Let $\{X_t\}_t$ be a time homogeneous Ito diffusion process such that $X_0=x$ that satisfies $$dX_t=b(X_t)dt+\sigma (X_t)dB_t $$ where B_t is the standard Brownian motion. If $Q^x$ is the distribution of this process, $$E_{Q^x}[f(X_{t_1})\cdot \cdot \cdot f(X_{t_k})]=E_{P^0}[f(X_{t_1}^x)\cdot \cdot \cdot f(X_{t_k}^x)]$$ for all bounded borel functions $f_1,...,f_k$ and $P^0$ is the distribution of the standard Brownian motion. I do not understand how this equality holds. I sort of get it conceptually but I do not see how this can be proved. Any help is appreciated.
2026-03-27 10:12:02.1774606322
$E_{Q^x}[f(X_{t_1})\cdot \cdot \cdot f(X_{t_k})]=E_{P^0}[f(X_{t_1}^x)\cdot \cdot \cdot f(X_{t_k}^x)]$ holds for an Ito diffusion
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