Theorem 4.1.3 Let N
i
≥ 2, k
i
≥ 1 i ∈
N
be integers tending to infinity as i
→ ∞ such that lim
i →∞
k
i
log N
i
= 0, 4.1.27
and assume that g
∗
∈
C
1
K . Then there exist ˆ
F
i t
t ≥0
-adapted processes ˆ B
i
= ˆ
B
i
t
t ≥0
and ˆ G
i
= ˆ G
i
t
t ≥0
with sample paths in
D R
d
[0, ∞,
D R
[0, ∞, re-
spectively, such that for each f ∈
C
2
K the process M
i
t
t ≥0
, given by
M
i
t : = f ˆX
i
t −
t α
ˆB
i,α
s
∂ ∂
x
α
f ˆX
i
s + ˆ
G
i
s
α ∂
∂ x
α
2
f ˆX
i
s ds
4.1.28 is an ˆ
F
i t
t ≥0
-martingale. Moreover, for each T 0 lim
i →∞
E
T
ˆB
i
t − c
∗
θ − ˆX
i
t
2
dt = 0.
4.1.29 and
lim
i →∞
E
T
ˆG
i
t − g
∗
ˆ X
i
t dt
2
= 0. 4.1.30
Formula 4.1.28 identifies ˆ B
i
and ˆ G
i
as the drift and the diffusion rate of the process ˆ
X
i
. Thus, formulas 4.1.29 and 4.1.30 show that these local character- istics of the process ˆ
X
i
converge, as i → ∞, and that their limits are universal in
the diffusion function g for single components. The fact that this happens for all N
i
, k
i
satisfying condition 4.1.27 is new even in the case of one-dimensional K . Theorem 4.1.3 is a universality result of the type we were originally after in den
Hollander and Swart [21]. The author believes that also the convergence in 4.1.25 holds under condition
4.1.27, but there are at present two technical obstacles to proving a result of that form.
The first difficulty comes from the fact that uniqueness of solutions to 4.1.16 for arbitrary g
= g
∗
, c = c
∗
and x = θ remains an open problem. The following
partial results are known. 1. Strong uniqueness is known to hold for θ
∈ K
◦
, c
∗
sufficiently large, and K satisfying mild regularity conditions Theorem 1.9 in Den Hollander
Swart [21].
2. Weak uniqueness is known to hold for K = {x ∈
R
d
: |x|
2
≤ 1}, in which case g
∗
x = 1 − |x|
2
d, and θ and c
∗
arbitrary Theorem 1.10 in Den Hollander Swart [21].
The second difficulty comes from the fact that even if uniqueness of Z
g
∗
, c
∗
θ
is known, the results in Theorem 4.1.3 are not sufficient to conclude that ˆ
X
i
con- verges to Z
g
∗
, c
∗
θ
. In particular, the type of convergence in 4.1.30 is not sufficient to show tightness of ˆ
X
i i
∈
N
in the topology of weak convergence in path space
D
K
[0, ∞. If instead of 4.1.30 we would have
lim
i →∞
E
T
ˆ G
i
t − g
∗
ˆ X
i
t
2
dt = 0,
4.1.31 then tightness of ˆ
X
i i
∈
N
would follow by Theorem 9.4 from Chapter 3 in Ethier Kurtz [16]. The author believes 4.1.31 to hold, but, as explained in Section 4.5,
the techniques in the present paper are not sufficient to show this. In future work we hope to establish the convergence of ˆ
X
i i
∈
N
, either by showing 4.1.31 to prove weak convergence in path space, or by using 4.1.30 to prove convergence
in some weaker sense.
4.2 Identification of the drift and diffusion rate of ˆ
X
i
In this section, we identify processes ˆ B
i
and ˆ G
i
such that 4.1.28 holds. In the following sections we then prove 4.1.29 and 4.1.30, which completes the proof
of Theorem 4.1.3.
Lemma 4.2.1 For each f
∈
C
2
K the process M
i
t
t ≥0
, given by M
i
t : = f ˆX
i
t −
t α
ˆB
i,α
s
∂ ∂
x
α
f ˆX
i
s + ˆ
G
i
s
α ∂
∂ x
α
2
f ˆX
i
s ds
4.2.1 with
ˆB
i
t : = σ
k
i
c
k
i
∞ n
=0
c N
i n
X
N
i
, k
i
+n+1
β
i
t − X
N
i
, k
i
β
i
t ˆ
G
i
t : = σ
k
i
1 N
k
i
i ξ
: ξ≤k
i
gX
N
i
ξ
β
i
t. 4.2.2
is an ˆ
F
i t
t ≥0
-martingale.
