A family of models of liquid on a two-dimensional lattice (2D lattice liquid models) have been proposed as primitive models of soft-material membrane. As a first step, we have formulated them as single-component, single-layered, classical particle systems on a two-dimensional surface with no explicit viscosity. Among the family of the models, we have shown and constructed two stochastic models, a vicious walk model and a flow model, on an isotropic regular lattice and on some honeycomb lattices of various sizes. In both cases, the dynamics is governed by the nature of the frustration of the particle movements. By simulations, we have found the approximate functional form of the frustration probability and peculiar anomalous diffusions in their time-averaged mean-square displacements in the flow model. The relations to other existing statistical models and possible extensions of the models are also discussed.