# Occupation time of SDE

Let $$b:\mathbb{R}^d\to\mathbb{R}^d$$ be locally Lipschitz and assume that, for any $$x\in\mathbb{R}^d$$ and any $$f\in C^{\infty}([0,1],\mathbb{R}^d)$$, the equation $$X_t^{x,f}=x+\int_0^t b(X_s^{x,f})\,ds+\int_0^t f(s)\,ds+W_t,\qquad t\in [0,1],$$ has a pathwise unique strong solution (this is for example the case if $$\langle b(x)-b(y),x-y\rangle\leq C-D|x-y|^2$$ and $$|b(x)|\leq C(1+|x|^N)$$ for some $$C,D,N>0$$). Here, $$W$$ is a $$d$$-dimensional Brownian motion. Is the following statement true? It is true for a globally Lipschitz drift $$b$$, but what about the locally Lipschitz case?

For any $$R>0$$, there are $$a,b>0$$ such that $$\mathbb{P}\left(\int_0^1 1_{\{|X^{x,f}_s|>R\}}\,ds\geq b\right)\geq a,$$ uniformly in $$x\in\mathbb{R}^d$$ and $$f\in C^{\infty}([0,1],\mathbb{R}^d)$$ (both deterministic).