The Lamperti transformation is commonly used to transform SDEs with state dependent coefficients into SDEs with constant diffusion.
For multidimensional processes there are some conditions on the drift that have to be valid before applying the transformation (e.g., drift should be the gradient of potential).
If we consider a general multidimensional SDE with constant drift $$ dX_t = f(X_t) dt + \sigma dW_t, $$
can we get the other way around, i.e., can we convert it to a driftless SDE with state dependent coefficient? I.e., obtain an equation like this $$dY_t = \sigma(Y_t) dW_t$$,
or this $$dY_t =g(Y_t) dt+ \sigma(Y_t) dW_t$$. (here $g()$ shall conform to the conditions of the Lamperti transform).
If yes, what are the conditions for this?