One-dimensional heat diffusion modelling and random walks on non-uniform grids

A probabilistic model for the diffusion of heat on one-dimensional spaces is developed. Specifically, grid points are arbitrarily placed on the real line and the heat particles are assumed to jump between these grid points in continuous time. This random walk by the heat particles is represented by a continuous-time Markov chain and state-transition intensities depends on the underlying, possibly non-uniform, spacing of the grid points to characterize the Markov chain. Several numerical examples are demonstrated to show the efficacy of the proposed framework for exemplary non-uniform grids.