Markov Processes Relat. Fields 18, 71–88 (2012)
º·
Markov
MP
Processes
&R
F
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and
Related Fields
c Polymat, Moscow 2012
°
Helffer – Sjöstrand Representation for
Conservative Dynamics∗
T. Bodineau and B. Graham
Département de mathématiques et applications, Ecole Normale Supérieure, UMR 8553,
75230 Paris cedex 05, France
Received October 20, 2011
Abstract. We consider a Helffer – Sjöstrand representation for the correlations
in canonical Gibbs measures with convex interactions under conservative Ginzburg – Landau dynamics. We investigate the rate of relaxation to equilibrium.
Keywords: Ginzburg – Landau dynamics, canonical Gibbs measure, random walk
representation, monotonicity
AMS Subject Classification: 60K35, 82C24
1. Introduction
The complicated interactions in particle systems lead to subtle correlations
which in some cases can be represented in terms of simpler quantities, e.g. random walk crossings [9]. A celebrated representation for the correlations of Gibbs
measures has been obtained by Helffer and Sjöstrand by means of the Witten
Laplacian [16, 17]. In this representation, the decay of correlations is related to
spectral properties and it can be studied by using spectral theory [13–15]. For
effective interface models, this representation triggered a probabilistic reinterpretation of the Witten Laplacian as the generator of a random walk coupled
to the evolution of the particle system [8, 12, 25]. For a large class of effective
interface models, the correlations between two sites x, y can be understood as
the total time spent at y by a random walk (in a random environment) starting from x. The strength of the Helffer – Sjöstrand representation is to relate
∗ We thank T. Funaki, G. Giacomin, B. Helffer, S. Olla, H. Spohn for very helpful discussions. TB acknowledges the support of the French Ministry of Education through the
ANR BLAN07-2184264 grant. BG acknowledges the support of the Fondation Sciences
Mathématiques de Paris
72
T. Bodineau and B. Graham
the equilibrium correlations to the behavior of the dynamics associated to the
model.
In this paper, we investigate the Helffer – Sjöstrand representation for the
correlations in canonical Gibbs measures, i.e. measures conditioned to have a
fixed mean density. This relates the equilibrium correlations in canonical measures to the conservative Ginzburg – Landau dynamics. In contrast to the nonconservative dynamics considered in the previous works [8, 16], fixing the total
density leads to a Witten Laplacian with a different structure: the gradients
are now replaced by gradient differences. In one-dimension, we will show that
for a class of Hamiltonian with convex interactions, (2.3), the correlations for
the canonical Gibbs measure can be interpreted as the occupation time of a
random walk evolving in a random environment coupled to the conservative
Ginzburg – Landau dynamics. Furthermore, the space-time correlations of the
dynamics are also encoded in the diffusive mechanism of the random walk. For
some specific dynamics, like the symmetric simple exclusion process, a similar
property for the correlations was obtained by duality [23]. In our model, the
random walk is not a consequence of a duality property but it is reminiscent of
the stochastic process considered in [8].
We will use the Helffer – Sjöstrand representation to study the relaxation
of the one-dimensional conservative Ginzburg – Landau dynamics. Equilibrium
fluctuations of the Ginzburg – Landau model and the convergence of the density
field to a generalized Ornstein – Uhlenbeck process have been obtained in [24,28].
The relaxation is also expected to occur at a microscopic scale: for initial data
η ∈ RZ sampled from the equilibrium measure h · iρ , the space-time correlation is
conjectured to obey the following scaling form for large t and |i| ( [27, page 177,
equation (2.14)])
hη0 (0); ηi (t)iρ ≈ p
χ(ρ)
2π q̂(ρ)t
³
exp −
i2 ´
,
2tq̂(ρ)
(1.1)
where q̂(ρ) is a diffusion coefficient and χ(ρ) = hη0 ; η0 iρ is the susceptibility. The
intuition behind the scaling (1.1) is that an initial fluctuation of the density at
the origin will diffuse in the course of time. By combining localization techniques and spectral gap bounds, sharp relaxation estimates of the type (1.1)
for functions supported in a neighborhood of the origin were derived (in any
dimension) for discrete models in [1–4, 18] and for continuous variables in [21].
For continuous variable models, the Helffer – Sjöstrand representation provides an alternative way to understand (1.1) as it relates the correlation between
the origin and a site i at time t to the probability that the random walk starting
at 0 touches the site i at time t. This random walk evolves in a random environment coupled to the evolution of the particle system, therefore its precise
limiting properties are difficult to study. By using the general theory of Aronson, De Giorgi, Nash, Moser for uniformly elliptic second order operators, some
Helffer – Sjöstrand representation for conservative dynamics
73
estimates can be obtained on the kernel of this random walk. This leads to
bounds on the correlation functions in (1.1) (see Section 4). In the spirit of [12],
more global relaxation estimates can be obtained by using the homogenization
theory of Kipnis, Varadhan [19] (see Section 5).
