Uniform convergence of action of Feller semigroup with $1$ variable

Question

Assume we have two subsets of the some euclidean spaces $X\subset \mathbb{R}^m$ and $Y\subset\mathbb{R}^n$ and a a Feller semigroup $(Q_t)_{t\geq 0}$ on $Y$. Suppose also that we have a continuous function $f:X\times Y\to\mathbb{R}$ such that for each $x\in X$, the function $f_x:=f(x,\cdot)$ is in $C_0(Y)$, the space of continuous functions on $Y$ vanishing at the infinity, and there exists a bounded open neighbourhood $U(x)$ of $x$ such that $f$ is (uniformly) bounded on $U(x)\times Y$.

Question. If we define $\varepsilon(t,x,y)$ by $$ \varepsilon(t,x', y)=\frac{1}{t}\int_0^t((Q_sf_{x'})(y)-f_{x'}(y))ds, $$ then does it follow $\varepsilon$ converges to $0$ as $t$ goes to $0$ uniformly on $x'\in U(x)$, where we may let $y$ fixed?

Answer 1

The answer is no, pretty much whatever your definition of Feller is. Take for $Q_t$ the heat semigroup on $\mathbb{R}$, $X = U = (0,1)$, and for example $f(x,y) = cos(y/x)\exp(-y^2)$. If you assume that $U$ is compact rather than just bounded, then that would be a different story and your conclusion would be true, at least if your notion of "Feller" implies a bit of stochastic continuity in time.