Small noise limits with irregular drift

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 $$\varepsilon \to 0$$?

• 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.
– user89404
Commented Feb 6, 2023 at 16:52

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"

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".