2. Ginzburg – Landau dynamics
Let Λ denote the one dimensional torus Λ = (Z/N Z) with nearest neighbor edges. We are going to study the relaxation properties of conservative
Ginzburg – Landau dynamics on RΛ
´
X ³ ∂H
√ ¡
¢
∂H
∀i ∈ Λ, dηi (t) =
(η) −
(η) dt + 2 dB(i,i+1) (t) − dB(i−1,i) (t) ,
∂ηj
∂ηi
j=i±1
(2.1)
where (B(i,i+1) (t))i∈Λ denote independent standard Brownian motions associated to each edge and we will consider Hamiltonians of the form
X
H(η) =
V1 (ηi ) + V2 (ηi + ηi+1 ).
(2.2)
i∈Λ
We will require convexity assumptions on the potentials, i.e. that there are
constants C± such that for
0 < C− 6 Vk00 ( · ) 6 C+ ,
k = 1, 2.
(2.3)
The dynamics (2.1) can be extended on Z (see [10, 28]). In Sections 4 and 5,
we will investigate the relaxation properties in the infinite volume limit and for
this, we will have to restrict to Hamiltonians without interactions, i.e. V2 = 0.
The Gibbs measure on RΛ associated to H will be denoted by µN and the
canonical Gibbs measure with mean density ρ by
³ ¯X
´
¯
µρ,N ( · ) = µN · ¯
ηi = |Λ|ρ .
i
We will write h · i to denote
P expectation with respect to µρ,N .
Λ
Let Sρ,N =
{η
∈
R
:
i ηi = |Λ|ρ}. In Λ, the dynamics (2.1) conserve the
P
total density i∈Λ ηi . We will see in (2.4) that µρ,N is a (reversible) invariant
measure.
The generator of the dynamics (2.1) is given by
LΛ =
X ³
−
i∈Λ
∂
∂ ´2 ³ ∂H
∂H ´³ ∂
∂ ´
−
+
−
−
.
∂ηi+1
∂ηi
∂ηi+1
∂ηi
∂ηi+1
∂ηi
Let B = {(i, i+1)}i∈Λ denote the set of oriented nearest neighbor bonds of Λ. It
is natural to associate a differential operator with each edge. For b = (i, j) ∈ B,
74
T. Bodineau and B. Graham
¯ to denote the vector
we will write ∂b to denote ∂/∂ηj − ∂/∂ηi . We will write ∇
¯
of all such operators: ∇ = (∂b )b∈B . The generator LΛ can now be written:
¯ ·∇
¯ + ∇H
¯ · ∇.
¯
LΛ = −∇
Unless otherwise stated, take ρ and N to be fixed quantities. Note that there
is an integration by parts formula
­
® ­
® ­
®
¯ = −g ∇f
¯ + f g ∇H
¯
f ∇g
,
(2.4)
where h · i denotes the expectation with respect to µρ,N . Thus LΛ is self-adjoint
with respect to µρ,N and the dynamics are reversible. The operator LΛ has a
self-adjoint extension with domain included in L2 (Sρ,N , µρ,N ).
3. Correlations for the canonical measure
3.1. Helffer – Sjöstrand representation
We will derive a formula similar to the Helffer – Sjöstrand representation
[13–15, 17] for the correlations under the canonical Gibbs measure µρ,N . We
follow a formalism similar to the one applied previously for the non-conservative
(Langevin) dynamics [16].
The correlation between two functions f, g under the measure µρ,N is defined by
­
®
hf ; gi = (f − hf i)(g − hgi) .
The operator LΛ has a spectral gap (see [6,20] or the Appendix for an alternative
proof) and thus for any smooth function g in C0∞ (Sρ,N , R) there exists a unique
inverse u in C0∞ (Sρ,N , R) such that
LΛ u = g − hgi .
Using integration by parts (2.4), one has:
­
®
hf ; gi = (f − hf i)LΛ u = E(f, u),
where the Dirichlet form E is defined by
­
®
¯ · ∇u
¯ .
E(f, u) = hf LΛ ui = ∇f
(3.1)
Let Hess H denote the (bond-wise) Hessian matrix
£
¤
Hess H = ∂b ∂c H b,c∈B .
