6. Outlook 66
6.2. Particle picture
We state another lemma, which can be proved in a similar way to the previous one by applying the bound on the derivative.
Lemma 6.1.4. There is a c=c(ν)<∞, s.t. for t >0, x, x0 ∈R Z
dy(∂xp(ν)t (x−y)−∂x0p(ν)t (x0−y))2 ≤ct−3/ν(t−2δ/ν|x−x0|δ∧1).
If one compares the spirit of these lemmas with Lemmas 9.6.2 of Section 9.6 and 9.3.4 of Section 9.3, one sees the 2 there is replaced by ν (setting α = 1, since we consider colored noise here). We do not know how many of the lemmas we see in Chapter 9, especially in Section 9.3, can be transferred to the stable setting, in colored or white noise. However one might guess that the following conjecture holds:
Conjecture 3. Pathwise uniqueness in C(R+, Ctem) holds for (6.1), ifα < ν(2γ− 1).
6.2. Particle picture
As already mentioned in Example 4.1.1, the large population limits for densities of branching particle systems are known to approach solutions of stochastic heat equations (5.1); [Daw75] or [Blo96] are two references. A similar convergence result holds true for certain interacting particle systems, called long-range voter models;
[MT94] or [Kli11] are two references. However, all of these results are limited to the noise coefficient being of the following form: σ(u) = c√
u, σ(u) = cu or σ(u) = cp
u(1−u) for a constant c. So, many of the SHE described by the class (5.1) do not have a so-called “particle picture”, i.e. there is no weakly converging density of a particle system. However, on page 326 of [MP92] an idea for density-dependent branching is given.
A classical weak convergence proof includes the two steps 1) Tightness; 2) Unique-ness of the limit. Even if tightUnique-ness results were possible, the lack of a uniqueUnique-ness result for the SHE made it difficult to obtain such convergence results. Now we have two such results: Theorem 4.3.3 for white noise and Theorem 5.3.1 for colored noise. So, it seems possible to attain weak convergence results as we describe in the following.
We will present a model which we conjecture to converge to a solution of (5.1) with diffusion coefficientσ(u) =uγand white noise. In order to describe the model, we need a certain class of offspring distributionsνγ0(u,·)∈ M1(Z+), u∈R+, γ0 ∈R, whose heuristic derivation we present in Chapter 11.
70 Outlook The model describes density-dependent branching with independent branching events. This will lead to a white noise SHE. Surely, it would be desirable to extend that to spatially correlated Gaussian noise (colored noise) as well, as the present work gives the required uniqueness result, but it is technically more demanding. So far, there are few convergence results (for example [Myt96] or [Stu03]) available for colored noise as in Definition 3.3.15.
For the description of the model we borrow notation from [Per02] and it is similar to Example 4.1.1.
Definition of the model
There are some properties of the class of offspring distributions. Let N be dis-tributed according to νγ0(u,·)∈ M1(Z+) for fixed u∈R+, γ0 ∈R.Then
E[N] = 1, E[(N−1)2] =uγ0. (6.2) Let n ∈N be the renormalization constant which is kept fixed throughout this definition.
We want to consider a population model in discrete time, where each individual gets a label α∈I =Z∗+ =S
n∈Z+Nn. Ifα=α0α1· · ·αn∈I, then we write|α|=n for the generation of individualα. Fori∈ {0,1, . . . ,|α|}we can writeα|i =α0. . . αi if we restrict α to its firsti+ 1 elements. This gives a natural ancestral relation:
Writeα≺β, wheneverα=β|i holds for a certaini∈ {0, . . . ,|β|}.Thenα is called an ancestor of β.
Our population inhabits a space E, which we suppose from now on to be R. Either we can think of E as a geographical space or a certain trait space of the individuals. Let A be the generator of a Feller semigroup on Cb(E) describing the motion of the individuals. Here we will specialize to the case A = 12∆. The corresponding semigroup (here the heat semigroup) is denoted by (St)t≥0.
Let M be a Poisson point process on Rwith intensity nµ, where µ∈ Mf(R) a finite Borel measure onR. Label its points in an arbitrary fashion with{1,2, . . . , kn} and denote the position of the points in R by xα0,1 ≤ α0 ≤ kn, where α0 is the label.
The population to be considered will evolve in time and thekninitial individuals labeled {1, . . . , kn} start at position xα0 for α0 ∈ {1,2, . . . , kn}. The evolution of the individuals is characterized by two meachanisms: First, motion in spaceE and second, resampling after discrete time steps of lengthn−1.
