I would like to know more about the two-dimensional processes derived from Brownian motion by the following stochastic differential equation $$dX_t = f(X_t) dt + \mathcal{R}(f(X_t)) dB_t$$ 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 irrotational (equivalently $\nabla \times f = 0$ or $f = \nabla g$ 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.