An SDE perspective on stochastic convex optimization and Rough paths

In the work “An SDE perspective on stochastic convex optimization”, they study the SDE version of gradient descent

$latex dX_{t}=-\nabla f(X_{t})+\sigma(X_{t},t)dW_{t},$

for convex $latex f$. One interesting question is to study this same question but for possibly more irregular potential $latex U(x)$ and volatility coefficient $latex \sigma(X_{t},t)$

$latex dX_{t}=-U(X_{t})+\sigma(X_{t},t)dW_{t},$

using the technology of Rough paths. For example, can we relax the regularity of $latex \sigma(X_{t},t)$ from Lipschitz to $latex \alpha$-Hölder for some $latex \alpha \in (\frac{1}{3},\frac{1}{2})$?