I'm considering a Markov process with a continuous state space. Let $V(x)$ be a differentiable function, $\Delta t$ a fixed time step, and, at every step, set
$$x_{n+1}= x_n-\alpha \frac{dV}{dx}\Big|_{x=x_n}\Delta t +\beta \xi_n \sqrt{\Delta t}$$
where $\xi_n$ are independent Gaussian variables with unit variance and zero mean. Assuming that $\Delta t $ is small, I'm looking for the stationary distribution.
I found that
$$\rho(x_{n+1}|x_n) = \frac{1}{2\beta \sqrt{\pi \Delta t}}e^{-\frac{1}{2}\big(\frac{x_{n+1}-(x_n-\alpha V'(x_n)\Delta t)}{\beta \sqrt{\Delta t}}\big)^2}$$
Letting $\pi(x)$ be the stationary distribution, we then have that
$$\pi(y) = \int_{-\infty}^\infty \pi(x)\rho(y|x)dx $$
The detailed balance condition tells us that
$$\pi(y) e^{\big(\frac{y\alpha V'(y) -x\alpha V'(y) +\alpha^2(V'(y))^2\Delta t}{\beta^2}\big)} = \pi(x) e^{\big(\frac{x\alpha V'(x)-y\alpha V'(x) +\alpha^2(V'(x))^2\Delta t}{\beta^2}\big)} $$
but I'm not sure if this is helpful. There's also some helpful information here. Maybe you do something along the lines of this post?