Hydrodynamic Limit for the Bak–Sneppen Branching Diffusions

Abstract

We prove a hydrodynamic limit for a system of N particles moving in an open domain $D\subseteq {\mathbb R}^{d}$ according to a diffusion and undergoing branching when one particle reaches the boundary. The particle at the boundary and another random particle are eliminated and replaced with two new particles created instantaneously at a random point with distribution$\gamma (dx)$ in D. The mechanism represents a hybrid between the Fleming–Viot branching and a mean-field version of the Bak–Sneppen fitness model where the absorbing boundary represents the minimal configuration, seen as biologically not viable. The limiting profile is the normalization of the solution of a heat equation with mass creation, a PDE with non-standard boundary conditions which was studied independently by one of the authors. Under stronger conditions, the limit solves a semi-linear parabolic equation of reaction–diffusion type with a reaction term depending directly on the flux balance that determines the mass creation. An outline of the tagged particle limit is included.

Appendix

Theorem 2, the main result of this paper, uses a partial differential equations result summarized in Theorem 1, which is proven in detail in [7].

The existence of the weak solution $(\nu _{t})_{t\ge 0}$ of Definition 1, Eq. (2.2)–(2.3) is based on the construction of the auxiliary branching process $(\zeta _{t})$ and relation (8.2). In short, when $\nu _{0}=\delta _{x}$, $x\in D$, the branching process is well defined and equals the expected value of the measure valued process starting with one particle $\nu ^{x}_{t} = E_{x}[\zeta _{t}]$, $t\ge 0$.

1.1 The Auxiliary Processes $Z_{t}$ and $\zeta _{t}$

In this section, we outline the construction of a particle system $Z_{t}$ having a random total number of particles $N_{t}$. This is a counting process resulting from branching upon reaching the boundary of the domain. In that sense, our dynamics, including the conservative process $(X^{N}_{t})$ given in (3.1), is intimately related to super-critical branching. See the comments in Sect. 3.3. This states that the expected value of the empirical measure, seen as finite measure-valued random trajectory, is the solution to (2.2)–(2.3). The formal construction, definition, and proof of the regularity properties of this process, as well as related questions to its evolution semigroup, are done in [7].

At $t=0$, a single particle is placed at a random point with distribution $m_{0}(dx)\in M_{1}(D)$. The particle, starts moving according to $(L, {\mathcal D}(L))$, until it reaches $\partial D$, when it dies. Instantaneously, two particles are born at the same random point in D chosen with distribution $\gamma$. All particles start afresh and continue an independent motion in D until the first one dies and the branching is repeated. We note that particles depend on each other only through ancestry, and not through their motion.

We shall make the convention that a particle hitting the absorbing boundary jumps, instead of being killed upon contact, which makes particle labelling easier. Then each particle has a Markovian motion once it is born, namely the Brownian motion with rebirth introduced in [8], also studied in [4, 11]. Under (1.4), the particle system is well defined, having a constant number of particles between branchings. The branching times for a strictly increasing sequence, since they never coincide; all with probability one. We assume it is defined on a filtered probability space, and built constructively, up to the limit of the strictly increasing sequence of branching times, denoted by $\tau ^{*}$, a stopping time in $[0, +\infty ]$.

The model can be easily generalized to have a random number K of offspring created at the recombination point, including a smaller number than one, leading to the possibility of dissipation of mass (e.g. K may be Poisson distributed), but we shall only consider a number of exactly $K=2$ for our purpose of representing the solution of (2.2)–(2.3).

The first particle is denoted $Z^{1}_{t}$, the second $Z^{2}_{t}$, and so on. Let the number of particles at time t be denoted $N_{t}$, which, only in this special case, coincides with the number of branchings—a feature that while convenient, is not essential to the construction.

In principle, $\tau ^{*}$ could be finite with positive probability, in which case the system is said explosive. In [7] it is shown that this is not the case. Corollary 1 to Theorem 1 in [7] (here Theorem 1 (i)) gives the exact bound $||\nu _{t}||\le ||\nu _{0}|| e^{\alpha _{*}t}$ for the total variation of the solution present in the regularity condition (2.5), where $\alpha =\alpha _{*}>0$ solves (3.13). This implies that $N_{t}$ has exponential moments up to the critical value $\alpha _{*}>0$.

