Consider the Ornstein-Uhlenbeck SDE: $$dX_t=-\gamma_t X_t dt + \sigma_t dB_t$$ where $\gamma_t$ and $\sigma_t$ are time-varying constants independent of $X_t$. Now assume I have two versions of the SDE with the only thing different being the initial distributions, $\hat{X}_0 \sim p_0(\hat{X})$ and $\tilde{X}_0 \sim q_0(\tilde{X})$. All else are the same.
Is there any relation I can write down between the marginal densities along the paths, $p_t(\hat{X})$ and $q_t(\tilde{X})$ that is related to how far apart the initial distributions are, for e.g., in terms of $D_{KL}(p_0||q_0)$? Intuitively (correct me if I'm wrong), the farther apart the initial distributions, the farther apart their marginals should be?
I was thinking along the lines of computing the ratio $p_t/q_t$, or equivalently the Radon-Nikodym derivative $dP/dQ$ via Girsanov's theorem, which would require me to transform one measure to the other, but I'm not sure if that's the right approach since the SDEs have identical forms.
Thank you very much! I'm a total newbie in stochastic calculus so please forgive me for my poor notations/reasoning.