{N} log E^{P_{N}} [exp αN ∫_{0}^{T} V^{6}(x(t), t)dt]

≤ lim sup_{{ϵ_{1}→0 \atop k→∞}} lim sup_{N→∞} \frac{1}{2N} log E^{P_{N}} [exp 2αN∫_{0}^{T} ∑_{i=1}^{N} \frac{∂J}{∂θ} (t, i/N)H(t, i/N)(ν^{i} − E^{π_{m{ϵ_{1} \atop i/N}}} [ν^{i}])dt]

+ lim sup_{{ϵ_{1}→0 \atop k→∞}} lim sup_{N→∞} \frac{1}{2N} log E^{P_{N}} [exp 2αN∫_{0}^{T} ∑_{i=1}^{N} |\frac{∂J}{∂θ} (t, i/N)||H(t, i/N)|M11_{\{|m_{i/N}^{ϵ_{1}} |>C\}}dt]

+ ϵ(1 + \sqrt{4M}) ∫_{0}^{T} ∫_{S} |H(t, θ)|dθdt

≤ 0 (by (2.39)) + ϵ (for sufficiently large C) + const.ϵ.

(3.18)

If F^{H} is the image of F̂^{H} under x ↦ μ, then

\begin{aligned}F^{H} = & (F^{1}(T) + ∫_{0}^{T} \frac{1}{2} ∑_{i=1}^{N}(J(t, i + 1/N) − J(t, i/N))H(t, i/N)ν^{i}(x(t))dt) \\ & + (F − F^{1}) − (∫_{0}^{T} V^{6}dt) + o(1). \end{aligned}  (3.19)

So (3.7) follows by Hölder’s inequality and (2.32). If I_{0}(μ(·, ·)) < ∞ then as in Lemma 2.2 we can calculate lim sup_{ϵ_{1},ϵ_{2}→0} sup_{J,G} F̂_{J,G,ϵ_{1},ϵ_{2}}^{H}(μ(·, ·)) > 0 unless m(t, θ) is a weak solution of (3.3). Therefore Q̂_{H,f} is concentrated on such solutions. It only remains to prove uniqueness for weak solutions of equation (3.3). It is shown in [V2] that â is continuous in m. A general result for equations of the form (3.3), whose coefficients are only known to be continuous, can be found in [BT].

IV. Lower bound

For a smooth m(·, ·) we can find a continuous H so that m is a weak solution of (3.3). We can also assume that for each 0 ≤ t ≤ T,

∫_{S} â(m(t, θ))H(t, θ)dθ = 0.  (4.1)

For each smooth local f we construct Q̂_{N,H,f} which by Theorem 3.1 converges weakly to the probability measure on X concentrated on m(·, ·). By the Cameron-Martin-Girsanov formula,

\begin{aligned}\frac{1}{N} log \frac{dQ_{N}}{dQ_{N,H,f}} = & \frac{1}{N} log \frac{f_{N}^{0}(x(0))}{f_{N}(x(0))} \\ & + \frac{1}{2N} ∑_{i=1}^{N} ∫_{0}^{T} H(t, i/N) (1 + D^{ii+1}ξ_{f}) \sqrt{a(x_{i}, x_{i+1})}dβ_{i} \\ & − \frac{1}{8N} ∑_{i=1}^{N} ∫_{0}^{T} H^{2}(t, i/N)a(x_{i}, x_{i+1}) (1 + D^{ii+1}ξ_{f})^{2} dt. \end{aligned}  (4.2)

Let

â_{f}(m) = E^{π_{m}}[a(x_{1}, x_{2}) (1+D^{12}ξ_{f})^{2}]  (4.3)

Then

lim_{N→∞} \frac{1}{N} log \frac{dQ_{N}}{dQ_{N,H,f}} = −I_{static}(m(0, ·)) − \frac{1}{8} ∫_{0}^{T} ∫_{S} â_{f}(m(t, θ))H^{2}(t, θ)dθdt  (4.4)

in Q_{N,H,f}-probability. Call the right hand side −I^{f}(μ(·, ·)). This provides us with a family of lower bounds: For each open U ⊂ X,

lim inf_{N→∞} \frac{1}{N} log Q̂_{N}(U) ≥ − inf_{μ(·,·)∈U} I^{f}(μ(·, ·)).  (4.5)

