Suppose Φ : R
+
× R
2
→ R. Define |Φ|
Lip
, respectively |Φ|
1 2
, to be the smallest element in R
+
such that
|Φs, x − Φs, y| ≤ |Φ|
Lip
|x − y|, ∀s ≥ 0, x ∈ R
2
, y ∈ R
2
, |Φs − t
N
, x − Φs, x| ≤ |Φ|
1 2
p t
N
, ∀ s ≥ t
N
, x ∈ R
2
, We will write
||Φ||
Lip
:= ||Φ||
∞
+ |Φ|
Lip
, ||Φ||
1 2
:= ||Φ||
∞
+ |Φ|
1 2
and ||Φ|| := ||Φ||
∞
+ |Φ|
Lip
+ |Φ|
1 2
. Obviously the definition of
||.||
Lip
also applies to functions from R
2
into R. Define P
N t
, t ≥ 0 as the semigroup of the rate−N random walk on S
N
with jump kernel p
N
.
Lemma 3.5. There exist δ
3.5
0, c
3.5
0 and for any T 0, there is a C
3.5
T , so that for all t ≤ T and any Ψ : R
2
→ R
+
such that ||Ψ||
Lip
≤ T , E
X
N t
Ψ
≤ e
c
3.5
t
X
N
P
N t
Ψ
+ C
3.5
log N
−δ
3.5
X
N
1 + X
N
1
2
. This lemma requires a key second moment estimate see Proposition 3.10 below, it is proved in
Subsection 5.3.
3.2 On the new drift term
If Φ : R
+
× R
2
→ R, let d
N ,3 s
Φ, ξ
N
:= log N
4
N X
x ∈S
N
Φs, x
ξ
N s
x f x, ξ
N s
2
− 1 − ξ
N s
x f
1
x, ξ
N s
2
,
so that 29 implies D
N ,3 t
Φ = R
t
d
N ,3 s
Φ, ξ
N
ds. When the context is obvious we will drop ξ
N
from the notation. For s
t
N
it will be enough to use the obvious bound |d
N ,3 s
Φ| ≤ 2log N
3
||Φ||
∞
X
N s
1. 41
On the other hand we are able, for s ≥ t
N
, to get good bounds on the projection of d
N ,3 s
Φ onto F
s −t
N
. Indeed on [s − t
N
, s], it is very unlikely to see a branching i.e. red or green arrows for the rate v
N
random walks making up the dual process coming down from a given x at time s, and therefore, the dynamics of the rescaled Lotka-Volterra model on that scale should be very close to
those of the voter model. Let
ˆ H
N
ξ
N s
−t
N
, x, t
N
:= ˆ E
ξ
N s
−t
N
ˆ B
x t
N
2
Y
i=1
1 − ξ
N s
−t
N
ˆ B
x+e
i
t
N
−1 − ξ
N s
−t
N
ˆ B
x t
N
2
Y
i=1
ξ
N s
−t
N
ˆ B
x+e
i
t
N
.
Lemma 3.6. There exists a constant C
3.6
such that for any s ≥ t
N
, and any Φ : R
+
× R
2
→ R, E
d
N ,3 s
Φ, ξ
N
| F
s −t
N
−
log N
4
N X
x ∈S
N
Φs − t
N
, x ˆ H
N
ξ
N s
−t
N
, x, t
N
≤ C
3.6
||Φ||
1 2
X
N s
−t
N
1log N
−6
. 1208
We prove Lemma 3.6 in Section 4. Let us now look a bit more closely at the term arising in Lemma 3.6. Note that in terms of the
voter dual, ˆ H
ξ
N s
−t
N
, x, t
N
disappears whenever the rate v
N
walk started at x, coalesces before time t
N
with either one or both of the rate v
N
walks started respectively at x + e
i
, i = 1, 2. The non zero contributions will come from two terms. The first corresponds to the event, which we will
denote {x | x + e
1
| x + e
2
}
t
N
, that there is no collision between the three rate v
N
walks started at x, x + e
1
, x + e
2
up to time t
N
. The second corresponds to the event, which we will denote {x | x + e
1
∼ x + e
