Abstract
In the paper we analyze singularly perturbed telegraph systems applying the newly developed compressed asymptotic method and show that the diffusion equation is an asymptotic limit of singularly perturbed telegraph system of equations. The results are applied to the random walk theory for which the relationship between correlated and uncorrelated random walks is explained in asymptotic terms.
Highlights
An asymptotic equivalence between uncorrelated and correlated random walks has been studied by many authors
The results can be divided into two streams: in the first one the probabilistic aspect of the problem is of primary interest and the results for the differential equations are obtained as a by-product and in the second one the situation is reversed
We combine recent advances in the asymptotic analysis of kinetic equations with an observation that the system of telegraph equation is a simplified Boltzmann equation to provide a unified treatment of various singularly perturbed telegraph systems showing their asymptotic equivalence with a diffusion equation
Summary
An asymptotic equivalence between uncorrelated and correlated random walks (or between the diffusion and telegraph equations) has been studied by many authors (see e.g. [14], a survey in [16] or recent paper [7]). An asymptotic equivalence between uncorrelated and correlated random walks (or between the diffusion and telegraph equations) has been studied by many authors In most cases known to us the asymptotic analysis is carried out for the second order telegraph equation. It is known that in the continuous limit of the correlated random walk one obtains a system of equations which can be reduced to the second order equation only for a restricted class of coefficients. We shall show that the diffusion approximation is valid under the sole assumption that the strength of correlations tends to zero allowing the speed of particles to remain finite. The reader is advised to consult the relevant references cited in the book ([1, 3, 4, 9, 10, 12])
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