One has
¯ = ∇L
¯ Λ u = LΛ ∇u
¯
∇g
(3.2)
Helffer – Sjöstrand representation for conservative dynamics
75
where LΛ denotes the ‘Witten-Laplacian’ operator defined on C0∞ (Sρ,N , RB ) by,
LΛ U (η) = LΛ ⊗ Id U (η) + (Hess H) · U (η),
U : RΛ → RB .
Here LΛ ⊗ Id denotes the identity matrix with diagonal elements equal to LΛ .
Note that the operator LΛ is also self-adjoint in L2 (Sρ,N , µρ,N ).
Combining (3.1) and (3.2), we therefore have for any f, g in C0∞ (Sρ,N , R)
­
®
¯ L−1 ∇g
¯ .
hf ; gi = ∇f
(3.3)
Λ
We remark that the Witten Laplacian LΛ and the identity (3.3) could have
been defined for more general Hamiltonians than (2.2), and in any dimension.
In the Appendix, we exploit (3.3) to estimate the spectral gap in dimension
d > 1. However the interpretation of LΛ as the generator of stochastic dynamics
(Section 3.2) requires the assumption (2.3) on the Hamiltonian H and the onedimensional structure.
3.2. The random walk representation
Let X(t) denote a continuous time random walk on the edges in B with jump
rates determined by the Hessian of H: X(t) steps from b to c at rate −∂b ∂c H.
Thus if X(t) = b = (i, i + 1), the non-zero jump rate to the bond b + k =
(i + k, i + k + 1) is given by (with k ∈ {−2, −1, 1, 2})
−∂b ∂b−2 H
−∂b ∂b−1 H
−∂b ∂b+1 H
−∂b ∂b+2 H
= V200 (ηi + ηi−1 ),
= V100 (ηi ),
= V100 (ηi+1 ),
= V200 (ηi+1 + ηi+2 ).
Thus LΛ can be interpreted as the generator of the joint evolution (η(t), X(t))
in Sρ,N × B. On this space, the representation (3.3) has a probabilistic interpretation.
Proposition 3.1. Let Eηb denote the expectation for the joint process starting
at η(0) = η and X(0) = b.
­
®
¯ L−1 ∇g
¯ =
hf ; gi = ∇f
Λ
Z∞ X
­
¡
¢®
∂b f (η)Eηb 1X(t)=c ∂c g(η(t)) dt.
(3.4)
0 b,c∈B
With no extra effort, we can extend the Hamiltonian to include terms such
as V3 (ηi + ηi+1 + ηi+2 ), V4 (ηi + ηi+1 + ηi+2 + ηi+3 ), and so on. In this case the
random walk would perform jumps from (i, i + 1) of length k with intensities
Vk00 (ηi + ηi−1 + · · · + ηi−k+1 ) and Vk00 (ηi+1 + ηi+2 + · · · + ηi+k ). For this class
76
T. Bodineau and B. Graham
of models, the variables η can be reinterpreted as the gradient field of effective interface models and therefore the representation (3.4) is equivalent to the
one derived for the non-conservative Ginzburg – Landau dynamics (see for example [8, Proposition 2.2]). However, looking directly at the non-conservative
dynamics changes the point of view. Below, we derive new identities for the
conservative Ginzburg – Landau dynamics.
It does not seem possible to extend this random walk representation to
dimensions d > 2. The matrix −Hess H cannot be interpreted (in an easy
way) as generating a continuous time Markov chain: many of the off-diagonal
elements are negative, so they cannot be interpreted as jump rates.
Proof of Proposition 3.1. The evolution under the semi-group generated by LΛ
can be rewritten as follow. For any g in C0∞ (Sρ,N , R)
³ hX
i´
¯
exp{−tLΛ }∇g(η)
= Eηb
1X(t)=c ∂c g(ηt )
.
b∈B
c
Applying LΛ yields,
µ ZT
ZT
¯
exp{−tLΛ }∇g(η)
dt = LΛ
LΛ
0
Eηb
0
hX
i ¶
1X(t)=c ∂c g(ηt ) dt
b∈B
c
¯
¯
= ∇g(η)
− exp{−T LΛ }∇g(η).
The space of gradients is invariant under the operator LΛ , and therefore
under the semi-group. Theorem A.1 implies that
³ kC ´­¡
­
®
¢2 ®
−
¯
¯
¯
∇g(η)
exp{−tLΛ }∇g(η)
6 exp −
t
∇g(η)
.
