Let $(B_t)$ be a Brownian motion started in $x$ and $f_1,\dots,f_n$ be $\mathbb{R}$-valued continuous bounded functions. Consider the map $x \mapsto \mathbb{E}^x \big( f_1(B_{t_1}) \cdots f_n(B_{t_n}) \big)$.
Is it true that this map is continuous?
My answer: Yes, write $$\mathbb{E} \big( f_1(B_{t_1}+x) \cdots f_n(B_{t_n}+x) \big).$$ Since all $f_i$ are bounded I can use dominated convergence to pull in the limit $\lim_{x \rightarrow y}$ and by the continuity of all $f_i$ the claim follows.
Is this correct? Because then this statement would be completely independent of Brownian motion and would hold for every stochastic process.