Let $B_t$ be a $d-$dimensional Gaussian process and we consider being given the SDE, $dY_t = - \nabla U(Y_t)dt + \sqrt{2}dB_t$. A Euler-Maruyama discretization of this will look like the Markov Chain, $X_{k+1} = X_k - \gamma_{k+1} \nabla U(X_k) + \sqrt{2 \gamma_{k+1}} G_{k+1}$ where $\gamma_k$ is a sequence of step-sizes which are either constant or decreasing to $0$ and $G_k$ is a sequence of i.i.d $d-$dimensional Gaussian random variables.
Given this for any $\gamma >0$ and measurable set $A$ of $\mathbb{R}^d$ one defines the following function as the ``Markov kernel associated to the Euler-Maruyama discretziation",
$$R_\gamma (x, A) = \frac{1}{(4\pi \gamma)^{\frac {d}{2}}} \int_A \exp \Big ( - \Vert y - x - \gamma \nabla U(x)\Vert^2/(4 \gamma) \Big ) dy$$
Can someone kindly explain where this $R$ function comes from and what is its significance?
I dont seem to be able to find this terminology or this construction in standard references I looked up. For example I cant locate this construction in SDE books like this, https://users.aalto.fi/~asolin/sde-book/sde-book.pdf or Markov Chain lecture notes like this, http://www.statslab.cam.ac.uk/~ps422/mixing-notes.pdf. Do these references explain this $R$ function using any different terminology?
$$R(X_k,\;\cdot\;)=\operatorname P\left[X_{k+1}\in\;\cdot\;\mid X_k\right]$$ is the conditional distribution of $X_{k+1}$ given $X_k$ and $R$ itself is what's being called a "regular version" of that conditional distribution. The "regularity" is the measurability of $R(\;\cdot\;,B)$ for all $B$.
But if you are asking why $R$ has this particular form: Well, as you can see immediately from the definition of the Euler-Maruyama discretization, the conditional distribution of $X_{k+1}$ given $X_k$ is the normal distribution with covariance operator $2\gamma_{k+1}I_d$ and mean $X_k-\gamma_{k+1}\nabla U(X_k)$. If you write this down, you arrive precisely by the formula you have given.