N2
Taking the limit T → ∞ yields
µ Z∞
Eηb
LΛ
hX
i ¶
1X(t)=c ∂c g(ηt ) dt
c
0
Hence
µ Z∞
¯
L−1
Λ ∇g
Eηb
=
0
can be substituted into (3.3).
¯
= ∇g.
b∈B
hX
c
i¶
1X(t)=c ∂c g(ηt ) dt
b∈B
2
The joint dynamics including the field η(t) and the random walk X(t) can
be extended to Z (see e.g. [10]). In Section 4, we will relate the relaxation to
equilibrium in Z to the fluctuation of the random walk X(t).
Helffer – Sjöstrand representation for conservative dynamics
77
4. Relaxation to equilibrium
In this section, we will consider the relaxationPof the dynamics on Z. We
focus only on systems with Hamiltonians H(η) = x V (ηx ) and with a potential V satisfying the convexity assumption (2.3). This is equivalent to setting
V1 = V and V2 = 0 in (2.2). Let h · iρ denote the corresponding invariant measure on Z with mean density ρ; h · iρ is a product measure and the marginal
probability density function can be written
Z −1 exp(−V (η) + αη),
where Z and α are constant chosen to give mass one and mean ρ.
The random walk representation provides a clear connection between the relaxation of the particle system and the diffusive mechanism conjectured in (1.1).
Proposition 4.1. For any i and t > 0, the following identity holds
­ 0
®
­
¡
¢®
V (η0 (0)); ηi (t) ρ = Eη(0,1) 1X(t)=(i,i+1) ρ .
(4.1)
In the Gaussian case V (x) = x2 /2, the jump rates are uniform and the
identity (4.1) coincides with the conjecture (1.1)
­
®
¡
¢
η0 (0); ηi (t) ρ = E(0,1) 1X(t)=(i,i+1) '
³ i2 ´
1
exp
−
,
4t
(4πt)1/2
where Xt is just a simple random walk. A similar relation also holds for the
space-time correlations of the symmetric simple exclusion process [23]. For
general potentials, (4.1) confirms the conjecture (1.1) as it explicitly relates the
relaxation to equilibrium to the relaxation of the random walk.
Relation (4.1) is non-symmetric, however upper and lower bound on the
relaxation of the two-point correlation function can be obtained.
Proposition 4.2. For any i and t > 0, one has
1
hη0 (0); ηi (t)iρ
1
® 6
6­ η
,
C+
C−
E(0,1) (1X(t)=(i,i+1) ) ρ
(4.2)
where the constants C± were introduced in (2.3).
The previous estimates (4.2) can be turned into quantitative bounds using
Aronson estimates for the transition kernel of strictly elliptic operators. Following [12], there exists c1 , c2 such that for t > 1
³
­ η ¡
¢®
c1
|i| ´
E(0,1) 1X(t)=(i,i+1) ρ 6 √ exp − √ ,
t
c1 t
(4.3)
78
T. Bodineau and B. Graham
and for |i| 6
√
t
­ η ¡
¢®
E(0,1) 1X(t)=(i,i+1) ρ >
c2
√ .
1∨ t
(4.4)
Proof of Proposition 4.1. First we will prove (4.1) for the dynamics in a finite
domain Λ and then we will pass to the limit Λ → Z for a fixed time t.
Given two functions f and g, and a time t > 0, we can apply formula (3.4)
to f and exp{−tLΛ }g = Pt (g) which is the semi-group at time t for the
Ginzburg – Landau dynamics
­
® ­
®
f (η); exp{−tLΛ }g(η) = f (η); Pt (g)(η)
Z∞ X
­
¡
¢®
=
∂b f (η)Eηb 1X(s)=c ∂c g(η(s)) ds,
t
b,c∈B
where h · i refers to the finite volume measure µρ,N with the canonical constraint.
For f (η) = V 0 (η0 ) and g(η) = ηi , this gives

00

if b = (−1, 0),

∂b f = V (η0 ),
00
¯
∇f = ∂b f = −V (η0 ), if b = (0, 1),


∂ f = 0
otherwise,
b
and


if b = (i − 1, i),

∂b g = 1,
¯ = ∂b g = −1, if b = (i, i + 1),
∇g


∂ g = 0
otherwise.
b
Thus,
­ 0
®
V (η0 (0)); ηi (t)
Z∞
­
¡
¢® ­
¡
¢®
= ds V 00 (η0 )Eη(−1,0) 1X(s)=(i−1,i) + V 00 (η0 )Eη(0,1) 1X(s)=(i,i+1)
t
­
¡
¢® ­
¡
¢®
− V 00 (η0 )Eη(−1,0) 1X(s)=(i,i+1) − V 00 (η0 )Eη(0,1) 1X(s)=(i−1,i) .
