Large time behaviour for the heat equation on $\mathbb{Z},$ moments and decay rates

The paper is devoted to understand the large time behaviour and decay of the solution of the discrete heat equation in the one dimensional mesh $\mathbb{Z}$ on ${\mathrm{\ell }}^p$ spaces, and its analogies with the continuous-space case. We do a deep study of the moments of the discrete gaussian kernel (which is given in terms of Bessel functions), in particular the mass conservation principle; that is reflected on the large time behaviour of solutions. We prove asymptotic pointwise and ${\mathrm{\ell }}^p$ decay results for the fundamental solution. We use that estimates to get rates on the ${\mathrm{\ell }}^p$ decay and large time behaviour of solutions. For the ${\mathrm{\ell }}^2$ case, we get optimal decay by use of Fourier techniques.