Let therefore {Yα,n : α0 ≤ kn, α ∈ I} be a collection of Feller-processes, with generator A, started atxα0 and stopped at time (|α|+ 1)n−1 each. We require that individuals share a common path up to their most recent common ancestor
Yα,n(t) =Yα||α|−1,n(t),0≤t≤ |α|
n
6.2 Particle picture 71 and that{Yα,n :|α|=k}are independent givenσ(Yβ,n:|β|< k) for a fixedk∈N. Note that the first requirement gives a tree structure.
Recursively, we define variablesτ and N. The n−1Z+-valued random variables τα,n will stand for the first point in time when an ancestor ofαhad no offspring or, even, there were not enough initial individuals. Thenα will be called “dead.” The Z+-valued random variables Nα,N stand for the number of offspring of individual α.
Let ¯α∈I with |¯α|=k∈Z+ be fixed. Assume that (τβ,n, Nβ,n) for β ∈I,|β|< k are given (this includes the casek= 0). Then let
τα,n¯ :=
0 if ¯α0> kn,
min{i+1n :Nα|¯i,n= 0,0≤i <|α|}¯ if set 6=∅ and ¯α0 ≤kn,
|¯α|+1
n otherwise.
For t∈[0,k+1n ) = [0,|¯α|+1n ) we define the following (random) relation
¯
α∼nt ⇔t∈[|¯α|n ,|¯α|+1n ) and t < τα,n¯ ,
indicating up to which time the direct ancestor of individual ¯α was alive. For B ∈ B(R) define for 0≤t < |¯α|+1n the renormalized number of alive individuals in the region B by
Xtn(B) = 1
n#{β :β ∼nt, Yβ,n(t)∈B}= 1 n
X
β∼nt
δYβ,n(t)(B) For anyy ∈Rdefine:
un(t, y) := 1 n
X
β∼nt
p1/n(Yβ,n(t)−y),
wherep1/n(·) is the 1-dimensional heat kernel at timen−1. The non-negative func-tion un describes an approximate density of individuals close to y.
Now, we have defined what is happening up to time t= |¯α|+1n −with individual ¯α.
At time |¯α|+1n a branching event happens with a certain offspring we define now.
Consider the Z+-valued random variable Nα,n¯ with distribution P
Nα,n¯ =·|{Yβ,n(|β|+ 1
n ),|β|=|¯α|, τβ,n= |β|+ 1 n }
=ν2γ−1
un((|α|¯ + 1
n )−, Yα,n¯ (|¯α|+ 1 n )),·
72 Outlook in the case τα,n¯ = k+1n . Here,νγ0 is a probability law onZ+ with the properties in (6.2); the definition is given in (11.2) for γ0 ∈R. We assume that conditionally on F(k+1)nn −1 = σ Yβ,n(s),|β| ≤k+ 1, s≤ (k+ 1)n−1, Nβ0,n,|β0| ≤ k
the collection {Nα,n :|α|=k+ 1} is independent. This completes the iterative description.
Convergence conjecture Let us write hf, gi=R
Rf(x)g(x)dx for the L2-scalar product. For the density uN as defined above, we conjecture the following fact:
Conjecture 4. Assume q = 1, γ ∈ (1/2,1) and assume that un0, u0 ∈ Ctem are deterministic for n ∈ N. If un0 → u0 in Ctem, then un is relatively compact in D(R+, Ctem). Each weak limit uis defined on a filtered probability space which also supports a white noise W such that for each φ∈Cc∞(R) we have almost surely for all t≥0
hu(t,·), φi=hu(0,·), φi+ Z t
0
hu(s,·),1
2φ00ids+ Z t
0
Z
R
√
λu(s, y)γφ(y)W(ds dy).
If γ∈(3/4,1),thenun converges weakly to u.
In order to show such a result, there are two things to do: 1) Tightness, 2) Identification of the limit.
Our idea of proof is based on the classic ways which one can use for measure-valued processes or densities, carried out e.g. in [MT94], [Blo96] or [Per02]. We give a more precise description of the idea in Section 11.2. Here, we will only consider part of what belongs to “identification of the limit:”
We will follow Perkins’ notation again and only write down the “branching mar-tingale” (defined in (11.4)), leading to the diffusion term in the SPDE:
Mtb,n(φ) = 1 n
X
s<t
X
α∼ns
φ(Ys+nα,n−1)(Nα,n−1)
for t ∈ n−1N. This is a discrete time martingale (E[N] = 1) and we calculate its quadratic variation:
E[(Ms+nb,n−1(φ)−Msb,n(φ))2|F¯s]
=n−2 X
α∼ns
φ(Ys+nα,n−1)2E[(Nα,n−1)2|F¯s]
=n−2 X
α∼ns
φ(Ys+nα,n−1)2λun(s+n−1, Yα,n(s+n−1))2γ−1.