Large deviations and gradient flows

0.23 MB·English

by  Stefan Adams

Large deviations and gradient flows
Stefan Adams∗ Nicolas Dirr† Mark Peletier‡ Johannes Zimmer§
January 24, 2012
2
1
Abstract
0
2 Inrecentwork[1]weuncoveredintriguingconnectionsbetweenOtto’scharacterisation
of diffusion as entropic gradient flow [16] on one hand and large-deviation principles
n
a describing the microscopic picture (Brownian motion) on the other. In this paper, we
J sketch this connection, show how it generalises to a wider class of systems, and comment

2 on consequences and implications.
2 Specifically, we connect macroscopic gradient flows with large deviation principles,

and point out the potential of a bigger picture emerging: we indicate that in some non-
]
P equilibrium situations, entropies and thermodynamic free energies can be derived via
A large deviation principles. The approach advocated here is different from the established
. hydrodynamic limit passage but extends a link that is well known in the equilibrium
h
situation.
t
a
m
1 Introduction
[

1
For systems in equilibrium, it is well known that the roles of energy and entropy can be
v
1 understoodrigorouslyintermsoflarge-deviation principles. Wedescribetwoexamplesbelow.
0 Recently, we showed how large-deviation principles also allow us to understand the role of
6
entropy in a specific non-equilibrium system [1]: the large-deviation behaviour of a system of
4
. independent Brownian particles connects rigorously to the entropy gradient-flow structure of
1
0 the diffusion equation. We explain this connection in Section 3.1.
2 The aim of this paper is to take this connection two steps further. The first step is to
1
extend the connection of [1], which was studied in a discrete-time context, to the case of
:
v continuous time. The second step is to discuss a variety of examples that illustrates the
i
X breadth of this phenomenon, and suggest a general principle that might hold across a wide
r range of systems.
a
In equilibrium systems, the connection is as follows. Let X (i = 1,2,...) be independent
i
and identically distributed stochastic variables with distribution µ on a state space . We
X
think of the X as positions of particles in the space , so that their concentration is given
i
by the empirical measure ρ := 1 n δ . Sanov’s Xtheorem (e.g., [7, Sec. 6.2]) states that
n n i=1 Xi
the random measure ρ satisfies the large-deviation principle
n
P
Prob(ρ ρ) exp[ nI(ρ)], as n , (1)
n
≈ ∼ − → ∞
∗Mathematics Institute, Universityof Warwick
†Cardiff School of Mathematics, Cardiff University
‡Departmentof Mathematics andComputerSciences andInstitutefor Complex Molecular Systems,Tech-
nische Universiteit Eindhoven
§Department of Mathematical Sciences, Universityof Bath
1
where the rate function I 0 is the relative entropy of ρ with respect to µ, which is
≥
f logf dµ if ρ µ and ρ = fµ,
I(ρ) = H(ρµ) := ≪
| (+R otherwise,
∞
This property illustrates how the relative entropy H(ρµ) characterises the probability of
|
observing a state ρ: higher relative entropy means smaller probability, as described by (1).
It also provides a rigorous version of the well-known thermodynamic principle that a system
aims to maximise its entropy (which corresponds to minimising H(ρµ), since the physical
|
entropy carries the opposite sign). For in the limit of large n, the characterisation (1) gives
vanishing probability to all states ρ except those for which I(ρ) = 0; in other words, only the
minimisers of I have non-vanishing probability.
This connection between entropy and large-deviation principles extends to systems in-
volving energy. In the appendix we show, for instance, how coupling a system with en-
ergy E to a heat bath with temperature θ changes the rate functional I to the free energy
(ρ) := H(ρµ)+(kθ)−1E(ρ)+constant:
F |
Prob(ρ ρ) exp[ n (ρ)], as n . (2)
n
≈ ∼ − F → ∞
In the same way as (1) explains why relative entropy is minimized, (2) explains why systems
coupled to a heat bath minimize their free energy: when n is large, only states ρ with near-
minimal free energy (ρ) will have finite probability.
F
As mentioned above, the central aim of this paper is to show how this connection between
entropy and free energies on one hand and large-deviation principles on the other extends
into the realm of non-equilibrium systems. We restrict our focus to the important class of
gradient flows, where this connection explains many aspects of these systems. Since the
entropy appears as the driving force of the process, we will occasionally call this functional
“energy” to conform with the standard terminology for gradient flows.
