Homogenization of diffusions on the lattice ${\mathbf{Z}}^d$ with periodic drift coefficients; Application of logarithmic Sobolev inequality

A homogenization problem of infinite dimensional diffusion processes indexed by ${\mathbf{Z}}^d$ having periodic drift coefficients is considered. By an application of the uniform ergodic theorem for infinite dimensional diffusion processes based on logarithmic Sobolev inequalities, an homogenization property of the processes starting from an almost every arbitrary point in the state space with respect to an invariant measure is proved. This result is also interpreted as solution to a homogenization problem of infinite dimensional diffusions with random coefficients, which is essentially analogous to the known ones in finite dimensions.