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.
Because of the restriction
α
x
α
= 1, the system of coordinates x
α α
=1,...,p
is overdetermined. The first p
− 1 coordinates suffice, and we may identify K
p
with the space
ˆ K
p
: =
x ∈ [0, ∞
p −1
:
p −1
α =1
x
α
≤ 1 .
1.1.21
A function ˆ f
∈
C
2
ˆ K
p
we can extend to a function f ∈
C
2
R
p
in such a way that f x
1
, . . . , x
p
= ˆfx
1
, . . . , x
p −1
∀x
1
, . . . , x
p −1
∈ ˆ K
p
, x
p
∈
R
. 1.1.22
This function f has the property that
∂ ∂
x
α
f x
1
, . . . , x
p
=
∂ ∂
x
α
ˆfx
1
, . . . , x
p −1
α = 1, . . . , p − 1
∂ ∂
x
p
f x
1
, . . . , x
p
= 0, 1.1.23
and hence we see that in terms of the restricted system of coordinates x
α α
=1,...,p−1
the operator A
p,q
must be written as A
p,q
ˆfx =
⊂{1,...,p} | |=q
g x
α,β ∈ \{p}
qδ
αβ
− 1
∂
2
∂ x
α
∂ x
β
ˆfx, 1.1.24
with
g x
=
γ ∈
x
γ
if p ∈
γ ∈ \{p}
x
γ
1 −
γ
x
γ
if p ∈ .
1.1.25
The system of coordinates x
α α
=1,...p−1
has the advantage that it is not overdeter- mined, but since it violates the symmetry between the colors, the formula for A
p,q
is more complicated.
1.1.3 Examples
We consider two examples of p-type q-tuple diffusion models in more detail. The first example is the p-type 2-tuple model. For this model formula 1.1.19 can be
written as A
p,2
f x =
⊂{1,...,p} | |=2
γ ∈
x
γ α,β
∈
2δ
αβ
− 1
∂
2
∂ x
α
∂ x
β
f x
=
α,β ∈
⊂{1,...,p} | |=2
∋α,β γ
∈
x
γ
2δ
αβ
− 1
∂
2
∂ x
α
∂ x
β
f x. 1.1.26
For α = β, the only set occuring in the second summation is = {α, β} and
the summand simplifies to −x
α
x
β ∂
2
∂ x
α
∂ x
β
f x. 1.1.27
For α = β, the summand in 1.1.26 can be written as
γ =α
x
α
x
γ
2 − 1
∂
2
∂ x
α 2
f x = x
α
1 − x
α ∂
2
∂ x
α 2
f x. 1.1.28
Thus, we can write A
p,2
as A
p,2
f x =
αβ
x
α
δ
αβ
− x
β ∂
2
∂ x
α
∂ x
β
f x. 1.1.29
It is easy to see that A
p,2
takes the same form in the restricted system of coordinates x
α α
=1,...,p−1
, only with the summation restricted to α, β = p.
The function w : ˆ K
p
→
R
p −1
⊗
R
p −1
given by w
αβ
x : = x
α
δ
αβ
− x
β
x ∈ ˆ
K
p
1.1.30 is called the Wright-Fisher diffusion matrix, and the diffusion process X associated
to A
p,2
is an example of a Wright-Fisher diffusion process. The behavior of this process is well-known. If we start it in a point where all colors are present, then
after a finite time one of the colors becomes extinct. It is clear from our resampling mechanism that once a color has become extinct it cannot reappear again, and
therefore the process X t
= X
α
t
α =1,...,p
moves from that moment on in the subspace
F
α
: = {x ∈ K
p
: x
α
= 0}, 1.1.31
where α is the color that has become extinct. F
α
is called the α-th face of the simplex K
p
. Note that F
α
is isomorphic to K
p −1
. After again a finite time a second color becomes extinct, and then another one, and the process moves in subspaces of
ever lower dimension until only one color is left. At that moment the process gets
stuck in one of the extremal points of the simplex. Thus, there is a finite stopping time τ such that almost surely
X τ + t = X τ ∈ {e
1
, . . . , e
p
} ∀t ≥ 0.
1.1.32 As a second example, we consider the p-type p-tuple model. For this model,
the only set occuring in the summation in 1.1.19 is = {1, . . . , p} and the
formula simplifies to A
p, p
f x =
p γ
=1
x
γ αβ
pδ
αβ
− 1
∂
2
∂ x
α
∂ x
β
f x. 1.1.33
The differential operator occuring here looks more transparant when expressed in different coordinates. We choose an orthonormal basis e
′ 1
, . . . , e
′ p
for
R
p
such that
e
′ p
=
1 √
p
1, 1, . . . , 1, 1.1.34
and we write x
′ 1
, . . . x
′ p
for the coordinates of a point x in this new basis: x
=
α
x
α
e
α
=
α
x
′ α
e
′ α
x ∈
R
p
. 1.1.35
Then
∂ ∂
x
′ p
f x = lim
ε →0
ε
−1
f x + εe
′ p
− f x =
1 √
p α
∂ ∂
x
α
f x 1.1.36
and
∂
2
∂ x
′ p
2
f x =
1 p
α ∂
∂ x
α
2
f x =
1 p
αβ ∂
2
∂ x
α
∂ x
β
f x. 1.1.37
We note that the Laplacian takes the same form in any orthonormal coordinate system:
f x =
α ∂
2
∂ x
α 2
f x =
α ∂
2
∂ x
′ α
2
f x. 1.1.38
Combining 1.1.37 and 1.1.38 we see that we can rewrite 1.1.33 as A
p, p
f x = p
p γ
=1
x
γ p
−1 α
=1 ∂
2
∂ x
′ p
2
f x. 1.1.39
Here
p −1
α =1
∂
2
∂ x
′ p
2
is the Laplacian in the plane given by the equation
α
x
α
= 1. Thus, A
p, p
is of the form A
p, p
=
p −1
α,β =1
w
αβ
x
∂
2
∂ x
′ α
∂ x
′ β
with w
αβ
x = δ
αβ
gx, 1.1.40
