Ergodicity and approximations of invariant measures for stochastic lattice systems with Markovian switching

This paper is concerned with the dynamics of stochastic lattice systems with Markovian switching. Based on the well-posedness of solutions, we first prove the ergodicity of invariant measures and show that the Markov chain facilitates the existence of invariant measures. In order to investigate numerical invariant measures, the convergence of invariant measures is considered between the underline systems and their finite-dimensional truncated systems. Due to this, we can use the backward Euler-Maruyama method to approximate invariant measures for such infinite-dimensional systems. This work provides a feasible path for the convergence of the finite-dimensional numerical invariant measures to the analytical invariant measure.