Mean-field models in the theory of random media.~III

L. V. Bogachev ; S. A. Molchanov

Teoretičeskaâ i matematičeskaâ fizika, Tome 87 (1991) no. 2, pp. 254-273

Résumé

In the mean-field (nonlocal) diffusion approximation, when the Laplacian $\Delta$ on the lattice $\mathbf Z^d$ is replaced by the corresponding operator $\overline\Delta_V$ in a volume $V\subset\mathbf Z^d$ $(|V|\to\infty)$ [1, 2], a study is made of the $t\to\infty$ asymptotics of the statistical moments (moment functions) $m_p=m_p(\mathbf x_1,\dots,\mathbf x_p, t)=\langle\psi(\mathbf x_1,t,\omega)\dots\psi(\mathbf x_p,t,\omega)\rangle$, $p=1,2,\dots,$ for the evolution equation $\partial\psi/\partial t=\varkappa\Delta_V\psi+\xi\psi$ with nonstationary random potential $\xi=\xi(\mathbf x,t,\omega)$. The case when $\xi$ represents Gaussian white noise (with respect to $t$) is considered in the paper. At the same time, the evolution equation in such a medium is understood in the sense of It$\operatorname{\hat o}$. In space, the potential $\xi$ is assumed either to be localized, $\xi(\mathbf x,t,\omega)=\delta(\mathbf x_0,\mathbf x)\xi(\mathbf x_0,t,\omega)$, or homogeneous, namely, $\delta$-correlated with respect to $\mathbf x$. Under these conditions, the exponent $\gamma_p=\displaystyle\lim_{t\to\infty}t^{-1}\ln m_p$ is calculated.