The general philosophy is illustrated by the diagram below.
dynamic rate functional thispaper gradient-flow structure
I or I ←−−−−−−−→ J or I
h h
large-deviationprinciple (3)
n→∞
x continuum limit x
stochastic n-particle system continuum evolution equation
y −−−−−n→−−∞−−−→ y
The bottom row in this diagram is the classical connection between a stochastic n-particle
system and its hydrodynamic limit: the typical case is that as n , the particle system
→ ∞
becomes deterministic, and the empirical measure of the particle system converges to the
solution of the (deterministic) continuum equation. Note that this statement concerns only
the typical behaviour of the particle system; large deviations are not captured.
In the left-hand column, a large-deviation principle characterises the behaviour in the
limit n in a different manner, in terms of a functional I or I of the time-dependent
h
→ ∞
system, as we shall see below. The right-hand column is the connection between an evolution
equation and the corresponding gradient-flow structure, when it exists.
The central statement of this paper is the double-headed arrow at the top. It provides
a connection between representations with more information on both sides: on the left-hand
side, the rate functional contains more information than just the most probable behaviour,
2
and on the right-hand side, the gradient-flow structure is an additional structure on top of
the equation itself.
In the following sections, we illustrate the double-headed arrow in a number of concrete
examples, first in the discrete-time approximation (Section 3) and then in continuous time
(Section 4). Section 5 generalises the argument to non-quadratic dissipations. Since the
implications of this connection are best appreciated once one has an overview of the breadth
of the phenomenon, we postpone most of the discussion of the consequences to Section 6.
The mathematical results described in this paper are not new, and mostly due to other
authors, such as Freidlin & Wentzell [10], Dawson & Ga¨rtner [5, 6], Feng & Kurtz [9], Kipnis,
Olla, & Varadhan [12] and others. Instead, we see the novelty of this paper in extracting
from these results the suggestion of a general principle connecting the broad class of gradient
flows with large deviations of stochastic processes. A particularly interesting aspect of this
connection is that thermodynamic quantities are derived in a non-equilibrium context.
2 The Wasserstein metric
Much of this paper centres on the Wasserstein metric and Wasserstein gradient flows. The
(quadratic) Wasserstein distance between two probability measures ρ and ρ with finite
0 1
second moments is [18]
d(ρ ,ρ )2 = inf x y 2q(dxdy), (4)
0 1
q ZRd×Rd| − |
where the infimum is taken over all q with marginals ρ and ρ , i.e., over all q satisfying
0 1
for any A Rd, q(A Rd) = ρ (A) and q(Rd A)= ρ (A).
0 1
⊂ × ×
We also need an incremental version of the Wasserstein distance. The Brenier-Benamou
formula [3] gives an alternative formulation of d as an infimum of curves of measures t ρ(t)
7→
such that ρ(0) = ρ and ρ(1) = ρ :
0 1
1
d(ρ ,ρ )2 = inf ∂ ρ(t) 2 dt. (5)
0 1 ρ:[0,1]→M1(Rd)Z0 k t kρ(t),∗
Here the local norm at a given point ρ is derived from an inner product (a local metric
ρ,∗
k·k
tensor) formally given by
(s ,s ) := ρ(x) p (x) p (x)dx, (6)
1 2 ρ,∗ 1 2
Rd ∇ ·∇
Z
where is the usual gradient in Rd, and the p solve the equation div(ρ p ) = s in Rd
i i i
∇ ∇
(see [5, 13] or [9, Sec. 9.4] for a rigorous definition).