We introduce the function
¡
¢
Fs (b, η) = Eηb 1X(s)=(i,i+1)
which is the probability that the walk starting at the edge b in an initial field η
is located at (i, i + 1) at time s.
Helffer – Sjöstrand representation for conservative dynamics
79
Using translation invariance,
Z∞
0
hV (η0 (0)); ηi (t)i =
­
£
¤®
ds V 00 (η1 ) Fs ((0, 1), η) − Fs ((1, 2), η)
t
­
£
¤®
+ V 00 (η0 ) Fs ((0, 1), η) − Fs ((−1, 0), η) .
As h · i is the invariant measure, hLΛ Fs ((0, 1), η)i = 0, and so
Z∞
0
hV (η0 (0)); ηi (t)i =
­
®
[LΛ Fs ]((0, 1), η) ds
t
Z∞
=−
­
®
∂s Fs ((0, 1), η) ds
t
­
®
­
®
= Ft ((0, 1), η) − lim Fs ((0, 1), η) .
s→∞
(4.5)
The random walk is uniformly distributed in the limit s → ∞, and so
hFs ((0, 1), η)i → |Λ|−1
where |Λ| is the number of sites in Λ.
For any fixed time t > 0, one can take the limit Λ → Z in (4.5). This
concludes the proof of (4.1).
2
Proof of Proposition 4.2. The dynamics have a monotonicity property.
Lemma 4.1. Let f, g : RZ → R be two locally defined functions that are
non-decreasing with respect to the coordinatewise partial order. For t > 0,
hf (η(0)); g(η(t))iρ > 0.
We postpone the proof of the Lemma and first use it to deduce Proposition 4.2. The function h(ξ) = V 0 (ξ)/C− − ξ is non-decreasing. By Lemma 4.1,
­
®
0 6 h(η0 (0)); ηi (t) ρ .
Hence, by the bilinearity of covariances,
­
η0 (0); ηi (t)
®
ρ
6
®
¢®
1 ­ 0
1 ­ η ¡
V (η0 (0)); ηi (t) ρ =
E(0,1) 1X(t)=(i,i+1) ρ .
C−
C−
The lower bound follows in the same way.
2
Monotonicity properties for non-conservative dynamics have been considered
in [11]. We provide below an alternative proof tailored for our model.
80
T. Bodineau and B. Graham
Proof of Lemma 4.1. By a standard approximation argument, it is sufficient to
show that for locally defined, increasing events A and B,
µρ (η(0) ∈ A
and η(t) ∈ B) > µρ (A)µρ (B).
(4.6)
Let η, η̃ represent two solutions to the SDE (2.1) such that
(i) η̃(0) has distribution µρ , and
(ii) η(0) has distribution µρ ( · | A).
We will write η̃(t) 6 η(t) if η̃i (t) 6 ηi (t) for all i ∈ Z. Note that (4.6) holds if
with probability one, η̃(t) 6 η(t).
The product measure µρ only differs from the conditional measure µρ ( · | A)
on the support of A. By Preston’s FKG inequality [26, Theorem 3], µρ ( · | A)
stochastically dominates µρ . In other words, there is a probability measure
under which η̃(0) 6 η(0) almost surely. We can then let η̃ and η evolve with the
same driving noise (Bb )b∈B .
Let ϕ(t) = η(t) − η̃(t) so that ϕ(0) > 0. By (2.1), if ϕi (t) < 0, then
X
d
ϕi (t) >
V 0 (ηj (t)) − V 0 (η̃j (t)) >
dt
j∼i
X
C+ ϕj (t).
(4.7)
{j∼i:ϕ(j)<0}
By [28, Theorem 2.1], for any r > 0, η and η̃ lie in the Hilbert space
n X
o
L2r = ζ :
ζi2 exp(−r|i|) 6 ∞ .
i∈Z
We can therefore define
A(t) =
X
2−|i| ϕi (t)2 1{ϕi (t)<0} .
i∈Z
2
Note that (d/dt)(t 1{t<0} ) = 2t 1{t<0} ; by (4.7) and the chain rule,
d
A(t) 6
dt
X
21−|i| C+ ϕi (t)ϕj (t).
{i∼j:ϕi (t),ϕj (t)<0}
Checking that for any ϕi−1 , ϕi , ϕi+1 ∈ R that
2−|i| ϕi (ϕi−1 + ϕi+1 ) 6 2−|i−1| ϕ2i−1 + 2−|i| ϕ2i + 2−|i+1| ϕ2i+1 ,
we obtain dA(t)/dt 6 6C+ A(t). Applying Gronwall’s lemma with
Zt
A(0) = 0,
A(t) 6
6C+ A(s) ds,
0
we find A(t) = 0 for all t > 0.
