\(N\) particles move according to independent Brownian motions in an open subset \(D\subset{\mathbb R}^d\). Whenever a particle hits the boundary of \(D\), it jumps to the current location of a uniformly selected particle within \(D\). Equivalently, the first particle is killed and another particle splits into two particles. It is shown that as \(N\to\infty\), the particle distribution density converges to the normalized heat equation in \(D\) with Dirichlet boundary conditions and the stationary distributions converge to the first eigenfunction of the Laplacian in \(D\) with the same boundary conditions. The model is closely related to that studied by the authors and D. Ingerman [J. Phys. A, Math. Gen. 29, No. 11, 2633-2642 (1996; Zbl 0901.60054)] using heuristic and numerical methods.