Consider a real-valued random variable $X$ with mean $0$ and variance $1$ and let $Z$ be an independent standard Gaussian random variable. Now consider the stochastic process $Y(t) = \sin(t) X + \cos(t) Z $ for $t \in [0, \frac{\pi}{2}]$. We will be interested in $S(Y(t))$, the relative entropy of $Y(t)$ with respect to the Gaussian, more specifically $ S(Y(t)) = \int \frac{d\mu}{d\nu} \log\left( \frac{d\mu}{d\nu}\right)d\nu$ where $ \nu $ is the standard Gaussian probability measure and $\mu$ is the probability measure induced by $Y(t)$.
From the entropy power inequalities for relative entropy, we can get upper bounds on $S(Y(t))$. Is there any general result or technique to lower bound the relative entropy of $Y(t)$ under reasonable conditions on $X$? Specifically, we are interested in the following case: $S(X) \geq \epsilon$ for some positive constant $\epsilon$, and so $S(Y(\pi/2)) \geq \epsilon$. Moreover, $S(Y(0))=0$. We would like to obtain a lower bound on $S(Y(t))$ as a function of $t$ as $t \rightarrow 0$. If needed, we can assume that $X$ is nice in various ways, e.g., it has bounded density and is sub-Gaussian.