2
Helffer – Sjöstrand representation for conservative dynamics
81
5. Diffusion coefficient
In Propositions 4.1 and 4.2, the relaxation to equilibrium was rephrased in
terms of the diffusion of the random walk X(t). We compute below a Central
Limit Theorem for this random walk.
We consider the Ginzburg – Landau dynamics on Z starting from the equilibrium measure h · iρ at density ρ. We will show that after rescaling, the random
walk X(t) converges to a Brownian motion. With ε > 0, let X ε (t) = εX(ε−2 t)
for t > 0. Let Y (t) denote a Brownian motion with Y (0) = 0 and variance
q(ρ) > 0 (defined in (5.1)),
E0 [Y (s)Y (t)] = (s ∧ t)q(ρ).
Let R1 = V 00 (η1 ) (respectively, R−1 = V 00 (η0 )) denote the jump rate of the
random walk X from (0, 1) to (1, 2) (respectively, (−1, 0)). For i ∈ Z, let τi
denote the shift operator that moves vertex i to 0,
τi ηj = ηi+j ,
j ∈ Z.
Theorem 5.1. Over any finite time interval [0, T ], as ε → 0, X ε (t) → Y (t)
weakly in the Skorohod space, with
©­
®
ª
q(ρ) = 2 inf (1 − f (η) + f (τ1 η))2 R1 ρ + E(f, f ) .
(5.1)
f
The infimum is taken over smooth, bounded local functions f . By abuse of
notation, E(f, f ) stands for the Dirichlet form (3.1) extended to Z.
This formula implies that 2C− 6 q(ρ) 6 2C+ . For the upper bound, set
f = 1. The lower bound follows by an application of the Cauchy – Schwarz
inequality (see [12, (4.14)]).
Combining Proposition 4.1 and Theorem 5.1, we get a weaker formulation
of (1.1). Let ϕ be a smooth function with compact support [−1, 1]. For any
x ∈ R and t > 0
D
³ t ´E
X
lim V 0 (η0 (0)); ε
ϕ(εi)ηi+bxεc 2
ε→0
ε
ρ
Z
³
y2 ´
1
dy ϕ(y − x) exp −
=p
,
2tq(ρ)
2πq(ρ)t
R
where b·c stands for the integer part.
Proof of Theorem 5.1. We follow the approach of Kipnis and Varadhan (see [12,
Section 4]) and consider the process viewed from the position of the random
walk X(t). Let η̃(t) denote the configuration η(t) as viewed from X(t):
if X(t) = (i, i + 1),
η̃(t) = τi η(t).
82
T. Bodineau and B. Graham
Note that the edge (0, 1) in η̃(t) corresponds to the edge X(t) in η(t).
We can write the displacement of the random walk X(t) as the sum of a
drift term and a martingale,
Zt
X(t) − X(0) =
j(η̃(s)) ds + M (t).
0
Here j(η̃(s)) = (R1 −R−1 )(η̃(s)) is the drift of the random walk X at time s. The
process M (t) is a martingale with respect to the family of σ-algebras generated
by (η(s), X(s))s∈[0,t] ; M (t) is cadlag with the same jumps as X(t); E(M (t)2 ) is
therefore t hR−1 + R1 iρ = 2t hR1 iρ . We will apply Kipnis and Varadhan’s [19,
Theorem 1.8] to the drift term. To do this, there are two conditions we must
check, see Lemmas 5.1 and 5.2. First some notation. Let L̃ denote the generator
of η̃(t),
X £
¤
L̃F (η) = LF (η) +
F (η) − F (τk η) Rk ,
k=±1
where L stands for the generator of the Ginzburg – Landau dynamics on Z.
Define a Dirichlet form Ẽ by
­
®
Ẽ(f, f ) = f L̃f ρ .
Note that by translation invariance, h · iρ = µρ is the invariant measure corresponding to both L and L̃, and
­
®
Ẽ(f, g) = E(f, g) + (f (η) − f (τ1 η))R1 (g(η) − g(τ1 η)) ρ .
(5.2)
Define dual norms k · k1 and k · k−1 by
kf k21 = Ẽ(f, f ),
©
ª
kgk2−1 = sup 2 hf giρ − kf k21 .
f
Note that kgk−1 = ∞ unless hgi = 0.
Lemma 5.1. The shifted process (η̃(s)) is time ergodic with respect to µρ .
