## 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

### MSC:

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

### Keywords:

lattice; infinitesimal generator

Zbl 0538.00017