Self-similar fragmentation

Self-similar mass fragmentation processes are characterised by – the index of self-similarityα∈R;

– a so-called dislocation measure ν on S = {s = (s_i)_i∈N | lim_i→+∞si = 0,1 ≥ sj ≥ si ≥ 0,∀j ≤i}which satisfies

ν(1,0,0, ..) = 0 and

Z

S(1−s)ν(ds)<+∞.

Ifν(S)<+∞then the dynamics is as follows:

– a block of massxremains unchanged for exponential periods of time with parameterx^αν(S);

– a block of massxdislocates into a mass partitionxs, wheres∈S, at rateν(ds);

– there are finitely many dislocations over any finite time horizon.

The last point is not verified whenν(S) = +∞. In this case, there is a countably infinite number of dislocations over any finite time horizon. So, whenν(S)< +∞, our setting capture this model with the following parameters:

G= 0, r(x) = x^αν(S), and for every continuous and bounded functionf,

Z 1 0

X

k≥0

p_k(x)

k

X

j=1

f(F_j^kx, θ)dθ =

Z

S

X

i≥0

f(s_ix)ν(ds) ν(S) . Hence, in this case we have

Gf(x) =x^αν(S)^X

i≥0



 Z

S

X

i≥0

f(s_ix)ν(ds)

ν(S) −f(x)



,

for every continuous and bounded f, and V : x 7→ x^p is an eigenevector. See [Ber06] for further details.

Limit theorems for some branching measure-valued processes

Chapter 6

Asymptotic estimates for the largest individual in a mitosis model

6.1 Introduction and statement of result

In this note, we consider a growth-fragmentation model, in continuous time, to represent dividing cells. The dynamics is as follows. The sizeXtof a cell, at timet, evolves according to the following stochastic differential equation

∀t≥0, dX_t= (µ+σ²)X_tdt+σ√

2X_tdBt, (6.1)

whereµ∈Randσ≥ 0. Ifσis equal to zero then the evolution is deterministic. At rater > 0, each cell splits into two offspring whose size is worth half of that of their mother. This model is entirely determined by the parametersµ, σ andr. The parametersµand σ depend on the ability of cells to ingest a common nutrient. The parameterris the division rate. We assume that is is constant. More generally, instead of size of cell, this model can represent some biological content which grows in the cells and is shared when the cells divide (for example proteins, nutriments, energy or parasite). The process we study is a Markov process on Galton Watson trees and [BDMT11,Clo11] give asymptotic results, under an ergodicity assumption, which is not fulfilled here. The article [Clo11] also shows that the empirical process converges, when the size of the population tends to infinity, to the following

Asymptotic estimates for the largest individual in a mitosis model

physiologically structured equation:

∀t≥0, ∀x≥0, ∂_tn(t, x) + (µ+σ²)∂_x(xn(t, x)) +rn(t, x) = 4rn(t,2x) +σ²∆n(t, x).

This type of equation has been recently studied in [DHKR12, DPZ09, Per07], for instance. From a probabilistic point of view, the behaviour of one cell is known. It is the exponential of a Lévy process and thus, using classical results [Ber96], we deduce its long time behavior. There is a duality: either the process explodes or it vanishes. Closely related, [BT11] studied a population of infected cell.

Between the division, the parasites grow following a Feller diffusion. The main difference between our model and theirs is the fact that, in our model, the size of the cells can not be null even if it tends to zero. Here, we are interested by the properties of the largest individual. This question has been intensively studied in the special case of Branching Brownian motion [Bra78,HHK06,McK75, Rob11] and fragmentation processes [BHK11,Ber04,Ber06]. Due to the spacial motion (exponential growth) and the non-local branching mechanism (the offspring do not appear at the position of their mother), the mathematical study is different. There is a competition between the exponential growth of size, between the division, and the multiplicative decreasing, at the division. One of purpose of our main results is to highlight this competition.

Let us give now a qualitative description of our model, which is rigorously defined in the next section. We consider a continuous time Yule tree, which is, a tree where each branch lives during an independent exponential time of mean1/rand then splits into two new branches. We denote by N_tthe size of the living population setV_t, at timet ≥ 0, and by(X_t^u)u∈Vt the size of the individual u∈Vt. We aim at determining how the maximumXevolves in the cell population:

∀t≥0, X_t= max

u∈Vt

X_t^u. Now, we can state our main results:

Theorem 6.1.1(Asymptotic estimates of the size of the largest individual). We have

t→+∞lim 1

t lnX_t = inf

α≥0µ+σ²α+r2e^−αln(2)−1

α .

The infimum in the right hand side of the last expression is attained in a unique α ≥ 0. This theorem gives the long time behavior of the extremal particle. To compare, we also give the mean behavior of the population:

Theorem 6.1.2(Mean behavior of the population). We have the following duality:

Asymptotic estimates for the largest individual in a mitosis model

(i) ifµ <2rln(2)then for allε >0, we have

t→+∞lim

card{u∈V_t|X_t^u ≥ε}

N_t = 0a.s.

(ii) Ifµ >2rln(2)then for every0≤κ < g−2rln(2) P lim sup

t→∞

card{u∈V_t:X_t^u ≥e^κt}

N_t >0

!

= 1 (6.2)

These two theorems give sharp estimates. Nevertheless, this model is too simple and it will be efficient to generalize the first one in one of the following generalization:

– the parameterσ is null butris not constant as in [DHKR12];

– the growth between two divisions is linear instead of exponential as in [Clo11];

– the growth between two divisions is described by a Feller diffusion as in [BT11].

Unfortunately, even ifrcan be constant in the two last models, these setting are very different to ours.

Our model is relatively close to branching random walk contrary to these models.

Outline:in the next section, we give more formal definitions based on measure-valued processes and state the asymptotic behavior for geometric Brownian motion with multiplicative jumps. This gives the asymptotic behavior of the size in a cell line. We introduce, in Section 6.2.2, the scaling property, and, in Section6.2.3, some martingales. Theorem 6.1.1is proved in Section6.3 and The- orem6.1.2 in Section 6.4. In the last section, we give some comments about the link between the fluctuation of the extremal particle and a F-KPP type equation.