I am looking for the steady state distribution of the following Poisson process:
$$d x(t) = -k_1(x(t)-k_2)dt + k_3dN(t)$$
where $k_1$, $k_2$ and $k_3$ are constants and the rate $\lambda(x)$ of the Poisson random variable $N(t)$ is given by:
$$\lambda(x) = \frac{-\nabla_xE(x) + k_1(x(t)-k_2)}{k_3}$$.
This Poisson process is designed to approximate gradient descent on $-\nabla_xE(x)$. I did some simulations and found that the mean of the steady-state distribution of this Poisson process is located at the of $x$ value for which $\nabla_xE(x)=0$. I also found that the variance of the steady state distribution linearly increases with mean of the steady-state distribution. The steady state distribution also looked like a Gamma distribution when $e^{-E(x)}/Z$ is a normal distribution. However, I am still looking for the analytical expression of the steady state distribution.
Any ideas on how I could find the steady state distribution of this Poisson process? Also, how is the steady state related to $e^{-E(x)}/Z$?