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Let $W$ be a standard $d$-dimensional Brownian motion.

Suppose $b: \mathbb R^d \to \mathbb R^d$ is measurable and bounded. Consider, for every $\varepsilon > 0$, the solution $X^\varepsilon$ on $[0, T]$ to the SDE

$$dX^\varepsilon_t = b(X_t) \, dt + \varepsilon \, dW_t, \, X_0 = 0$$

It is known that the SDE admits weak solutions for every $\varepsilon$.

Question: Do the solutions $X^\varepsilon_t$ converge in law in $C[0, T]$ to some (possibly nondeterministic) limit as $\varepsilon \to 0$?

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    $\begingroup$ I think there is no general result, but you can check Section 1.4 of "Random Perturbation of PDEs and Fluid Dynamic Models" by Franco Flandoli. $\endgroup$
    – user89404
    Commented Feb 6, 2023 at 16:52

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Indeed as a follow-up to the comment, there is a wide literature of studying ODEs

$$\frac{dx(t)}{dt}=b(t,x(t))$$

which can have existence but not uniqueness eg. for $b(x):=2sgn(x)\sqrt{|x|}$, whereas for for any bounded Borel function $b$, the following SDE has a strong solution

$$dx^{\epsilon}(t)=b(x^{\epsilon}(t))dt+\epsilon dW_{t}.$$

In terms of convergence, here is one result for the bounded measurable case from "On limiting values of stochastic differential equations with small noise intensity tending to zero"

enter image description here

Other references include

  • "Random Perturbation of Some Multi-dimensional Non-Lipschitz Ordinary Differential Equations"
  • "On Stochastic Differential Equations with Locally Unbounded Drift"
  • and the book mentioned by Toyomu "Random Perturbation of PDEs and Fluid Dynamic Models".
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