| Abstract: |
| In this talk,
we are concerned with a modified Euler scheme for the SDE under consideration, where the drift is of super-linear growth and dissipative merely outside a closed ball.
By adopting the synchronous coupling, along with the construction of an equivalent metric, the $L^2$-Wasserstein ergodicity of the modified Euler scheme is addressed provided that the diffusivity is large enough. In particular, as a by-product, the $L^2$-Wasserstein ergodicity of the projected Euler scheme and the tamed Euler algorithm is treated under much more explicit conditions imposed on drifts. The theory derived on the
$L^2$-Wasserstein ergodicity has numerous applications on various aspects. In addition to applications on Poincar\`{e} inequalities, concentration inequalities for empirical averages, and bounds concerning KL-divergence, in this paper we present another two potential applications. One concerns the $L^2$-Wasserstein error bound between the exact invariant probability measure and the numerical counterpart corresponding to the projected Euler scheme and the tamed Euler recursion, respectively. It is worthy to emphasize that the associated convergence rate is improved greatly in contrast to the existing literature. Another is devoted to the strong law of large numbers of additive functionals related to the modified Euler algorithm, where the observable functions involved are allowed to be of polynomial growth. |
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