Suppose that $\phi(t,x):[0,\infty)\times \mathbb{R}^d\rightarrow \mathbb{R}^d$ is a flow. Is it possible to extend $\phi$ to a family of stochastic flows $\{\Phi(t,x,\sigma)\}_{\sigma \in [0,1]}$ such that

- For every $\sigma_0 \in (0,1]$, $\Phi(t,x,\sigma_0):[0,\infty)\times \mathbb{R}^d\rightarrow \mathbb{R}^d$ is a stochastic flow,
- $\Phi(t,x,0)=\phi(t,x)$,
- For every $\sigma_0 \in[0,1]$, $\lim\limits{\sigma \downarrow \sigma_0} \Phi(t,x,\sigma)=\Phi(t,x,\sigma_0)$ almost-surely?
- For $\sigma>0,x \in \mathbb{R}^d$, $\lim\limits_{t \mapsto \infty}\Phi(t,x,\sigma_0) \in \mathbb{R}^d$?