Heat diffusion modelling with random walks on triangular lattices

In order to approximate heat diffusion in two dimensions, we view diffusion processes as the random motion of particles and model the behavior of each particle with a continuous-time Markov chain. The infinitesimal generator of each Markov chain is characterized by the distances between lattice points imposed on a given two-dimensional surface. We derive requirements for the mean and the covariance matrix of a Markov chain and present simulations to demonstrate how a large number of Markov chains behave in the proposed framework on an exemplary lattice.