Weak Error Estimates of Ergodic Approximations for Monotone Jump-diffusion SODEs

We first derive the exponential ergodicity of the stochastic theta method (STM) with $\theta \in (1\mathrm{/}2,1]$ for monotone jump-diffusion stochastic ordinary differential equations (SODEs) under a dissipative condition. Then we establish the weak error estimates of the backward Euler method (BEM), corresponding to the STM with $\theta =1$. In particular, the time-independent estimate for the BEM in the jump-free case yields a one-order convergence rate between the exact and numerical invariant measures, answering a question left in {\it Z. Liu and Z. Liu, J. Sci. Comput. (2025) 103:87}.