Geometric ergodicity of discrete-time approximations to multivariate diffusions (Q1431539)

The stochastic differential equations of the form \[ dX(t)= - \nabla H(X(t))dt + \sigma d W(t), \] where \(W(t), t \geq 0\), is a \(k\)-dimensional Brownian motion, \(\sigma > 0\) and \(H:\mathbb R^k \to\mathbb R\) is some potential are studied. The discrete-time Markov chain is considered as an approximation for this equation. Criteria for the discrete scheme to be geometrically ergodic is found. The use of these criteria is illustrated by examples. From the derived results it follows that the considered approximation has a better behavior than the Euler discretization for such equations.