Proof. If exp{−L̃t}f = f for all t > 0, then L̃f = 0 and therefore Ẽ(f, f ) = 0.
As V 00 > C− ,
­
®
(5.3)
0 = Ẽ(f, f ) > C− (f (η) − f (τ1 η))2 ρ .
Hence f is translation invariant. The stationary measure is ergodic, so f is
constant with probability one.
2
Lemma 5.2. The drift function is finite in the dual norm: kjk−1 < ∞.
Helffer – Sjöstrand representation for conservative dynamics
83
Proof. A sufficient condition [19, 1.14] for kjk−1 < ∞ is that,
∀f ∈ D(L̃),
hf jiρ 6 Ckf k1 .
By (5.3) and translation invariance,
X
­
®
hf (η)j(η)iρ =
hf (η)kRk iρ = R1 (f (η) − f (τ1 η)) ρ
k=±1
­ ®1/2 ­
®1/2
6 R12 ρ (f (η) − f (τ1 η))2 ρ
­
®1/2
6 C+ (f (η) − f (τ1 η))2 ρ
−1/2
6 C+ C−
kf k1 .
2
We now seek to determine the diffusion coefficient q(ρ). The random walk
X(t) is antisymmetric [7]; on the time interval [0, T ], the law of (X(t), j(t)) is
equal to the law of (X(T − t) − X(T ), j(T − t)), but the displacement of the
random walk part is in the opposite direction. This symmetry implies that,
·
¸
Zt
E X(t) j(s) ds = 0.
0
This allows us to expand 1t E(Mt2 ) = 1t E((X(t) −
Rt
0
j(η̃(s)) ds)2 ):
·µ Zt
¶2 ¸
1
1
2
2 hR1 iρ = E (X(t) ) + E
j(η̃(s)) ds
.
t
t
0
By [19, Remark 1.7 and Theorem 1.8],
·µ Zt
¶2 ¸
1
lim E
j(η̃(s)) ds
= 2kjk2−1 .
t→∞ t
0
By (5.2),
©
ª
q(ρ) = 2 hR1 iρ − 2 sup 2 hf j(η)iρ − Ẽ(f, f )
f
ª
­
®
= 2 inf hR1 iρ − 2 hf (R1 − R−1 )iρ + (f (η) − f (τ1 η))2 R1 ρ + E(f, f ) .
©
f
The variational formula (5.1) now follows as hf R−1 iρ = hf (τ1 η)R1 iρ .
2
84
T. Bodineau and B. Graham
6. Conclusion
In this paper, we derived the Helffer – Sjöstrand representation for the equilibrium correlations in canonical Gibbs measures. For a class of one-dimensional
Hamiltonian, this representation can be reinterpreted in terms of a stochastic
process describing the joint evolution of the conservative Ginzburg – Landau
dynamics and a random walk coupled to this dynamics. Using the random
walk analogy, the diffusive relaxation of the Ginzburg – Landau dynamics can
be related to the diffusive behavior of this random walk (Proposition 4.1). In
this way several bounds on the return to equilibrium for the Ginzburg – Landau
dynamics are obtained (4.3), (4.4).
To sharpen the estimates in this paper and to derive the precise relaxation to
equilibrium conjectured in (1.1), one would need to prove a local central limit
theorem for the random walk in the Helffer – Sjöstrand representation. This
seems to be a challenging task. It would be also interesting to obtain a stochastic
interpretation of the Witten Laplacian in higher dimensions. This would provide
a straightforward approach to derive relaxation bounds for Ginzburg – Landau
dynamics in dimension d > 2.
Appendix. Spectral Gap
Sharp bounds on the spectral gap have been derived for conservative Ginzburg – Landau type dynamics in [5, 6, 20]. In this appendix, we show how
to recover these bounds by using the Witten Laplacian formalism when the
potential is strictly convex. We will follow the probabilistic approach devised
in [22].
In this appendix, we will relax some assumptions on the dynamics and we
suppose that it is defined in dimension d > 1. Let Λ denote the d-dimensional
torus (Z/N Z)d . Let B denote the set of oriented nearest neighbor edges of Λ;
¯ = (∂b )b∈B , the
how the edges are oriented will not be important. With ∇
definitions of LΛ and LΛ extend naturally to this higher dimensional setting.
Consider the Hamiltonian
X
X
H(η) = H1 (η) + H2 (η) =
V1 (ηx ) +
V2 (ηx + ηy ).
x
(x,y)∈B
We will assume that V1 is strictly convex, with V100 > C− > 0, and that V2 is
convex.
