Small noise limits with irregular drift

Question

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 $\varepsilon \to 0$?

Answer 1

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".