Higher-dimensional generalizations Overview of the three articles
It is not clear from our definitions what function g
∗
could be a fixed shape under F
c
, or in fact whether such a function exists at all. In the one-dimensional case, the fact that gx
= x1 − x is a fixed shape follows from the equilibrium conditions for the time evolution of the first and second moments of solutions to
the martingale problem for A
g,c x
. We found out that this proof immediately extends to the case that
K = {x ∈
R
d
: |x| ≤ 1}
g
∗
x = 1 − |x|
2
. 1.2.32
In view of the interpretation of our model as explained in section 1.1, this case is rather unsatisfactory. We would rather like to be able to treat the p-type p-tuple
model see section 1.1.3 or the 4-type 2 + 2-tuple model see section 1.1.7. For
the latter, K = [0, 1]
2
and the diffusion function gx
= x
1
1 − x
1
x
2
1 − x
2
1.2.33 arises in a natural way as the continuum limit of a discrete model. A natural idea
would be to see if this function is a fixed shape under F
c
. But this turns out not to be the case.
We found out that there is no explicit formula for F
c
in dimensions d ≥ 2. This
has the following reason. The equilibrium ν
g,c x
solves the equation ν
g,c x
|A
g,c x
f = 0
∀ f ∈
C
2
K , 1.2.34
where we use the notation µ| f :=
K
f xµd x 1.2.35
for any probability measure µ on K and any function f ∈
C
K . If ν
g,c x
has a sufficiently differentiable density, and also g is sufficiently smooth, then after an
integration by parts we can rewrite 1.2.34 as
A
g,c x
†
ν
g,c x
| f = 0 ∀ f ∈
C
2
K , 1.2.36
so that for the density ν
g,c x
we find the partial differential equation A
g,c x
†
ν
g,c x
= −
α ∂
∂ y
α
cx
α
− y
α
+
α ∂
2
∂ y
α 2
gy ν
g,c x
y = 0.
1.2.37 In vector notation we can write this equation in the form
∇ · T
g,c x
ν
g,c x
= 0, 1.2.38
where · denotes inner product and
∇ :=
∂ ∂
x
1
, . . . ,
∂ ∂
x
d
T : = T
1
, . . . , T
d
T
α
f y : =
− cx
α
− y
α
+
∂ ∂
y
α
gy f y.
1.2.39
The vector T can be interpreted as the expected flux, i.e., it measures the net trans- port of particles at each point in the domain. Equation 1.2.38 now says that the
divergence of the flux is zero. In some cases, the equilibrium even turns out to solve the stronger equation
T
α
ν
g,c x
= 0 α
= 1, . . . , d 1.2.40
i.e., the flux is zero. In this case we say that the equilibrium is reversible. This comes from the fact that the process, started in ν
g,c x
, is symmetric with respect to time reversal. A reversible equilibrium defines an L
2
-space of square-integrable with respect to ν
g,c x
functions, on which the operator A
g,c x
is self-adjoint, which has many technical advantages. In the one-dimensional case i.e., K is an interval,
the equilibrium is always reversible, and equation 1.2.40 can easily be solved explicitly, leading to the explicit formula 1.2.20 for F
c
. However, in the higher- dimensional case one can show that the equilibrium is for most choices of the
parameters not reversible and all we have for ν
g,c x
is equation 1.2.34, which we do not know how to solve explicitly.
The idea that finally allowed us to get some control on F
c
, at least on a heuristic level, was to try an expansion in c
−1
. For large c, the equilibrium ν
g,c x
is sharply peaked around the point x. A simple moment calculation shows that for sufficiently
differentiable g:
F
c
gx = gx + c
−1 1 2
gx gx +
O
c
−2
. 1.2.41
Here : =
α ∂
2
∂ x
α 2
is the Laplacian. If we want the right-hand side to be a multiple of g for all c
∈ 0, ∞, then g has to be a solution of the Dirichlet equation gx
= λ 1.2.42
for some λ ∈
R
. With boundary conditions gx = 0 for x ∈ ∂ K this equation has
a unique solution for each λ. Thus, we found out that the only possible candidate for a fixed shape are the multiples of the function g
∗
∈
C
2
K
◦
∩
C
K , defined as the unique solution of
−
1 2
g
∗
x = 1
on K
◦
g
∗
x = 0
on ∂ K . 1.2.43