Continuous time random walks and the Cauchy problem for the heat equation

Abstract

We deal with anomalous diffusions induced by continuous time random walks - CTRW in ℝn. A particle moves in ℝn in such a way that the probability density function u(·, t) of finding it in region Ω of ℝn is given by ∫Ωu(x, t)dx. The dynamics of the diffusion is provided by a space time probability density J(x, t) compactly supported in {t ≥ 0}. For t large enough, u satisfies the equation u(x, t) = [ (J− δ) * u] (x, t) , where δ is the Dirac delta in space-time. We give a sense to a Cauchy type problem for a given initial density distribution f. We use Banach fixed point method to solve it and prove that under parabolic rescaling of J, the equation tends weakly to the heat equation and that for particular kernels J, the solutions tend to the corresponding temperatures when the scaling parameter approaches 0.