2
}
t
N
, that the rate v
N
walks started at x + e
1
, x + e
2
coalesce before t
N
, but that both do not collide with the walk started at x up to time t
N
. For convenience, and when the context is clear, we will drop the subscript from these two notations. We can now write recall e
= 0, ˆ
H
N
ξ
N s
−t
N
, x, t
N
= ˆ E
h ξ
N s
−t
N
ˆ B
x t
N
1 − ξ
N s
−t
N
ˆ B
x+e
1
t
N
− 1 − ξ
N s
−t
N
ˆ B
x t
N
ξ
N s
−t
N
ˆ B
x+e
1
t
N
1
{x|x+e
1
∼x+e
2
}
tN
i + ˆ
E ξ
N s
−t
N
ˆ B
x t
N
+ 2
2
Y
i=0
ξ
N s
−t
N
ˆ B
x+e
i
t
N
− X
≤i j≤2
ξ
N s
−t
N
ˆ B
x+e
i
t
N
ξ
N s
−t
N
ˆ B
x+e
j
t
N
1
{x|x+e
1
|x+e
2
}
tN
= ˆ E
h ξ
N s
−t
N
ˆ B
x t
N
− ξ
N s
−t
N
ˆ B
x+e
1
t
N
1
{x|x+e
1
∼x+e
2
}
tN
i + ˆ
E h
ξ
N s
−t
N
ˆ B
x t
N
1
{x|x+e
1
|x+e
2
}
tN
i + ˆ
E
2
2
Y
i=0
ξ
N s
−t
N
ˆ B
x+e
i
t
N
− X
≤i j≤2
ξ
N s
−t
N
ˆ B
x+e
i
t
N
ξ
N s
−t
N
ˆ B
x+e
j
t
N
1
{x|x+e
1
|x+e
2
}
tN
=: F
N 1
s − t
N
, x, t
N
+ F
N 2
s − t
N
, x, t
N
+ F
N 3
s − t
N
, x, t
N
42
Lemma 3.7. There is a constant C
3.7
such that the following hold for any u, v ≥ 0.
log N
4
N X
x ∈S
N
Φv, xF
N 1
u, x, t
N
≤ C
3.7
|Φ|
Lip
log N
−6
X
N u
1, 43
log N
4
N X
x ∈S
N
Φv, xF
N 2
u, x, t
N
− log N
3
ˆ P
{0 | e
1
| e
2
}
t
N
X
N u
Φv, . ≤ C
3.7
|Φ|
Lip
log N
−6
X
N u
1, 44
log N
4
N X
x ∈S
N
Φv, xF
N 3
u, x, t
N
≤ C
3.7
||Φ||
∞
X
N u
1. 45
There exist δ
3.7
, η
3.7
∈ 0, 1 such that log N
4
N X
x ∈S
N
Φv, xF
N 3
u, x, t
N
≤ C
3.7
||Φ||
∞
1 t
N
log N I
N η
3.7
u + X
N u
1log N
−δ
3.7
, 46
where I
N η
u := RR
1
{0|x− y|
p
t
N
log N
1 −η
}
d X
N u
xd X
N u
y. 1209
We prove Lemma 3.7 in Section 4.
Remark 3.8. If we set s = t
N
in 42 and u = 0 in the above Lemma and combine 42,43, 44, and 46 we see there is a
δ
3.8
0 and C
3.8
so that for any ξ
N
∈ S
N log N
4
N
P
x
Φv, x ˆ
H ξ
N
, x, t
N
− ξ
N
xP{0|e
1
|e
2
}
t
N
≤ C
3.8
kΦk
Lip
h
1 t
N
log N
I
N η
3.7
0 + log N
−δ
3.8
X
N
1 i
. 47
We now deduce two immediate consequences of the above. Firstly, for any s ≥ t
N
, combining Lemma 3.6, 42, the first three estimates of the above Lemma with u = s
− t
N
, and 6, we obtain E
d
N ,3 s
Φ, ξ
N
| F
s −t
N
≤ C
48
||Φ||X
N s
−t
N
1. 48
This, along with 41, allows us to bound the total mass of this new drift term and conclude that
E
D
N ,3 t
1
≤ C
48
E
Z
t−t
N +
X
N s
1ds
+ 2log N
3
E Z
t ∧t
N
X
N s
1ds
. 49
Secondly, estimates 43, 44, 46 used with u = s −t
N
allow us to refine the estimate of Lemma 3.6 on the conditional expectation of d
N ,3 s
Φ. That is we have :
Lemma 3.9. There is a positive constant C
3.9
such that for any s ≥ t
N
, E
d
N ,3 s
Φ | F
s −t
N
− log N
3
ˆ P
{0 | e
1
| e
2
}
t
N
X
N s
−t
N
Φs − t
N
, . ≤ C
3.9
||Φ||log N
−δ
3.7
X
N s
−t
N
1 + C
3.7
||Φ||
∞
t
N
log N I
N η
3.7
s − t
N
.
3.3 A key second moment estimate