By the following lemma we obtain the lower bound (1.22) by optimising (4.3) over f. Lemma 4.1. I(μ(·, ·)) = inf_{f} I^{f}(μ(·, ·)) Proof. Let ϵ > 0. Since m(t, θ) is bounded, we can choose f(t, θ, x_{−r}, ..., x_{r}) so that

1 ≥ \frac{â(m(t, θ))}{â_{f}(m(t, θ))} > 1 − ϵ.  (4.6)

Since H is continuous, and ∫_{S} âHdθ = 0,

\frac{1}{8} ∫_{0}^{T} ∫_{S} H^{2}(t, θ)â_{f}(m(t, θ))dθdt = sup_{φ}\{\frac{1}{2} ∫_{0}^{T}∫_{S} H(t, θ)â(m(t, θ))\frac{∂φ}{∂θ}(t, θ)dθdt  (4.7)

− \frac{1}{2} ∫_{0}^{T}∫_{S} (\frac{∂φ}{∂θ}(t, θ))^{2} â(m(t, θ))\frac{â(m(t, θ))}{â_{f}(m(t, θ))}dθdt\}

Since m is a weak solution of (3.3),

\frac{1}{2} ∫_{0}^{T} ∫_{S} â(m)H \frac{∂φ}{∂θ}dθdt = − ∫_{S} mφdθ|_{0}^{T} + ∫_{0}^{T} ∫_{S} m\frac{∂φ}{∂θ}dθdt − \frac{1}{2} ∫_{0}^{T} â(m)\frac{∂h′}{∂θ} \frac{∂φ}{∂θ}dθdt.

(4.8)

Therefore,

I_{dynamic}(m(·, ·)) ≤ \frac{1}{8} ∫_{0}^{T} ∫_{S} H^{2}(t, θ)â_{f}(m(t, θ))dθdt < (1 − ϵ)^{−1}I_{dynamic}(m(·, ·)). (4.9)

Since ϵ > 0 is arbitrary this proves the lemma.

To complete the argument it remains to show that for any μ(·, ·) with I(μ(·, ·)) < ∞ we can find a sequence of smooth functions μ^{k}(·, ·) → μ(·, ·) with I(μ^{k}(·, ·)) → I(μ(·, ·)). To do this we first approximate m(t, θ) by m^{C}(t, θ) = −C ∨m(t, θ)∧C. Then we convolute with a smoothing kernel φ_{ϵ}(t, θ), and use the fact that for |m| ≤ C, b(m) = â(m)h′′(m) is continuous and bounded away from both 0 and ∞.

References

[BT] Bénilan, P. and Touré, H., Sur l’équation générale u_{t} = φ(u)_{xx} − ψ(u)_{x} + v, C. R. Acad. Sci. Paris, t.299, Série 1.no 18, 1984.

[DV] Donsker, M., and Varadhan, S. R. S., Large deviations from a hydrodynamic scaling limit, Comm. Pure Appl. Math. 42 , 243–270, 1989.

[GPV] Guo, M. Z., Papanicolaou, G. C., and Varadhan, S. R. S., Nonlinear diffusion limit for

a system with nearest neighbor interactions, Comm. Math. Phys. 118, 31–59, 1988.

[V1] Varadhan, S. R. S., Large Deviations and Applications, CBMS-NSF Regional Conference Series in Applied Mathematics 46, SIAM, 1984.

[V2] Varadhan, S. R. S., Nonlinear diffusion limit for a system with nearest neighbor interactions II, Proc. Taniguchi Symp., 1990, Kyoto.

Jeremy Quastel Department of Mathematics, University of California Davis, CA 95616; e-mail: quastel@ucdmath.ucdavis.edu