info:eu-repo/semantics/article
Continuous time random walks and the Cauchy problem for the heat equation
Fecha
2018-10Registro en:
Aimar, Hugo Alejandro; Beltritti, Gastón; Gomez, Ivana Daniela; Continuous time random walks and the Cauchy problem for the heat equation; Springer; Journal d'Analyse Mathématique; 136; 1; 10-2018; 83-101
0021-7670
CONICET Digital
CONICET
Autor
Aimar, Hugo Alejandro
Beltritti, Gastón
Gomez, Ivana Daniela
Resumen
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.