Convergence of the drift
where the covariance of two K -valued random variables X and Y is defined as CovX, Y :
= E[X · Y ] − E[X] · E[Y ] 4.4.6
with x · y :=
α
x
α
y
α
the usual inner product on
R
d
. A covariance calculation as in Swart [41] gives that for
ξ ≤ k
i ∂
∂ s
C
s
ξ =
η
a
k
i
−1 N
i
η − ξ[C
s
η − C
s
ξ ]
+2dδ
0,ξ
E[gX
N
i
β
i
t + s] − 2
c N
i k
i
C
s
ξ , 4.4.7
where a
k N
is the k-block interaction kernel a
k N
ξ :
=
k l
=ξ
1 N
l
c N
l −1
. 4.4.8
Using our assumption about local equilibrium, we set
∂ ∂
s
C
s
ξ = 0 in 4.4.7 and
we assume that E[gX
N
i
ξ
β
i
t +s] does not depend on s. Now we can solve C
s
ξ in terms of E[gX
N
i
ξ
β
i
t + s] and a random walk on
i
: = {ξ ∈
N
i
: ξ ≤ k
i
− 1} 4.4.9
that jumps from site ξ to site η with rate a
k
i
−1 N
i
η − ξ and that is killed in each site
with rate
c N
i
k
i
. Indeed, denoting by P
i t
η − ξ the probability that this random
walk moves from site ξ to site η in time t, we have the representation C
s
ξ = d E[gX
N
i
β
i
t + s]
∞
P
i t
ξ dt.
4.4.10 Note that with probability one the random walk is eventually killed, so that the
integral on the right-hand side is finite. Picking ξ = 0, we get
VarX
N
i
β
i
t + s = dµ
i
E[gX
N
i
β
i
t + s]
4.4.11 with
µ
i
: =
∞
P
i t
0dt 4.4.12
the expected time the random walk starting in 0 spends at the origin. STEP 3: It turns out that we can also express the expectation of any harmonic
function of X
N
i
ξ
β
i
t + s in terms of the above random walk. Indeed, we have the
representation see Swart [41], Lemma 3.1.6 in this dissertation E[ f X
N
i
β
i
t + s] = E
f ˆθ +
ξ
P
i s
ξ [X
N
i
ξ
β
i
t − ˆθ]
4.4.13
for any function f ∈
C
2
K
◦
∩
C
K satisfying
α ∂
∂ x
α
2
f x = 0
x ∈ K
◦
. 4.4.14
Formula 4.4.13 says that harmonic functions of a component evolve under the semigroup associated with the evolution in 4.4.3 as if the diffusion function g is
zero.
The assumption of local equilibrium now leads to the relation E[ f X
N
i
β
i
t + s] = f ˆθ,
4.4.15 which may be described by saying that the ‘harmonic mean’ of X
N
i
β
i
t + s is ˆθ.
We next note that the function x
→ dg
∗
x + |x − ˆθ|
2
4.4.16 is harmonic. Therefore, combining 4.4.15 and 4.4.11, we find that
µ
i
E[gX
N
i
β
i
t + s] = g
∗
ˆ θ
− E[g
∗
X
N
i
β
i
t + s].
4.4.17 STEP 4: We will show that µ
i
∼ σ
k
i
i → ∞. Hence 4.4.17 becomes
σ
k
i
E[gX
N
i
β
i
t + s] ∼ g
∗
ˆ θ
− E[g
∗
X
N
i
β
i
t + s]
i → ∞. 4.4.18
Since σ
k
i
tends to infinity and the right-hand side of 4.4.18 is bounded by g
∗ ∞
, it follows that E[gX
N
i
β
i
t + s] tends to zero as i → ∞. This means that, with
high probability, the components X
N
i
ξ
β
i
t + s of the system are concentrated near
the boundary of K , i.e., the system clusters. Since g
∗
is continuous on K and zero on ∂ K , it follows that also E[g
∗
X
N
i
β
i
t + s] tends to zero as i → ∞. Hence,
using 4.4.18 once more we see that lim
i →∞
σ
k
i
E[gX
N
i
β
i
t + s] = g
∗
ˆ θ .
4.4.19 STEP 5: We now consider the k
i
-block {ξ ∈
N
i
: ξ ≤ k
i
}. The k
i
− 1-blocks that the k
i
-block consists of, N
i
in total, all reach equilibrium on the time scale T
i
, and they do so independently of each other. Hence we expect a law of large numbers to apply. In particular, we expect that
lim
i →∞
Var N
−k
i
i ξ
: ξ≤k
i
σ
k
i
gX
N
i
ξ
β
i
t + s
= 0. 4.4.20