Note: We denote by $\mathcal L(U)$ the Lebesgue measure of a set $U$.
Let $\mu: \mathbb R^d \to \mathbb R^d$ and $\sigma: \mathbb R^{d} \to \mathbb R^{d \times d}$ be Borel functions.
Suppose the following assumptions hold (Assumption A):
$\mu$ is Lipschitz continuous with Lipschitz constant $L > 0$.
$\lvert \mu(0) \rvert < M$ for some constant $M > 0$.
$\sigma$ is locally Lipschitz continuous.
For some constant $C > 0$, $\xi^{T} \sigma(x) \sigma(x)^{T} \xi \geq C |\xi|^2$ for all $x, \xi \in \mathbb R^d$.
$\sup_{ij} |\sigma_{ij}| < D$ for some constant $D > 0$.
Let $W$ be a standard d-dimensional Brownian motion, and let $X$ be the solution to the SDE
$$dX_t = \mu(X_t) \,dt + \sigma(X_t) \,dW_t$$
with initial condition $X_0 = 0$.
Question: Fix $T > 0$. For every $\varepsilon, h > 0$, does there exist a $\delta > 0$ depending only on $\varepsilon, h, L, C, D$ and $M$ satisfying the following property?
For all $\mu, \sigma$ satisfying Assumption A, and all Borel subsets $U$ of $\mathbb R^d \setminus \{0\}$ with $\mathcal L(U) < \delta$, we have
$$\mathbb P\left(\int_{0}^T \mathbf 1_{U} (X_s) ds > h\right) < \varepsilon.$$
