Describe discrete-time Langevin dynamics by its stationary distribution

Question

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?

Answer 1

I don't think so. Counterexample:

State space = $\{0,1,2\}$

$p_1: x_{t+1}=x_t+1 $ mod $ 3$

$p_2: x_{t+1}=x_t-1 $ mod $ 3$

so $\Sigma=0$.

Then the same $\rho(x_t)$ belongs to both $p_1$ and $p_2$.