The p-type q-tuple model
The process X
N
t takes values in the discrete p − 1-dimensional simplex
K
N p
: =
x ∈
1 N
N
p
:
α
x
α
= 1 .
1.1.9 We can write the generator of X
N
as G
N
f x =
y ∈K
N p
Ŵ
N
x, y f y, 1.1.10
where Ŵ
N
x, y =
N
2 ⊂{1,...,p}
| |=q γ
∈
x
γ α,β
∈ α
=β
δ x
+
1 N
e
α
−
1 N
e
β
, y
− δx, y .
1.1.11
As N tends to infinity, we expect the process X
N
to converge to a diffusion process X on the p
− 1-dimensional simplex K
p
: =
x ∈ [0, ∞
p
:
α
x
α
= 1 .
1.1.12 In order to find out what the generator of X could be, we have to calculate the
moments of the kernel Ŵ
N
in 1.1.11 up to leading order in N . It is immediately clear from the definition that the zeroth moment of Ŵ
N
is zero:
y ∈K
N p
Ŵ
N
x, y = 0.
1.1.13 For the first moment, we note that the process X
N
is a martingale: in our resam- pling procedure the expected increase in the number of balls of any color is zero.
This implies that for all α = 1, . . . , p
y ∈K
N p
Ŵ
N
x, yy
α
− x
α
= 0. 1.1.14
For the second moments a small calculation yields
y ∈K
N p
y
α
− x
α
y
β
− x
β γ ,η
∈ γ
=η
δ x
+
1 N
e
γ
−
1 N
e
η
, y
=
γ ,η ∈
γ =η
1 N
2
δ
αγ
δ
βγ
+ δ
αη
δ
βη
− δ
α,γ
δ
β,η
− δ
α,η
δ
β,γ
= 2
N
2
1
{α,β∈ }
qδ
αβ
− 1, 1.1.15
so that for all α, β = 1, . . . , p
y ∈K
N p
Ŵ
N
x, yy
α
− x
α
y
β
− x
β
=
⊂{1,...,p} | |=q
∋α,β γ
∈
x
γ
2 qδ
αβ
− 1 .
1.1.16
Finally, we have
y ∈K
N p
Ŵ
N
x, y |y − x|
3
=
O
N
−1
, 1.1.17
uniformly in x as N → ∞. Using a Taylor expansion one can now check
1
that for all f
∈
C
2
R
p
lim
N →∞
sup
x ∈K
N p
G
N
f x − A
p,q
f x =
0, 1.1.18
where we have defined A
p,q
f x : =
⊂{1,...,p} | |=q
γ ∈
x
γ α,β
∈
qδ
αβ
− 1
∂
2
∂ x
α
∂ x
β
f x 1.1.19
By definition, the domain of A
p,q
is
D
A
p,q
: =
C
2
K
p
, 1.1.20
the space of real functions on K
p
that can be extended to a function in
C
2
R
p
. Formula 1.1.18 makes clear that the definition of A
p,q
f does not depend on the choice of this extension.
We conclude from 1.1.18 that if the jump process X
N
converges to a diffusion process X , then the generator of X has to be an extension of A
p,q
. It would carry too far for this introduction to prove the convergence of X
N
to X , but we take 1.1.18 as our motivation to study solutions to the martingale problem for A
p,q
. It can be shown that this martingale probem is well-posed.
2 1
Compare [16], Theorem 1.1 of chapter 10.
2
Uniqueness of solutions to the martingale problem for A
p,q
can be shown with the help of techniques mentioned in the proof of Example 3.1.8 below. Convergence of X
N
to X in the sense of weak convergence on path space
D
K
p
[0, ∞ is a non-trivial problem, even when uniqueness of
solutions to the martingale problem for A
p,q
is known. The problem is to show tightness for X
N
. It is sufficient if the closure of A
p,q
generates a Feller semigroup, see [16], section 8 of chapter 4. It is known that the closure of A
p,2
generates a Feller semigroup, see [16], Theorem 2.8 of chapter 8.