A Wasserstein gradient flow is a gradient flow of an energy with respect to the Wasser-
E
stein metric structure. A curve of measures t ρ(t) is a solution of such a gradient-flow
7→
equation if its time derivative ∂ ρ, in the sense of distributions, satisfies
t
δ
(∂ ρ(t),s ) = E(ρ(t))s dx for all s and all t > 0, (7)
t 2 ρ(t),∗ 2 2
− Rd δρ
Z
3
where δ /δρ is the variational derivative of . A straightforward calculation shows that this
E E
is equivalent to the equation
δ
∂ ρ= divρ E . (8)
t
∇ δρ
(cid:16) (cid:17)
By analogy with gradients in Riemannian geometry, this suggests to define the Wasserstein
gradient of a functional as
E
δ
grad (ρ) := divρ E . (9)
W E − ∇ δρ
(cid:16) (cid:17)
Below we shall also use more general versions of this structure. Replacing ρ above by a
general diffusion matrix D(ρ), we define
(s ,s ) := D(ρ(x)) p (x) p (x)dx, where s = divD(ρ) p . (10)
1 2 D(ρ),∗ 1 2 i i
Rd ∇ ·∇ ∇
Z
Repeatingtheconstructionabove,itfollowsthattheD-Wassersteingradient ofafunctional
E
is characterised by the equation
δ
∂ ρ= divD(ρ) E . (11)
t
∇ δρ
(cid:16) (cid:17)
Gradient flows have natural time-discrete approximations, constructed in an iterative
manner:
For given approximation ρ at time (k 1)h, choose ρ at time kh
k−1 k
−
1
as minimiser of the functional ρ d(ρ,ρk−1)2+ (ρ). (12)
7→ 2h E
This is essentially a backward-Euler discretisation, as can berecognised by comparing it with
the Rd-gradient-flow x˙ = E(x). For this equation the backward-Euler discretisation is
−∇
constructed by solving
1
(x x )= E(x ),
k k−1 k
h − −∇
for x , which is equivalent to minimising
k
1
x x x 2+E(x). (13)
k−1
7→ 2h| − |
Note the similarity between (13) and (12): in both expressions the first term measures the
distance between old and new states, while the second term favours a reduction of the func-
tional respectively E.
E
3 Discrete time
We can now formulate the first example.
4
3.1 A system of independent Brownian particles
We consider n independent Brownian particles X (t) in Rd, with deterministic initial posi-
n,i
tions X (0) = x , each hopping to a new position X (h) at time h > 0 with a Gaussian
n,i n,i n,i
probability with mean x and variance1 2h.
n,i
As in the equilibrium case discussed above, we describe this system by the empirical
measure ρ (t) := 1 n δ at a given time t, and we assume that the initial measure
n n i=1 Xn,i(t)
ρ (0) converges to a given measure ρ0 as n . In the limit of large n, the probability
n P → ∞
of this jump process attaining any ρ1 at time t = h is again characterised in terms of a
large-deviation principle,
Prob(ρ (h) ρ1) exp[ nI (ρ1)], (14)
n h
≈ ≈ −
where the rate functional I has an explicit expression that can be derived from Stirling’s
h
formula (see [1] for the expression; in [1], I is only the limit of a sequence of rate functionals,
h
but can be shown to be a rate functional in its own right [14, 17]).
The main result of [1] is that
I K as h 0, (15)
h h
≈ →
where
1 1 1
K (ρ1;ρ0) := d(ρ0,ρ1)2+ Ent(ρ1) Ent(ρ0). (16)
h
4h 2 − 2
Here d is the Wasserstein distance defined above, and
flogfdx if ρ and ρ= f ,
Ent(ρ) := H(ρ ) = ≪ L L
|L (R+ otherwise,
∞
is the relative entropy of ρ with respect to the Lebesgue measure . Therigorous formulation
L
of (15) is a Gamma-convergence result of I to K after both have been desingularised.
h h
The functional K has the same form as the functional in (12), since the term Ent(ρ0)/2
h
does not influence the minimisation with respect to ρ1. Therefore the time-discrete approxi-
mation that one constructs with this K is an approximation of the Wasserstein gradient flow
h
of the entropy Ent, which is the diffusion equation [11]
∂ ρ = ∆ρ in Rd. (17)
t
This is the connection referred to above: the large-deviation behaviour of the system of
particles is represented by the rate functional I , and this functional is asymptotically equal
h
to the functional K that defines the gradient-flow formulation of the diffusion equation. The
h
approximation result (15) therefore creates a link between the gradient-flow structure of the
deterministic limit equation on one hand and the large-deviation behaviour of the system
of particles on the other. The same result can be shown for Gaussian measures on the real
line [8]. In the rest of this paper we shall see many more versions of such connections.
1In this paper, we consider Brownian particles with generator ∆, rather than (1/2)∆, and therefore the
transition kernel is (4πh)−d/2exp−|x−y|2/4h.
5
Consequences Whilemostof thediscussionisdeferredtoSection 6, wementionhereafew
consequences of the fact (15) that the large-deviation rate functional I and the constructing
h
functional K of the gradient flow are equal in the limit h 0.
h
→
First, theconstructionof atime-discreteapproximation (12)tothediffusionequation(17)
was motivated in [11] by analogy with the backward-Euler discretisation (13). This is an
indirect and purely mathematical motivation, which explains neither the reason for the ap-
pearance of the entropy and the Wasserstein distance in K , nor the reason for minimising
h
just this combination.
