Let $(X_t)_{t \geq 0}$ be a diffusion process with dynamics governed by the stochastic differential equation \begin{equation} dX_t = b(X_t)dt + \sigma(X_t)dW_t, ~~ X_0 = x_0, \end{equation} where $b,\sigma$ are Lipschitz and $(W_t)_{t \geq 0}$ is a standard $d$-dimensional Wiener process. Assume $(X_t)_{t \geq 0}$ possesses a unique invariant probability measure $\mu(\cdot)$.

The Euler-Maruayama approximation of this SDE with step size $h$ is a Markov chain defined recursively as $Y_0 = x_0$, and for $n \in \mathbb{Z}^+$ \begin{equation} Y_{(n+1)h}|Y_{nh} = Y_{nh} + b(Y_{nh})h + \sqrt{h}\sigma(Y_{nh})\varepsilon, ~~ \varepsilon \sim \mathcal{N}(0,I_{d \times d}). \end{equation}

I can see that under certain conditions $\{Y_i\}_{i \in \mathbb{Z}^+}$ will also possess a unique invariant probability measure, $\mu^h(\cdot)$. I've seen certain results discussing how the ergodicity properties of $(X_t)_{t \geq 0}$ and $\{Y_i\}_{i \in \mathbb{Z}^+}$ can differ (e.g. scenarios in which the first can be exponentially ergodic but the second transient, see: http://projecteuclid.org/euclid.bj/1178291835).

What I've been unable to find are results discussing how $\mu^h(\cdot)$ and $\mu(\cdot)$ differ. Can we make some statement about how 'close' the two measures are, in total variation \begin{equation} \| \mu^h(\cdot) - \mu(\cdot) \|_{TV} \end{equation} or some other appropriate distance. And can we say anything about the distance \begin{equation} \|\mathbb{E}_{\mu^h}f - \mathbb{E}_{\mu}f\| \end{equation} for some suitable class of functions $f$, assuming both processes are defined on the same space which has norm $\|\cdot\|$? I've searched around online a bit, the closest I've come is here http://www.newton.ac.uk/preprints/NI03065.pdf, where the result is alluded to, which makes me think it might be fairly well known. So if anyone could point me in the right direction or explain why it's obvious then that would be great, thanks!

Apologies if this isn't an appropriate question for overflow, also asked here on math.stackexchange https://math.stackexchange.com/questions/751991/invariant-measure-of-euler-maruyama-discretisation-of-an-ito-diffusion