  We explore the relationship between recursive distributional equations and
convergence results for finite difference schemes of parabolic partial
differential equations (PDEs). We focus on a family of random processes called
symmetric cooperative motions, which generalize the symmetric simple random
walk and the symmetric hipster random walk introduced in [Addario-Berry,
Cairns, Devroye, Kerriou and Mitchell, arXiv:1909.07367]. We obtain a
distributional convergence result for symmetric cooperative motions and, along
the way, obtain a novel proof of the Bernoulli central limit theorem. In
addition, we prove a PDE result relating distributional solutions and viscosity
solutions of the porous medium equation and the parabolic $p$-Laplace equation,
respectively, in one dimension.
