Reference request: showing that solution of an Ito SDE stays bounded with positive probability

Question

Assume that we have a (well-posed) Ito SDE of the form $$\mathrm{d} X_t = b(X_t)\,\mathrm{d} t + \sigma(X_t)\,\mathrm{d}W_t $$ where $b \colon \mathbb{R}^d \to \mathbb{R}^d$, $\sigma \colon \mathbb{R}^d \to \mathbb{R}^d \times \mathbb{R}^d$, and $W_t$ is a $d$-dimensional Brownian motion. I am wondering under what kind of assumptions on the drift $b$ and the diffusion $\sigma$ can we show that $$\mathbb{P}\left(\sup_{t\geq 0} \|X_t\| \leq R\right) > 0$$ for some constant $R < \infty$? Or in other words, under what kind of assumptions on the drift $b$ and the diffusion $\sigma$ can we show that there is a positive probability such that the trajectory of $X_t$ does not leave some large enough balls? Currently I am only aware of the so-called Stroock-Varadhan support theorem showing that solutions of certain Ito SDEs stay within certain (large enough) finite balls almost surely for all time. But I am not aware of any literature which are related to my question. I would appreciate any pointers to contemporary research papers or lecture notes.