FUNDAMENTALNAYA I PRIKLADNAYA MATEMATIKA

(FUNDAMENTAL AND APPLIED MATHEMATICS)

2009, VOLUME 15, NUMBER 2, PAGES 3-21

Random process in a homogeneous Gaussian field

V. I. Alkhimov

Abstract

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We consider a random process in a spatial-temporal homogeneous Gaussian field $V($q,t) with the mean $$EV=0 and the correlation function $W(|$q - q'|,|t - t'|) º E[V(q,t)V(q',t')], where $$q Î R^d, $t$Î R⁺, and $d$ is the dimension of the Euclidean space $$R^d. For a "density" $G(r,t)$ of the familiar model of a physical system averaged over all realizations of the random field $V$, we establish an integral equation which has the form of the Dyson equation. The invariance of the equation under the continuous renormalization group allows using the renormalization group method to find an asymptotic expression for $G(r,t)$ as $r\; \to$¥ and $t\; \to$¥.

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Last modified: December 22, 2009