Trimmed branching random walk and a free obstacle problem
Rami Atar, Leonid Mytnik, Gershon Wolansky
TL;DR
The paper analyzes a trimmed branching random walk on an ε-grid with density-dependent removal, where branching triggers removal from the most populated site. In the hydrodynamic limit (first $N\to\infty$, then $\varepsilon\to0$), the particle density $u$ is governed by the parabolic free obstacle problem $\partial_t u = L^*u+u-\beta$, with $L^*=\Delta-\nabla\cdot(b\cdot)$ and a measure-valued removal term $\beta$ supported on the argmax set of $u$, while preserving total mass $|u|_1=1$. The authors develop a three-step program: (i) derive a lattice ODE for fixed $\varepsilon$ describing $(u^{(ε)},\Lambda^{(ε)})$, with a unique solution; (ii) prove tightness and convergence of the particle system to the lattice dynamics, and then (iii) obtain uniform continuity estimates via a coupling to pass to the continuum PDE; finally, they prove PDE uniqueness to ensure convergence to a single limit. They also present explicit stationary solutions illustrating flat-top vs sharp-top argmax sets and discuss open problems, including extending well-posedness beyond Lipschitz drift and connecting to gradient-flow frameworks. Overall, the work provides a rigorous link between density-dependent selection in particle systems and a free obstacle PDE, with implications for understanding macroscopic selection mechanisms in interacting particle models.
Abstract
Consider $N$ particles performing random walks on the $ε$-grid $(εZ)^d$, $ε>0$ 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 $ε\to0$, 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 $L^*=Δ-\nabla\cdot(b\,\cdot)$, the hydrodynamic limit, as $N\to\infty$ followed by $ε\to0$, is characterized in terms of a parabolic free obstacle problem \[ \partial_t u=L^*u+u-β\] where $β$ 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 $β$, the removal measure, for which $t\mapstoβ(R^d\times[0,t])$ 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-$ε$ 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.