An L2-approximation method for construction and smoothing estimates of Markov semigroups for interacting diffusion processes on a lattice

We develop an [Formula: see text]-approximation strategy to study Markov semigroups generated by an infinite system of elliptic diffusion processes on a lattice. The proposed dynamics incorporate nearest neighbor interactions influencing diffusivity, which has received little attention so far as a mathematical problem. We prove the existence and the smoothness of Markov semigroups by extending the well-known pointwise estimation techniques such as the finite speed of propagation property and the Lyapunov function methods.