The connection between K and I , however, gives a direct motivation. By (14)–(15),
h h
K (ρ;ρ0) is a measure of the likelihood of observing a state ρ after time h. For large n,
h
the characterisation (14) implies that only the global minimiser of I , and therefore of K , is
h h
observedwithnon-vanishingprobability. Thestochasticminimisation(14)ofI thusbecomes
h
converted into an absolute minimisation of K .
h
Secondly, in the limit h 0, the proof that I K explains the origin of the two terms
h h
→ ≈
of K . The entropy arises from the indistinguishibility of the particles after transforming to
h
an empirical measure. The origin of the Wasserstein cost functional x y 2 in (4) can be
traced back to the exponent of the term e−|x−y|2/4h in the Gaussian tra|ns−itio|n probability of
the Brownian particles. We return to this issue in Section 6.
4 Continuous time
Theconstruction in the previous section is discrete in time: the rate function I describes the
h
probability distribution of the state ρ (h) at time h > 0. A continuous-time large-deviation
n
principle, where one considers deviations from a whole path of empirical measures for a fixed
terminaltime, providesadifferentkindofinsight, andmay beeven closer tothegradient-flow
formulation. We start with some preliminaries.
4.1 An alternative formulation of the gradient-flow structure
In a formal sense, Wasserstein gradient flows and many others can be written in the form
δ
∂ ρ= M E, (18)
t ρ
− δρ
where is the ‘energy’ functional driving the evolution, and M a ρ-dependent symmetric
ρ
E
mapping2. In the case of Wasserstein gradient flows, for instance,
M ξ = divρ ξ,
ρ
− ∇
as follows by comparing (8) with (18). Taking this case of Wasserstein gradient flow as an
example, we shall encounter the equation (18) in a different form, connected to the functional
J given by
1 T δ 2
J(ρ) := (ρ(T)) (ρ(0))+ ∂ ρ 2 + E dt, (19)
E −E 2Z0 (cid:20)k t kρ,∗ (cid:13)−δρ(cid:13)ρ(cid:21)
2Thiswayofwritingthegradientflowhighlightsthefactthatagradient(cid:13)(cid:13)flowis(cid:13)(cid:13)aninstanceofaGENERIC
evolution, in which the conservativeevolution term is absent [15].
6
where
ξ 2 := ξM ξdx = ρ ξ 2
k kρ Rd ρ Rd |∇ |
Z Z
and the norm 2 is the norm defined in (6). The norms and are dual norms,
k·kρ,∗ k·kρ k·kρ,∗
and has the alternative characterisation
ρ,∗
k·k
sξdx
s := supZRd .
ρ,∗
k k ξ
ξ6=0 ρ
k k
By writing the energy difference (ρ(T)) (ρ(0)) as
E −E
T δ T δ
(ρ(T)) (ρ(0)) = E ∂ ρdxdt = M E,∂ ρ dt,
t ρ t
E −E Z0 ZRd δρ Z0 (cid:16) δρ (cid:17)ρ,∗
using the inner product defined in (7), the functional J in (19) can now be written as
1 T δ 2
J(ρ) = ∂ ρ+M E dt.
t ρ
2 δρ
Z0 (cid:13) (cid:13)ρ,∗
(cid:13) (cid:13)
(cid:13) (cid:13)
This expression shows that J is non-negativ(cid:13)e. It also imp(cid:13)lies that if ρ satisfies J(ρ) = 0, then
equation (18) holds at almost each time 0 < t < T; therefore
ρ is a Wasserstein gradient flow of J(ρ) = 0. (20)
E ⇐⇒
In the examples of this paper, J is a large-deviation rate functional, and this equivalence
is the connection between the large-deviation behaviour, given by J, and the gradient-flow
structure of the limiting equation.
IfwetakefortheoperatorM in(18)nottheWasserstein operatorbutageneraloperator,
ρ
then we find a similar statement:
ρ is a solution of the ( ,M )-gradient-flow (18) J (ρ) = 0, (21)
ρ M
E ⇐⇒
where
1 T δ 2
JM(ρ) := E(ρ(T))−E(ρ(0))+ 2Z0 (cid:20)k∂tρk2Mρ−1 +(cid:13)−δEρ(cid:13)Mρ(cid:21) dt, (22)
and the two norms are defined, at least formally, by (cid:13) (cid:13)
(cid:13) (cid:13)
ξ 2 := ξM ξdx,
k kMρ Rd ρ
Z
sξdx
kskMρ−1 := sξu6=p0 ZRkdξkMρ = ZRdsMρ−1sdx = kMρ−1sk2Mρ.
