Large time behaviour for the heat equation on 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 Z on ℓp 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 ℓp decay results for the fundamental solution. We use that estimates to get rates on the ℓp decay and large time behaviour of solutions. For the ℓ2 case, we get optimal decay by use of Fourier techniques.