Let's consider stochastic dynamics with discrete time step $t$ and states $x_t\in\mathbb{R}^d$ that evolve as
$$ \mathrm{p}(x_{t+1}|x_t)=\mathcal{N}(x_t+f(x_t), \Sigma). $$
Moreover, we assume the existence of a unique stationary distribution $\rho(x)$ that satisfies
$$ \int\mathrm{p}(x_t|x_{t-1})\rho(x_{t-1})\mathrm{d}x_{t-1}=\rho(x_t). $$
Can $f$ be written as a gradient descent on $\ln\rho$ transformed by a positive semi-definite matrix $R(x)$,
$$ \mathrm{p}(x_{t+1}|x_t)=\mathcal{N}(x_t+R(x_t)\nabla_{x_t}\ln\rho(x_t),\Sigma)\,? $$
Do I additionally need to assume ergodicity or detailed balance for this to work?