I would like to know more about the two-dimensional processes derived from Brownian motion by the following stochastic differential equation (in the Ito sense)

$$dX_t = f(X_t) dt + \mathcal{R}(f(X_t)) dB_t$$

where

- $X_t$ is the two-dimensional stochastic process
- $f$ is a smooth vector field
- $\mathcal{R}$ is the linear map which rotates a vector through a quarter turn
- $B_t$ is a standard one-dimensional Brownian motion

So $X_t$ follows the field lines of $f$ apart from a lateral "shake" whose intensity is proportional to the strength of the field $f$ at $X_t$.

In particular I am curious about what is known when $f$ is conservative (i.e. $f = \nabla g$ for some real-valued field $g$). In this case, can the trajectory of $X_t$ self intersect?

References to further properties of this process would also be appreciated.