Proof of Lemma 4.2.1: Fix a function f
∈
C
2
K and for N ≥ 2 and k ≥ 1 define
f
N,k
∈
C
2 fin
K by
f
N,k
x : = f x
k
= f 1
N
k ξ
: ξ≤k
x
ξ
. 4.2.3
Then the process M
N,k
t
t ≥0
is a martingale, where M
N,k
t : = f X
N,k
β
N,k
t −
β
N,k
t
A f
N,k
X
N
s ds = f ˆX
N,k
t −
t
β
N,k
A f
N,k
X
N
s ds. 4.2.4
Here β
N,k
A f
N,k
x = σ
k
N
k η,α
∞ l
=1
c N
l −1
[x
l,α η
− x
α η
]
∂ ∂
x
α η
f x
k
+σ
k
N
k η,α
gx
η ∂
∂ x
α η
2
f x
k
. 4.2.5
The first term on the right-hand side in 4.2.5 can be written as σ
k
N
k α
∞ l
=1
c N
l −1
1 N
k η
: η≤k
[x
l,α η
− x
α η
]
∂ ∂
x
α
f x
k
= σ
k
N
k α
∞ l
=k+1
c N
l −1
[x
l,α
− x
k,α
]
∂ ∂
x
α
f x
k
= σ
k
c
k α
∞ n
=0
c N
n
[x
k +n+1,α
− x
k,α
]
∂ ∂
x
α
f x
k
, 4.2.6
where in the first equality we use that for l ≤ k
1 N
k η
: η≤k
[x
l η
− x
η
] =
1 N
k η
: η≤k
1 N
l ζ
: ζ −η≤l
[x
ζ
− x
η
] =
1 N
k +l
η :
η≤k ζ
: ζ ≤k
ζ −η≤l
[x
ζ
− x
η
] = 0
4.2.7
and for l k 1
N
k η
: η≤k
[x
l η
− x
η
] =
1 N
k η
: η≤k
[x
l
− x
η
] = [x
l
− x
k
]. 4.2.8
The second term on the right-hand side in 4.2.5 can be written as σ
k
N
k
1 N
2k η
: η≤k
gx
η α
∂ ∂
x
α
2
f x
k
. 4.2.9
Inserting 4.2.9 and 4.2.6 into 4.2.5, inserting 4.2.5 in 4.2.4 and defining M
i
: = M
N
i
, k
i
we arrive at Lemma 4.2.1.
4.3 Convergence of the drift
We start by giving an upper bound on the speed with which block averages change in time. In the following lemma we consider the process
X
N
t + s
s ≥0
= X
N ξ
t + s
ξ ∈
N
s ≥0
, 4.3.1
conditioned on the event X
N
i
β
i
t = x,
4.3.2 with x
∈ K
N
. We choose a regular version of the conditional expectation E
x
[ · ] := E[ · |X
N
t = x],
4.3.3 with the property that under the conditional law, the process in 4.3.1 solves the
martingale problem for the operator A in 4.1.10 with initial condition x.
Lemma 4.3.1 There exists a constant M such that for all integers N ≥ 2 and
k ≥ 1, for all t, s ≥ 0 and for all x ∈ K
N
E X
N,k
t + s − x
k 2
X
N
t = x
≤ M s
N
k
. 4.3.4
Proof of Lemma 4.3.1: A calculation similar to the proof of Lemma 4.2.1 then gives
E
x
X
N,k
t + s − x
k 2
=
s
du 1
N
2k ξ
: ξ≤k
2d E[gX
N ξ
u] +
s
du
∞ l
=k+1
c N
l −1
2E
x α
X
N,l,α
t + u − X
N,k,α
t + u
X
N,k,α
t − x
k,α
≤ 2ds
N
k
g
∞
+ 2R
2
s c
N
k ∞ n
=0
c N
n
≤ 2d
g
∞
+ 4R
2
s N
k
, 4.3.5
where R is a constant such that |x − y| ≤ R for all x, y ∈ K and in the last step
we use that N ≥ 2 and c 1.
Proof of formula 4.1.29: Since the terms with n
≥ 1 in 4.2.2 tend to zero uniformly as i
→ ∞, it suffices to show that lim
i →∞
E
T
σ
k
i
c
k
i
X
N
i
, k
i
+1
β
i
t − ˆX
i
t − c
∗
[θ − ˆX
i
t]
2
dt = 0.
4.3.6 Since σ
k
c
k
→ c
∗
as k → ∞ recall 4.1.15 and 4.1.23, it thus suffices to show
that lim
i →∞
E
T
X
N
i
, k
i
+1
β
i
t − θ
2
dt = 0.
4.3.7 We use Lemma 4.3.1 to estimate X
N
i
, k
i
+1
= θ E
T
X
N
i
, k
i
+1
β
i
t − θ
2
dt ≤ T sup
≤s≤β
i
T
E X
N
i
, k
i
+1
s − θ
2
≤ M β
i
T
2
N
k
i
+1 i
. 4.3.8
Since T is fixed, the right-hand side tends to zero provided that lim
i →∞
β
i
N
k
i
+1 i
= 0. 4.3.9
Inserting β
i
= σ
k
i
N
k
i
i
and σ
k
i
∼ c
−k
i
c1 − c, we find that this condition
amounts to lim
i →∞
c
k
i
N
i
= ∞. 4.3.10
But the latter holds for any c ∈ 0, 1 because of condition 4.1.27.
4.4 Convergence of the diffusion rate
4.4.1 Strategy of the proof
In this section the essential ideas behind Theorem 4.1.3 will have to come in. In particular, we will need to explain how the universal large space-time diffusion
function g
∗
arises and why the scaling of time with the factor σ
k
i
N
k
i
is the correct one. Before we embark on the calculations that will give us the convergence in
4.1.30, we outline the heuristics of the proof. STEP 1: We fix a t
≥ 0 and look at the process X
N
i
ξ
β
i
t + s
ξ :
ξ≤k
i
−1, s∈[0,T
i
]
, 4.4.1
