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
and g : K
p
→ [0, ∞ some function. We express this by saying that the diffusion matrix w is isotropic. The process X associated to A
p, p
is an isotropic diffusion. Such an isotropic diffusion with zero drift is just a time-transformed Brownian
motion. The behavior of X is as follows. In a finite time one of the types gets extinct and, as is clear from our resampling mechanism, after that time the fre-
quencies of all other colors remain fixed. Thus, there is a finite stopping time τ such that almost surely
X τ + t = X τ ∈ ∂ K
p
∀t ≥ 0, 1.1.41
where ∂
K
p
: =
p α
=1
F
α
= {x ∈ K
p
: x
α
= 0 for some α}. 1.1.42
The behavior of general p-type q-tuple diffusions with 2 q p is similar to that of the two examples above. One by one colors become extinct, until only
q − 1 colors are left and the process comes to a halt.
1.1.4 The p-type q-tuple model with migration
Once again consider an urn with balls of p colors, but this time let us assume that the number of balls is not fixed. Instead, we introduce the following migration
mechanism. We assume that with rate cρ our urn receives balls from some large reservoir, where the proportions of the different colors are fixed to θ
α α
=1,...,p
= θ
∈ K
p
. Moreover, we assume that each ball in our urn dissapears from the urn with rate c. Here c, ρ
∈ 0, ∞ and it is easy to see that the expected number of balls in our urn tends to ρ as time tends to infinity.
In addition to this migration mechanism, we assume that the q-tuples present in our urn are at any given moment subject to the resampling mechanism described
in section 1.1.1 with rate qq − 1ρ
2 −q
. We write Y
ρ α
t for the number of balls of color α present in our urn at time t, and we define
X
ρ α
t : =
1 ρ
Y
ρ α
t. 1.1.43
The generator of the process X
ρ
is given by compare 1.1.8 and 1.1.11 G
ρ
f x = ρ
2 ⊂{1,...,p}
| |=q γ
∈
x
γ α,β
∈ α
=β
f x +
1 ρ
e
α
−
1 ρ
e
β
− f x
+
α
cρθ
α
f x +
1 ρ
e
α
− f x + cx
α
f x −
1 ρ
e
α
− f x .
1.1.44
Here the first term results from the resampling mechanism see formula 1.1.5. One can check that for every f
∈
C
2
[0, ∞
p
and for every compact subset C ⊂
[0, ∞
p
lim
ρ →∞
sup
x ∈C∩
1 ρ
N
p
G
ρ
f x − A
p,q
f x − B
θ, c
f x =
0, 1.1.45
where A
p,q
is defined as in 1.1.19 and B
θ, c
f x : = c
α
θ
α
− x
α ∂
∂ x
α
f x. 1.1.46
Thus we expect the process X
ρ
to converge, as ρ → ∞, to a process X with
generator G, where G is an extension of the operator A
p,q
+ B
θ, c
. It is possible to prove existence of solutions to the martingale problem for A
p,q
+ B
θ, c
, and one can check that for a solution X with initial condition X 0
∈ K
p
X t ∈ K
p
∀t ≥ 0 1.1.47
almost surely. This means that in the limit of large ρ, the total number of balls in the urn is approximately fixed to ρ. For a function f
∈
C
2
K
p
we define A
p,q
f + B
θ, c
f by extending f to a function in
C
2
[0, ∞
p
, and we can see that the result does not depend on the choice of the extension.
1.1.5 Uniqueness problems
Remarkably, it is not known whether the diffusion processes introduced in the last section are well-defined. Namely, it is not known in general whether solutions to
the martingale problem for A
p,q
+ B
θ, c
are unique. The standard result about strong uniqueness of solutions to stochastic differential equations does not apply, because
it is not possible to find a Lipschitz continuous root of the diffusion matrix occuring in A
p,q
see section B.2. The problems occur at the boundary ∂ K
p
of the simplex. It is possible to represent solutions to the martingale problem for A
p,q
+ B
θ, c
as solutions to a stochastic differential equation of the form
d X
α
t = cθ
α
− X
α
tdt +
β
σ
αβ
X td B
β
t, 1.1.48
where the function σ is locally Lipschitz on K
p
\∂ K
p
but not Lipschitz at ∂ K
p
. Therefore strong uniqueness of solutions to 1.1.48 can be shown only up to the
first hitting time of ∂ K
p
. If c
= 0, then solutions to 1.1.48 are martingales, and hence after the first hitting time of ∂ K
p
the process stays in one of the faces F
α
. Since these faces are
isomorfic to K
p −1
, it is then possible to prove strong uniqueness of solutions to 1.1.48 by induction. For this technique, see Example 3.1.8 below.
For c sufficiently large, it is possible to show that solutions to the martingale problem for A
p,q
+ B
θ, c
never hit the boundary ∂ K
p
, so that solutions to 1.1.48 are strongly unique for all time. For this technique, see Theorem 2.2.9 below.
However, for c positive but not too large the process X with probability one reaches the boundary ∂ K
p
in a finite time. In fact, it hits ∂ K infinitely often in a finite time, each time bouncing back from it. About processes with such behavior
very little is known. Certainly the standard strong uniqueness results do not apply, except in the one-dimensional i.e. 2-type case.
For the p-type 2-tuple model, there are several ways to circumvent this prob- lem. For example, it is possible to find a root σ that is lower-triangular. This
corresponds the fact that each subselection of the colors is itself a Markov process following a Wright-Fisher diffusion. Thus, the idea is that one can first prove
strong uniqueness for one color, using one-dimensional techniques, then prove strong uniqueness for the second color conditional on the first one, and so on.
In another approach, one can prove weak uniqueness for the p-type 2-tuple model with migration by more or less explicitly calculating all moments of X t.
Here one uses the fact that A
p,2
+ B
θ, c
maps a polynomial of degree n into a polynomial of degree at most n. Thus, the time evolution of all moments up to
n-th order is described by a closed system of equations, that is easily seen to have a unique solution.
3
This technique has the advantage that it also proves that the closure of A
p,2
+ B
θ, c
generates a Feller semigroup. For general p-type q-tuple models, one can see that A
p,q
maps a polynomial of degree n into a polynomial of degree n
− 2 + q, and hence for q ≥ 3 the time evolution of moments up to n-th order cannot be expressed in a closed system.
It seems that duality techniques involving moments that are known to work in certain other models also fail here, and uniqueness of solutions to the martingale
problem for A
p,q
+ B
θ, c
with p, q ≥ 3, for general θ ∈ K
p
and c 0, is still an open problem.
1.1.6 Interacting p-type q-tuple models
We now consider a collection of urns, indexed by a finite or countable Abelian group , with
group operation i
+ j inverse
−i unit element
0. 1.1.49
3
In fact, this solution can be represented in terms of a dual process, which is sometimes handy in calculations. But here this duality is not essential.