We now discuss a number of examples.
7
4.2 Continuous-time large deviations for the diffusion equation
Taking the same system of particles as in Section 3.1, the continuous-time large-deviation
principle for that system of Brownian particles is as follows. Fix a terminal time T > 0 and
consider the whole path [0,T] (Rd) of empirical measures [0,T] t ρ (t). Then the
1 n
→ M ∋ 7→
probability that the entire curve ρ () is close to some other ρ() is characterised as [5, 13] as
n
· ·
a pathwise large-deviation principle,
Prob(ρ ρ) exp[ nI(ρ)],
n
≈ ∼ −
where now
1 T
I(ρ) := ∂ ρ ∆ρ 2 dt. (23)
2 k t − kρ(t),∗
Z0
This rate function I has the structure of J in (19). Using the fact that
δEnt
∆ρ= divρ ,
∇ δρ
(cid:16) (cid:17)
we find that
1 T δEnt 2
I(ρ) = Ent(ρ(T)) Ent(ρ(0))+ ∂ ρ 2 + dt.
− 2Z0 hk t kρ,∗ (cid:13)− δρ (cid:13)ρi
ThereforetheEntropy-Wassersteingradientflowisconnectedto(cid:13)thelarge-(cid:13)deviationbehaviour
(cid:13) (cid:13)
of a system of stochastic particles, in the sense of (20). We discuss this further in Section 6.
4.3 Diffusive particles with interactions
We extend the previous example by including interaction of the particles with a background
potential Ψ and with each other via an interaction potential Φ, and modelled by Itˆo stochas-
tic differential equations. Specifically, we take the microscopic system of n particles to be
described by
n
1
dX (t) = Ψ(X (t))dt Φ(X (t) X (t))dt+√2 dW (t), (24)
i i i j i
−∇ − n ∇ −
j=1
X
where for each i, W is a Brownian motion in Rd. The hydrodynamic limit of this system is
i
the equation
∂ ρ =∆ρ+divρ Ψ+ρ Φ . (25)
t
∇ ∗
Thelarge-deviation ratefunctionaldescribingfluctuationsofthesystemisgivenby(see[9,
(cid:2) (cid:3)
Theorem 13.37], and also [5] for weakly interacting diffusive particle systems)
1 T 2
I(ρ) := ∂ ρ ∆ρ divρ Ψ+ρ Φ dt, (26)
t
2 Z0 (cid:13) − − ∇ ∗ (cid:13)ρ,∗
which again can be written as (cid:13) (cid:2) (cid:3)(cid:13)
(cid:13) (cid:13)
1 T δ 2
I(ρ) = (ρ(T)) (ρ(0))+ ∂ ρ 2 + F dt,
F −F 2Z0 (cid:20)k t kρ,∗ (cid:13)δρ(cid:13)ρ(cid:21)
where the free energy is given by the sum of entropy and pot(cid:13)entia(cid:13)l energy,
F (cid:13) (cid:13)
1
(ρ) := Ent(ρ)+ ρΨ+ ρ(ρ Φ) . (27)
F ZRd(cid:20) 2 ∗ (cid:21)
Indeed equation (25) is the Wasserstein gradient flow of the functional .
F
8
4.4 The Symmetric Simple Exclusion Process
The diffusion equation (17) is the continuum limit for various stochastic processes, one of
which is the system of Brownian particles described above. Here we briefly describe the
symmetric simple exclusion process, which has the same limiting equation in a parabolic
scaling. However, it has a different large-deviation behaviour, which gives rise to a different
gradient flow.
Consider a periodic lattice T = 0,1/n,2/n,...(n 1)/n and its continuum limit, the
n
{ − }
flat torus T = R/Z. Each lattice site contains zero or one particle; each particle attempts to
jump from to a neighbouring site with rate n2/2, and they succeed if the target site is empty.
