Diffusive behavior of a random walk in a random medium. (English) Zbl 0549.60075

Stochastic analysis, Proc. Taniguchi Int. Symp., Katata & Kyoto/Jap. 1982, North-Holland Math. Libr. 32, 105-119 (1984).
[For the entire collection see Zbl 0538.00017.]
Let \(Z^ d_{\epsilon}\) be the simple cubic d-dimensional lattice of span \(\epsilon\) and \(\Delta_ j^{\epsilon^{\pm}}\) be difference operators along \(e_ j\) coordinate unit vector. Let \((L_{\epsilon}f^{\epsilon})(x,t,\omega)\equiv - \sum^{d}_{j=1}(\nabla_ j^{\epsilon -}a_ j\nabla_ j^{\epsilon +}f^{\epsilon})(x,t,\omega)\) and \((a_ jf)(x)=a_ j(x/\epsilon,\omega)f(x,t,\omega)\) where \(a_ j(x/\epsilon,\omega)\) is a possible realization of the medium conductivities, on the bond from \(x\in Z^ d_{\epsilon}\) in the \(e_ j\) direction for every \(\omega\). And \(P_{\epsilon}(x,y,t,\omega)\equiv (\delta_ x,e^{L_{\epsilon}t}\delta_ y)=(e^{L_{\epsilon}t}\delta_ y)(x,\omega)\). The author obtains \[ K_ 1(t^{{1\over2}}- \epsilon)\leq\sum_{x\in Z^ d_{\epsilon}}| x-y| P_{\epsilon}(x,y,t,\omega)\leq K_ 2(t^{{1\over2}}+\epsilon) \] with \(K_ 1\), \(K_ 2\) independent of \(\omega\),x,y and \(\epsilon\), if there are two constants \(\alpha\) and \(\beta\) such that \(0<\alpha\leq a_ j(x/\epsilon,\omega)\leq\beta <\infty\) uniformly in \(x\in Z^ d_{\epsilon}\) and \(\omega\in\Omega \). This estimate is Nash’s estimate in the lattice case, the latter is \(K_ 1t^{{1\over2}}\leq\int | x| T(x,t)dx\leq K_ 2t^{{1\over2}}\) where T is a bounded solution of a linear parabolic equation.
The author also shows that the Markov process \(x_{\epsilon}(t)\) corresponding to \(P_{\epsilon}\) with \(\Pr ob(x_{\epsilon}(0)=x)=1\), \(x\in Z^ d_{\epsilon}\), converges to the diffusion process on \(R^ d\) whose infinitesimal generator is \(\sum^{d}_{i,j=1}q_{ij}(\partial /\partial x_ i)(\partial /\partial x_ j)\), the constants \(q_{ij}\) are given.
Reviewer: Y.Ge


60J70 Applications of Brownian motions and diffusion theory (population genetics, absorption problems, etc.)
60J60 Diffusion processes


Zbl 0538.00017