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.)

  • $\begingroup$ This is a fantastic example, thanks Martin :) $\endgroup$ – AIM_BLB Feb 13 at 20:23

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