# Convergence of Stochastic Flow but not Flow

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 $$\sigma>0,x \in \mathbb{R}^d$$, $$\lim\limits_{t \mapsto \infty}\Phi(t,x,\sigma_0) \in \mathbb{R}^d$$?

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.)