Local Estimates for Parabolic Difference Operators

We prove a local maximum principle and weak Harnack inequality for parabolic difference inequalities analogous to previous work on elliptic difference inequalities. As applications, we have discrete analogues of the Hölder and Harnack estimates of Krylov and Safonov and results pertaining to the stability of discrete scheme for nonlinear parabolic problems, extending previous work on elliptic equations.