We define the configuration ρ : T 0,1 such that ρ (k/n) = 1 if there is a particle at
n n n
→ { }
site k/n, and zero otherwise. For this system the large deviations are characterised by the
rate function [12]
1 T
I(ρ) := ∂ ρ ∂ ρ 2 dt, (28)
2 k t − xx kρ(1−ρ),∗
Z0
where the norm is given by (10) with D(ρ) = ρ(1 ρ). This functional can be
ρ(1−ρ),∗
k·k −
written as
1 T δEnt 2
I(ρ) = Ent (ρ(T)) Ent (ρ(0))+ ∂ ρ 2 + mix dt,
mix − mix 2 Z0 hk t kρ(1−ρ),∗ (cid:13)− δρ (cid:13)ρ(1−ρ)i
(cid:13) (cid:13)
where the mixing entropy Ent is defined as (cid:13) (cid:13)
mix
Ent (ρ) := ρlogρ+(1 ρ)log(1 ρ) .
mix
Rd − −
Z
(cid:2) (cid:3)
This is true since ∂ ρ is the ‘ρ(1 ρ)’-Wasserstein gradient of Ent , by
xx mix
− −
ρ δEnt
mix
∂ ρ = ∂ ρ(1 ρ)∂ log = ∂ ρ(1 ρ)∂ (ρ)
xx x x x x
− − − 1 ρ − − δρ
(cid:18) − (cid:19) (cid:18) (cid:19)
(compare this to (11)). Therefore I is of the form (22), with operator
M ξ := divρ(1 ρ) ξ,
ρ
− ∇
andtheequation∂ ρ = ∂ ρis(also)thegradientflowofEnt withrespecttothis‘ρ(1 ρ)’-
t xx mix
−
Wasserstein structure .
ρ(1−ρ),∗
k·k
5 Further generalisations
The arguments of the integrals in (5), (23), (26), and (28) are quadratic. This arises from
a parabolic rescaling and the central limit theorem, and it leads to a gradient flow with a
(formal)inner-productstructure,orequivalently, toalinear operatorM in(18). Othertypes
ρ
of randomness lead to non-quadratic gradient-flow structures, as we now describe.
A close inspection of the arguments of Section 4.1 shows that they hinge on the inequality
1 1 δ 2
∂tE(ρ) ≥ −2k∂tρk2Mρ−1 − 2 −δEρ Mρ,
(cid:13) (cid:13)
(cid:13) (cid:13)
(cid:13) (cid:13)
9
together with the observation that equality holds if and only if ∂ ρ = M δ /δρ. This
t ρ
− E
can be generalised by introducing a Legendre pair of convex functions ψ and ψ∗, where the
ρ ρ
subscriptρserves to indicate that they may dependon ρ, in thesame way as theoperator M
ρ
does; in this context, ψ , is often called dissipation potential. In terms of this pair we then
ρ
derive that
δ δ
∂ (ρ(t)) = E∂ ρ ψ∗(∂ ρ) ψ E ,
tE δρ t ≥− ρ t − ρ δρ
Z (cid:16) (cid:17)
and equality holds if and only if
δ
∂ ρ ∂ψ E . (29)
t ρ
∈ −δρ
(cid:16) (cid:17)
The case of the M-gradient flow (29) corresponds to
1 1
ψ∗(ξ) := ξ 2 and ψ (s) := s 2 .
ρ 2k kMρ ρ 2k kMρ−1
The obvious generalisation of (20) then is
ρ is a solution of the ( ,ψ)-gradient-flow (29) J (ρ) = 0, (30)
ψ
E ⇐⇒
where J is given by
ψ
T δ
J (ρ) := (ρ(T)) (ρ(0))+ ψ∗(∂ ρ)+ψ E dt. (31)
ψ E −E ρ t ρ −δρ
Z0 (cid:20) (cid:16) (cid:17)(cid:21)
5.1 Birth-death processes
Asimpleexampleofastochasticprocesswithnon-quadraticdissipationψandacorresponding
generalised gradient flow is a birth-death process, which is a continuous-time jump process
on Z. The system may only jump to neighbours, from position k with rate a to k+1 and
k
with rate b to k 1. We construct a continuum limit by defining the new stochastic variable
k
−
U by rescaling time t and position k(t) with n:
n
k(nt)
U (t) := .
n
n
Astandardargumentgivesthelarge-deviation behaviourforU intermsoftheratefunctional
n
(see [4] for a finite-lattice proof of the claims made below). If we choose the jump rates so
that
a = αe−E′(k/n) and b = αe+E′(k/n)
k k
for α > 0 and some smooth function : R R, then the rate functional is
E →
T
I(u) = L(u(t),u′(t))dt,
Z0
with
v+√v2+4α2
L(u,v) = vlog v2+4α2 +αe−E′(u)+αe+E′(u).
2αexp( ′(u)) −
−E
p
Writing
v+√v2+4α2
ψ∗(v) = vlog v2+4α2,
2α −
p
10