Persistence exponent of the diffusion equation in dimensions

We consider the d-dimensional diffusion equation ∂tφ(x,t)=Δφ(x,t) with random initial condition, and observe that, when appropriately scaled, φ(0,t) is Gaussian and Markovian in the limit d→0. This leads via the Majumdar–Sire perturbation theory to a small d expansion for the persistence exponent θ(d). We find