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).



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