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 $\sigma>0,x \in \mathbb{R}^d$, $\lim\limits\_{t \mapsto \infty}\Phi(t,x,\sigma\_0) \in \mathbb{R}^d$?



Convergence of Stochastic Flow but not Flow
Take the solution to
$$
dx = \tanh^3(x)\,dt + \sigma x \, dW\;.
$$
For $\sigma > 0$, solutions diverge if you start anywhere except at $0$.
For any $\sigma \neq 0$ on the other hand, solutions converge to $0$ almost surely. (The cube makes sure that solutions behave like $dx = \sigma x \,dW$ once $x$
is sufficiently close to $0$, so they converge exponentially fast.)


