From this paper, if the deterministic dynamics of $x_t$ is $dx_t=v_tdt$ where $v_t=\nabla\log\pi-\nabla\log\mu_t$ with $\mu_t$ denotes the law of $x_t$ and $\pi$ is a distribution depending on $x_t$, then the Langevin dynamic of $x_t$ is given by $$ dx_t=\nabla\log\pi(x_t)dt+\sqrt{2}dB_t $$ where $B_t$ is a Brownian motion.
Question: Assume that $g$ is a nice function. I would like to ask if the deterministic dynamics of $y_t$ is $$dy_t=u_tdt, \quad u_t=g(y_t)[\nabla\log\pi(y_t)-\nabla\log\mu_t(y_t)]$$ then what is the corresponding Langevin dynamics of $y_t$.