The domain of A is the space of all
C
2
-functions on K
p
that depend on finitely many coordinates only, and a point x
∈ K
p
we denote as x
= x
α i
α =1,...,p
i ∈
. 1.1.57
Infinite systems of interacting diffusion processes of this type and their generaliza- tions are the main subject of study in this dissertation.
For definiteness let us write down the operator A above in restricted coordi- nates. With ˆ
K
p
as in 1.1.21 and ˆ K
p
the space of all points x = x
i i
∈
with x
i
∈ ˆ K
p
, we have for any
C
2
-function ˆ f that depends on finitely many of the x
i
only A ˆ
f x : =
i αβ
w
p,q αβ
x
i ∂
2
∂ x
α i
∂ x
β i
f x +
i j,α
a j − ix
α j
− x
α i
∂ ∂
x
α i
f x, 1.1.58
where for any x ∈ ˆ
K
p
and α, β = 1, . . . , p − 1
w
p,q αβ
x =
⊂{1,...,p} | |=q
∋α,β
g xqδ
αβ
− 1, 1.1.59
with g x as in 1.1.25. A short look at section 1.1.3 learns us that in particular
w
p,2 αβ
x = x
α
δ
αβ
− x
β
w
p, p αβ
x =
γ
x
γ
1 −
γ
x
γ
pδ
αβ
− 1. 1.1.60
1.1.7 Other models
We briefly mention here some diffusion models that are closely related to the p- type q-tuple models. We start with two models with non-compact state space.
Feller’s branching diffusion Consider an urn with balls of one color. With
rate one each ball is with equal probabilities replaced by either two or zero balls ‘critical branching’. In the right scaling, this process converges to a diffusion on
[0, ∞ whose generator extends the operator
A f x : = x
∂
2
∂ x
2
f x. 1.1.61
Extensions to more colors are immediate. These models can be viewed as a limit of the p-type 2-tuple model when one of the types occupies almost all the urn.
It is easy to see that the evolution of the rare types can then be approximated by independent critical branching.
Mutually catalytic branching This model has been introduced by Dawson
and Perkins [14]. Consider an urn with balls of two colors, subject to the following resampling mechanism. With rate one, each pair 2-tuple of balls is selected. If
they have different colors, then one ball is selected and with equal probability this ball is replaced by two or zero balls of the same color. In the right scaling, we
expect this process to converge to a diffusion on [0,
∞
2
whose generator extends the operator
A f x : = x
1
x
2 ∂
2
∂ x
1 2
+
∂
2
∂ x
2 2
f x. 1.1.62
This is an isotropic model. It can be viewed as a limit of the 3-type 3-tuple model when one of the types is much more common than the other two. The evolution of
the two rare colors can then be approximated by mutually catalytic branching. It is known that the martingale problem for the operator A in 1.1.62 is well-posed,
also when a migration term B
θ, c
as in 1.1.46 is added to 1.1.62. In fact, it is easy to see that all the moments of the process can be calculated. However, since the
moment problem is not well-posed in this non-compact setting, one has to do a bit more to prove uniqueness of solutions to the martingale problem. This is achieved
in [14] by means of a self-duality of the model, due to Mytnik.
The 4-type 2 + 2-tuple model Consider an urn with balls of four colors. With
rate one each quadruple of balls is selected. If all the colors are different, then with equal probabilities either the ball with color 1 is replaced by a ball of color 2 or
vice versa, or the ball of color 3 is replaced by a ball of color 4 or vice versa. In this way, the proportion of the colors 1 plus 2 with respect to the colors 3 plus 4 is
not changed. We start in a situation where the total number of balls of the colors 1 plus 2 equals the total number of balls of the colors 3 plus 4, and we denote by
Y
α
t the number of balls of color α at time t. In the right scaling, we expect the process Y
1
, Y
3
to converge to a diffusion on [0, 1]
2
whose generator extends the operator
A f x : = x
1
1 − x
1
x
3
1 − x
3 ∂
2
∂ x
1 2
+
∂
2
∂ x
3 2
f x. 1.1.63
This is an isotropic model, similar in spirit to mutually catalytic branching. In fact, in the limit that the colors 2 and 4 are much more common than the colors 1
and 3, we recover the mutually catalytic model. When a migration term B
θ, c
as in 1.1.46 is added to the diffusion in 1.1.63, it is not known whether solutions to
the corresponding martingale problem are always unique.
Composition-dependent resampling This is not really one model, but a recipe
by which one can produce a whole collection of other models. One considers the situation where the rate with which the various resampling mechanisms take
place depends on the whole composition of the urn. This includes all the models discussed so far. For example, if in the 3-type 2-tuple model one lets the rate at
which one color replaces another depend linearly on the amount of the third color present, then one arrives at the 3-type 3-tuple model. If in the 2-type 2-tuple model
one lets the resampling rate depend on the amount of both types present in the urn, then one finds the operator
A f x : = x
2
1 − x
2 ∂
2
∂ x
2
, 1.1.64
known as Kimura’s random selection model. One of the main goals in this disser- tion is to show that for infinite systems of interacting diffusions such modifications
of the diffusion function do not influence the behavior of the system on large scales, both in space and time.
1.2 Overview of the three articles
1.2.1 Renormalization theory
Renormalization theory is one of the most succesful techniques for understanding universal large scale behavior of interacting particle systems, at least on the level
of heuristic and non-rigorous calculations. The basic idea of the theory is quite simple. First, one needs to find a way to describe a system on a series of ever larger
scales. A usual way is to group the particles into blocks, consisting of a particle and a few of its neighbours, then group these blocks into larger blocks, and so on. With
each scale is associated a set of variables describing the system as if viewed from ever larger distances, where details of the local behavior become ever less visible.
For example, the first set of variables may give the precise state of each particle, the second set only the average value of all particles in a block, and the third set
only averages over blocks of blocks, etc. Each time one goes to a larger scale, the probability law describing the new variables is a marginal of the law describing the
old variables. Thus, in principle, one has a map describing how to go from the old variables to the new larger scale variables. This map is called a renormalization
transformation. If it is the case that under iteration of this transformation different local laws converge to one and the same global law, then one has universal behavior
on large scales.
In practice it is not so easy to realize this renormalization scheme. In order to make it work, one needs an efficient way to describe the probability law of
the renormalized variables. However, it often happens that while the law of the local variables has nice properties, the renormalized law has not for example, it
may be non-Markovian or non-Gibbsian. In such cases a rigorous study of the renormalization transformation is very hard and frequently impossible.
