Trimmed branching random walk and a free obstacle problem

TL;DR

This paper studies a particle system with branching and spatial selection, deriving a hydrodynamic limit described by a free obstacle PDE, and introduces novel techniques for analyzing the resulting free boundary problem.

Contribution

It establishes a new connection between branching random walks with spatial selection and a class of free obstacle PDEs, using innovative analytical methods.

Findings

Hydrodynamic limit characterized by a free obstacle PDE.

Existence and uniqueness of solutions for the PDE.

Discussion of open problems like flat versus sharp top solutions.

Abstract

Consider $N$ particles performing random walks on the $\epsilon$-grid $(\epsilon Z)^d$, $\epsilon>0$ with branching and density-dependent selection: When one of the particles branches, a particle is removed from the most populated site. The walks are assumed to be asymptotic, as $\epsilon\to0$, to diffusion processes of the form \[ dX_i(t)=b(X_i(t))dt+\sqrt{2}dW_i(t), \] for $b$ a given vector field. Denoting $L^*=\Delta-\nabla\cdot(b\,\cdot)$, the hydrodynamic limit, as $N\to\infty$ followed by $\epsilon\to0$, is characterized in terms of a parabolic free obstacle problem \[ \partial_t u=L^*u+u-\beta \] where $\beta$ is a measure on $R^d\times[0,\infty)$ supported on $\{(x,t):u(x,t)=|u(\cdot,t)|_\infty\}$. Here, the unknowns are $u$, the mass density, and $\beta$, the removal measure, for which $t\mapsto\beta(R^d\times[0,t])$ is prescribed. This is analogous to the well-understood relation between particle systems with spatial selection and free boundary problems, but the techniques require quite different ideas. The key ingredients of the proof include PDE uniqueness for continuous densities and a uniform-in-$\epsilon$ estimate on modulus of continuity of prelimit densities. The work gives rise to open problems such as ``flat top'' versus ``sharp top'' solutions, which are discussed based on concrete examples.