Let $X_t$ be a $d\geq 3$-dimensional diffusion process solving the SDE $$ dX_t = b(X_t) dt + \sigma W_t, \qquad X_0 =x \in \mathbb{R}^d $$ where $\sigma>0$ $b$ smooth and Lipschitz, and let $D$ be the ball about $0\in \mathbb{R}^d$ of radius $\delta>0$. Are there known (non-trivial) upper estimates on $$ \mathbb{P}\left( X_t \not \in D\, (\forall t \in [0,1]) \right). $$ granted that $x \not \in D$. Suppose further that $b(x)=0$ iff $x=0$ and that $b$ is "small".
I would expect that such an estimate would depend on $\sigma,\delta$ and the distance of $x$ from $D$...