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Let $f:\mathbb R^3 \to \mathbb R^3$ be a smooth Lipschitz function (bounded if needed), and $W_t$ a $3$-dimentional Brownian motion. Consider the SDE on $\mathbb R^3 \times \mathbb S^2$ (where $\mathbb S^2$ is the sphere).

\begin{cases} \mathrm{d}X_t = f(X_t) \mathrm{d}t + \mathrm{d} W_t\\ \mathrm{d}s_t = (\mathrm{d}f(X_t) s_t - \langle s_t, \mathrm{d}f(X_t) s_t \rangle s_t) \mathrm{d}t, \end{cases} where $\mathrm{d}f(x)$ is the derivative of $f$ at $x$.

I would like to know is there any condition on $f,$ which makes the above SDE, transitive on $B(0,1) \times \mathbb{S}^2$ (where $B(0,1)$ is the ball of radius $1$ centered in $0$), in the following sense: For every $(x,\theta)\in B(0,1)\times \mathbb S^2$ and open set $U\subset B(0,1)\times \mathbb S^2,$ there exists $t= t(x,\theta,U)$ such that $$\mathbb E\left[\mathbb 1_U (X_t,s_t) \mid (X_0,s_0)= (x,\theta)\right] >0$$

So far, I was not able to figure what condition would imply this property. Can anyone help me?