Denote the empirical measure

$$\begin{aligned} \zeta _{t}=\sum _{i=1}^{N_{t}}\delta _{Z_{t}}. \end{aligned}$$

(8.1)

This is a finite measure-valued Markov process, i.e. living on ${\mathbb D}([0, \infty ), M_{F}(D))$. In the technical construction, the state space is not the whole $M_{F}(D)$, but a strict subset denoted $M_{0}(D)$, the space of discrete measures on D. This aspect is not important in the present paper. For a detailed construction we point to [7].

Based on the exponential estimate on $N_{t}$ we define the expected value $\nu ^{x}_{t}(dx)$ of the empirical measure of the process $(Z_{t})_{t\ge 0}$ starting with one particle at x. Technically, we should denote this initial point by the non-random delta measure $\delta _{x}$, for consistency with the measure valued setup. We can see that $x\rightarrow \nu ^{x}_{t}(dx)$ is continuous in the topology of weak convergence and then the second integral in (8.2) is well defined.

For a bounded test function $\phi$ and a probability measure $\nu _{0}(dx)=v_{0}(x)dx\in M_{1}(D)$, we put

$$\begin{aligned} \langle \nu ^{x}_{t}, \phi (t, \cdot )\rangle := E_{x}[\sum _{j=1}^{N_{t}}\phi (t, Z^{j}_{t})] \,, \qquad \nu ^{v_{0}}_{t} = \int _{D} v_{0}(x) \nu ^{x}_{t}dx\,. \end{aligned}$$

(8.2)

Then, by uniqueness, the function $\nu ^{v_{0}}_{t}$ is the stochastic representation of the weak solution $\nu _{t}$ of the heat equation with particle creation at $\gamma (dx)$ satisfying (2.2)–(2.3). The solution has the regularity properties of Theorem 1. Moreover, if the total mass is $n_{t}=\langle \nu _{t}, 1\rangle$, then $n_{t}>0$ for all $t\ge 0$ and if $\nu _{0}\in M_{1}(D)$ then $n_{t}>1$ for $t>0$, as well as differentiable with continuous derivative. This justifies the definition $\ln n_{t} = \int _{0}^{t}a_{s} ds$.

1.2 The Strong Solution

In terms of analytic properties, a strong solution exists only outside the support of $\gamma$. In that case we can formulate the boundary conditions for the density, i.e. the forward equation, in terms of the flux balance. The second main result in this paper, Theorem 3, is based on the following result.

Theorem 5

(Theorem 3 in [7]) Let $L=\frac{1}{2}\Delta$, $\gamma (dx)=\delta _{c}(dx)$ for some $c\in D$ and $\nu _{0}(dy)=v_{0}(y)dy$, $v_{0}\in C(\bar{D})$. Then the solution from Theorem 1 has a density, i.e. $\nu _{t}(dy)=v(t, y)dy$, integrable in the space variable for any $t\ge 0$ with $v\in C([0, \infty )\times \overline{D}\setminus \{c\}) \cap C^{1, 2}((0, \infty )\times D\setminus \{c\})$ which is a solution of $\partial _{t}v = \frac{1}{2}\Delta v$ on $D\setminus \{c\}$ with $v(t, y)|_{\partial D}=0$ satisfying the flux balance condition (3.10).

We conclude with a simple example in the one dimensional case. Let $D=(0,1)$, $\partial D=\{0,1\}$, $\gamma =\delta _{c}$, $c\in (0, 1)$ and $L=\frac{1}{2}\frac{d^{2}}{d y^{2}}$ with $\nu _{0}(dx)=v_{0}(x)dx$. Then $L=L^{*}$, $\nu _{t}(dy)=v(t, y)dy$ with $v(0+, \cdot )=v_{0}(\cdot )$ and v has continuous time derivative. In addition, one can verify directly that for any $t>0$, v is smooth in $(0, c)\cup (c, 1)$ and satisfies the boundary conditions

$$\begin{aligned} \forall t>0\qquad&v(t, c-)=v(t, c+)\,, \quad v(t, 0)=v(t,1)=0\nonumber \\&(v'(t, c+)-v'(t, c-))+2 v'(0)=0\,. \end{aligned}$$

(8.3)

A similar case (with reflection at $x=1$) is studied in [17] with some additional considerations on the quasi-stationary distribution.