Let λ = λ(ρ, N, d) denote the spectral gap of the operator LΛ with respect
to µρ,N ,
E(f, f )
λ = inf
.
f ⊥1 hf ; f i
Helffer – Sjöstrand representation for conservative dynamics
85
The Dirichlet form E was introduced in (3.1). Define also
­
®
¯ · LΛ ∇f
¯
∇f
® .
λ̃ = inf ­ ¯
¯
f ⊥1
∇f · ∇f
Theorem A.1. There is a constant k such that for all N and ρ,
λ > λ̃ >
kC−
.
N2
The scaling N 2 is of the correct order as it characterizes the diffusive behavior
of the conservative dynamics. The assumption of strict convexity should only
be seen as a limitation in the method of proof.
Proof. We will first show that λ > λ̃. Recall that Pt = exp{−tLΛ } denotes the
semi-group associated with the dynamics. As P0 f = f and P∞ f = hf i,
­
®
hf ; f i = f [P0 f − P∞ f ] =
Z∞
­
®
f LΛ Pt f dt
0
Z∞
=
­
®
Pt/2 f LΛ Pt/2 f dt =
Z∞
E(Pt/2 f, Pt/2 f ) dt.
0
0
Let F (t) = E(Pt f, Pt f ). Then by (3.2)
­
®
­
®
¯ t f · ∇L
¯ Λ Pt f = −2 ∇P
¯ t f · LΛ ∇P
¯ tf .
F 0 (t) = −2 ∇P
For any u,
­
®
­
®
¯ · LΛ ∇u
¯ > λ̃ ∇u
¯ · ∇u
¯ = λ̃E(u, u).
∇u
With u = Pt f , F (t) 6 exp(−2tλ̃)F (0) and the bound on λ follows:
Z∞
hf ; f i =
Z∞
exp(−tλ̃)F (0) dt = λ̃−1 E(f, f ).
F (t/2) dt 6
0
0
¯ we need,
We will now show the lower bound for λ̃. With F = ∇f
hF · LΛ F i = hF · LΛ ⊗ Id F i + hF · (Hess H)F i ≥
The term hF · LΛ ⊗ Id F i is equal to
P
b
kC−
hF · F i .
N2
E(∂b f, ∂b f ) > 0, thus
hF · LΛ F i > hF · (Hess H)F i
Let S denote either {x} (for x ∈ Λ) or {x, y} (for (x, y) ∈ B). If S = {x}, let
W = V1 (ηx ); if S = {x, y}, let W = V2 (ηx + ηy ). Let vS (b) = +1 if b points in
86
T. Bodineau and B. Graham
to S, let vS (b) = −1 if b points out from S, and let vS (b) = 0, if b points neither
into or out from S. Note that ∂b ∂c W = vS (b)vS (c)W 00 and
F · (Hess W )F > C−
³X
vS (b)∂b f
´2
.
b
Hence,
F · (Hess H1 )F > C−
d
XµX
x
¶2
∂(x,x+ei ) f − ∂(x−ei ,x) f
i=1
and F · (Hess H2 )F > 0. It is sufficient now to show that,
d
XµX
x
¶2
>
∂(x,x+ei ) f − ∂(x−ei ,x) f
i=1
k
(F · F ).
N2
(A.1)
Let ϕx = ∂f /∂ηx , and let
g(ϕ) =
d
XµX
x
¶2
−ϕx−ei + 2ϕx − ϕx+ei
,
X
h(ϕ) =
i=1
(ϕx − ϕy )2 .
(x,y)∈B
Inequality (A.1) is equivalent to g(ϕ) > (k/N 2 )h(ϕ). Let Q = [qxy ] denote the generator matrix for the
P rate-1 nearest-neighbor simple random walk
on Λ: qxy = 1 iff x ∼ y and y qxy = 0. Q has |Λ| eigenvalues 0 = λ1 >
λ2 > . . . > λ|Λ| , and corresponding eigenvectors ψ1 , . . . , ψ|Λ| . Q has a uniform stationary distribution, so we can assume that the eigenvectors form an
orthonormal basis for `2 (RΛ ). The spectral gap of the walk |λ2 | is known to be
P|Λ|
at least k/N 2 with k a constant. Write ϕ = i=1 ai ψi . Then
|Λ|
X
X
2
g(ϕ) =
(ϕQ)(x) =
a2i λ2i ,
x
i=2
and
h(ϕ) = −
X
x
and so g(ϕ) > |λ2 |h(ϕ).
ϕ(x)(ϕQ)(x) > |λ2 |
|Λ|
X
a2i ,
i=2
2
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