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Let $m: \mathcal{B}(\mathbb{R}^m) \rightarrow [0,1]$ be a probability measure.

Let $f: \mathbb{R}^n \times \mathbb{R}^m \rightarrow \mathbb{R}^n$ be measurable, and continuous in the first argument.

Consider the process $$ \begin{array}{rcl} X_{k+1} & = & f( X_k, Y_k ) \\ Y_{k+1} & \sim & m(\cdot) \end{array} $$ where $Y_{k+1}$ is an extraction of $m$, and $k = 0, 1, ...$.

I am wondering under which conditions on $m$ the above process has a Feller stochastic kernel.

Comment. The notion of Feller stochastic kernel is also given here: Feller continuity of the stochastic kernel.

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Example C.7 here states that if $F:X\times Y\to X$ is a measurable function where $X$ and $Y$ are Borel spaces, with $F(\cdot,y)$ being continuous on $X$ for any fixed $y$, and $y_t\sim m$ where $m$ is arbitrary, then $$ x_{t+1} = F(x_t,y_t) $$ is weakly (Feller) continuous Markov process. Your assumptions seem to fit, if I got what you meant by continuity in the first argument.

I also believe that the fact shall hold also for any topological space $X$ and any measurable $Y$, since the proof only uses the fact that $v(F(\cdot,y))$ is a continuous functions of $x$ whenever $v(\cdot)$ is, and the bounded convergence theorem.

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