*from math.stackexchange*

I'm currently looking at stochastic differential equations with irregular coefficients such as $W^{1,p}_{loc}$. If I have \begin{align} dX_t=b(X_t)dt+\sigma dW_t, \end{align} where $b\in W^{1,1}_{loc}$ and $\sigma$ is a constant, I can write this SDE as a ODE for every Brownian path $w$ by defining $Y_t=X_t-w_t$ and a new vector field $b^w=b(Y,t)=b(Y_t+w_t)$, so the ODE is \begin{align} dY=b^w(t,Y)dt \end{align} with initial condition $Y_0=y$. Since $b^w$ has Sobolev regularity I can then apply DiPerna-Lions theory (1989), which guarantees the existence and uniqueness of the flow of the ODE.

My question is now what happens if $\sigma$ is not a constant,

\begin{align}
dX_t=b(X_t)dt+\sigma(X_t)dW_t.
\end{align}
Apparently the above argument is NOT correct in general. Is there any ways that the diffusion coefficients $\sigma(X_t)$ can be absorbed into the Brownian motion? or maybe there are some conditions on $\sigma$ under which the above argument still hold?

Many thanks!!