Police and robber game on infinite chessboard

It is considered two variants of the game "Policeman and a robber" on an infinite chessboard that is a graph giving a regular partition of the plane into squares. Heuristic and precise definitions of the concepts "the initial state is winning for the pursuer" and "the initial state is winning for the evader" are formulated. Then, criteria for determining if a given initial state is winning for the pursuer or for the evader is given.