I stumbled upon the following problem, which intuitively makes sense but I don't know how to rigorously show it. Let $Y$ be a continuous Markov process with values in $\mathbb{R}^d$. Fix $N\in\mathbb{N}$ and $T\geq 0$ What can be said about the ratio
$$\frac{\mathbb{P}^0(\vert Y_T\vert\leq \epsilon )}{\mathbb{P}^0(\vert Y_T\vert\leq \frac{N-1}{N}\epsilon )}=\mathbb{P}^0\bigg(\vert Y_T\vert<\frac{N-1}{N}\epsilon\bigg\vert \vert Y_T\vert<\epsilon\bigg)^{-1}$$ as $\epsilon\downarrow 0$?
In particular, is there an upper bound?